LMFDB - Embedded newform 469.1.bl.a.167.1

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			167.1
Root			$0.235759 - 0.971812i$ of defining polynomial
Character	$\chi$	$=$	469.167
Dual form			469.1.bl.a.132.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0623191 - 1.30824i) q^{2} +(-0.712131 + 0.0680003i) q^{4} +(-0.888835 - 0.458227i) q^{7} +(-0.0530529 - 0.368991i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})$	\(q+(-0.0623191 - 1.30824i) q^{2} +(-0.712131 + 0.0680003i) q^{4} +(-0.888835 - 0.458227i) q^{7} +(-0.0530529 - 0.368991i) q^{8} +(0.841254 - 0.540641i) q^{9} +(-0.0311250 + 0.0899299i) q^{11} +(-0.544078 + 1.19136i) q^{14} +(-1.18186 + 0.227786i) q^{16} +(-0.759713 - 1.06687i) q^{18} +(0.119589 + 0.0351146i) q^{22} +(0.437742 - 1.80440i) q^{23} +(-0.142315 + 0.989821i) q^{25} +(0.664127 + 0.265876i) q^{28} +(0.888835 + 1.53951i) q^{29} +(0.283763 + 1.16969i) q^{32} +(-0.562319 + 0.442213i) q^{36} +(-0.981929 + 1.70075i) q^{37} +(0.481929 + 1.05528i) q^{43} +(0.0160499 - 0.0661584i) q^{44} +(-2.38786 - 0.460222i) q^{46} +(0.580057 + 0.814576i) q^{49} +(1.30379 + 0.124497i) q^{50} +(0.601300 - 1.31666i) q^{53} +(-0.121926 + 0.352283i) q^{56} +(1.95865 - 1.25875i) q^{58} +(-0.995472 + 0.0950560i) q^{63} +(0.357685 - 0.105026i) q^{64} +(0.981929 - 0.189251i) q^{67} +(0.651174 - 0.0621796i) q^{71} +(-0.244123 - 0.281733i) q^{72} +(2.28618 + 1.17861i) q^{74} +(0.0688733 - 0.0656706i) q^{77} +(-1.78153 + 0.713215i) q^{79} +(0.415415 - 0.909632i) q^{81} +(1.35052 - 0.696241i) q^{86} +(0.0348346 + 0.00671382i) q^{88} +(-0.189030 + 1.31473i) q^{92} +(1.02951 - 0.809616i) q^{98} +(0.0224357 + 0.0924813i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/469\mathbb{Z}\right)^\times$.

$n$	$269$	$337$
$\chi(n)$	$-1$	$e\left(\frac{16}{33}\right)$

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 $

$2$	−0.0623191	−	1.30824i	−0.0623191	−	1.30824i	−0.786053	−	0.618159i	$-0.787879\pi$
$2$	−0.0623191	−	1.30824i	−0.0623191	−	1.30824i	0.723734	−	0.690079i	$-0.242424\pi$
$3$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$3$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$4$	−0.712131	+	0.0680003i	−0.712131	+	0.0680003i
$5$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$5$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$6$		0			0
$7$	−0.888835	−	0.458227i	−0.888835	−	0.458227i
$7$	−0.888835	−	0.458227i	−0.888835	−	0.458227i
$8$	−0.0530529	−	0.368991i	−0.0530529	−	0.368991i
$9$	0.841254	−	0.540641i	0.841254	−	0.540641i
$10$		0			0
$11$	−0.0311250	+	0.0899299i	−0.0311250	+	0.0899299i	−0.959493	−	0.281733i	$-0.909091\pi$
$11$	−0.0311250	+	0.0899299i	−0.0311250	+	0.0899299i	0.928368	+	0.371662i	$0.121212\pi$
$12$		0			0
$13$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$13$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$14$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$15$		0			0
$16$	−1.18186	+	0.227786i	−1.18186	+	0.227786i
$17$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$17$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$18$	−0.759713	−	1.06687i	−0.759713	−	1.06687i
$19$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$19$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$20$		0			0
$21$		0			0
$22$	0.119589	+	0.0351146i	0.119589	+	0.0351146i
$23$	0.437742	−	1.80440i	0.437742	−	1.80440i	−0.142315	−	0.989821i	$-0.545455\pi$
$23$	0.437742	−	1.80440i	0.437742	−	1.80440i	0.580057	−	0.814576i	$-0.303030\pi$
$24$		0			0
$25$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$26$		0			0
$27$		0			0
$28$	0.664127	+	0.265876i	0.664127	+	0.265876i
$29$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$29$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$30$		0			0
$31$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$31$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$32$	0.283763	+	1.16969i	0.283763	+	1.16969i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.562319	+	0.442213i	−0.562319	+	0.442213i
$37$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$37$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$41$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$42$		0			0
$43$	0.481929	+	1.05528i	0.481929	+	1.05528i	0.981929	+	0.189251i	$0.0606061\pi$
$43$	0.481929	+	1.05528i	0.481929	+	1.05528i	−0.500000	+	0.866025i	$0.666667\pi$
$44$	0.0160499	−	0.0661584i	0.0160499	−	0.0661584i
$45$		0			0
$46$	−2.38786	−	0.460222i	−2.38786	−	0.460222i
$47$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$47$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$48$		0			0
$49$	0.580057	+	0.814576i	0.580057	+	0.814576i
$50$	1.30379	+	0.124497i	1.30379	+	0.124497i
$51$		0			0
$52$		0			0
$53$	0.601300	−	1.31666i	0.601300	−	1.31666i	−0.327068	−	0.945001i	$-0.606061\pi$
$53$	0.601300	−	1.31666i	0.601300	−	1.31666i	0.928368	−	0.371662i	$-0.121212\pi$
$54$		0			0
$55$		0			0
$56$	−0.121926	+	0.352283i	−0.121926	+	0.352283i
$57$		0			0
$58$	1.95865	−	1.25875i	1.95865	−	1.25875i
$59$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$59$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$60$		0			0
$61$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$61$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$62$		0			0
$63$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$64$	0.357685	−	0.105026i	0.357685	−	0.105026i
$65$		0			0
$66$		0			0
$67$	0.981929	−	0.189251i	0.981929	−	0.189251i
$67$	0.981929	−	0.189251i	0.981929	−	0.189251i
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.651174	−	0.0621796i	0.651174	−	0.0621796i	0.235759	−	0.971812i	$-0.424242\pi$
$71$	0.651174	−	0.0621796i	0.651174	−	0.0621796i	0.415415	+	0.909632i	$0.363636\pi$
$72$	−0.244123	−	0.281733i	−0.244123	−	0.281733i
$73$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$73$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$74$	2.28618	+	1.17861i	2.28618	+	1.17861i
$75$		0			0
$76$		0			0
$77$	0.0688733	−	0.0656706i	0.0688733	−	0.0656706i
$78$		0			0
$79$	−1.78153	+	0.713215i	−1.78153	+	0.713215i	−0.786053	+	0.618159i	$0.787879\pi$
$79$	−1.78153	+	0.713215i	−1.78153	+	0.713215i	−0.995472	+	0.0950560i	$0.969697\pi$
$80$		0			0
$81$	0.415415	−	0.909632i	0.415415	−	0.909632i
$82$		0			0
$83$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$83$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$84$		0			0
$85$		0			0
$86$	1.35052	−	0.696241i	1.35052	−	0.696241i
$87$		0			0
$88$	0.0348346	+	0.00671382i	0.0348346	+	0.00671382i
$89$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$89$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$90$		0			0
$91$		0			0
$92$	−0.189030	+	1.31473i	−0.189030	+	1.31473i
$93$		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	$0.666667\pi$
$98$	1.02951	−	0.809616i	1.02951	−	0.809616i
$99$	0.0224357	+	0.0924813i	0.0224357	+	0.0924813i
$100$	0.0340387	−	0.714560i	0.0340387	−	0.714560i
$101$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$101$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$102$		0			0
$103$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$103$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$104$		0			0
$105$		0			0
$106$	−1.75998	−	0.704590i	−1.75998	−	0.704590i
$107$	−0.759713	+	0.876756i	−0.759713	+	0.876756i	−0.995472	−	0.0950560i	$-0.969697\pi$
$107$	−0.759713	+	0.876756i	−0.759713	+	0.876756i	0.235759	+	0.971812i	$0.424242\pi$
$108$		0			0
$109$	0.223734	−	1.55610i	0.223734	−	1.55610i	−0.500000	−	0.866025i	$-0.666667\pi$
$109$	0.223734	−	1.55610i	0.223734	−	1.55610i	0.723734	−	0.690079i	$-0.242424\pi$
$110$		0			0
$111$		0			0
$112$	1.15486	+	0.339098i	1.15486	+	0.339098i
$113$	−0.981929	−	0.189251i	−0.981929	−	0.189251i	−0.327068	−	0.945001i	$-0.606061\pi$
$113$	−0.981929	−	0.189251i	−0.981929	−	0.189251i	−0.654861	+	0.755750i	$0.727273\pi$
$114$		0			0
$115$		0			0
$116$	−0.737654	−	1.03589i	−0.737654	−	1.03589i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.778934	+	0.612561i	0.778934	+	0.612561i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.186393	+	1.29639i	0.186393	+	1.29639i
$127$	−0.738471	−	0.380708i	−0.738471	−	0.380708i	0.0475819	−	0.998867i	$-0.484848\pi$
$127$	−0.738471	−	0.380708i	−0.738471	−	0.380708i	−0.786053	+	0.618159i	$0.787879\pi$
$128$	0.233975	+	0.676026i	0.233975	+	0.676026i
$129$		0			0
$130$		0			0
$131$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$131$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$132$		0			0
$133$		0			0
$134$	−0.308779	−	1.27280i	−0.308779	−	1.27280i
$135$		0			0
$136$		0			0
$137$	−0.452418	+	0.132842i	−0.452418	+	0.132842i	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−0.452418	+	0.132842i	−0.452418	+	0.132842i	0.0475819	+	0.998867i	$0.484848\pi$
$138$		0			0
$139$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$139$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$140$		0			0
$141$		0			0
$142$	−0.121926	−	0.848016i	−0.121926	−	0.848016i
$143$		0			0
$144$	−0.871098	+	0.830590i	−0.871098	+	0.830590i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.583610	−	1.27793i	0.583610	−	1.27793i
$149$	−0.841254	−	0.540641i	−0.841254	−	0.540641i	0.0475819	−	0.998867i	$-0.484848\pi$
$149$	−0.841254	−	0.540641i	−0.841254	−	0.540641i	−0.888835	+	0.458227i	$0.848485\pi$
$150$		0			0
$151$	−1.67489	−	0.159932i	−1.67489	−	0.159932i	−0.786053	−	0.618159i	$-0.787879\pi$
$151$	−1.67489	−	0.159932i	−1.67489	−	0.159932i	−0.888835	+	0.458227i	$0.848485\pi$
$152$		0			0
$153$		0			0
$154$	−0.0902048	−	0.0860101i	−0.0902048	−	0.0860101i
$155$		0			0
$156$		0			0
$157$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$157$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$158$	1.04408	+	2.28621i	1.04408	+	2.28621i
$159$		0			0
$160$		0			0
$161$	−1.21590	+	1.40323i	−1.21590	+	1.40323i
$162$	−1.21590	−	0.486774i	−1.21590	−	0.486774i
$163$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$163$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$167$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$168$		0			0
$169$	0.235759	+	0.971812i	0.235759	+	0.971812i
$170$		0			0
$171$		0			0
$172$	−0.414955	−	0.718724i	−0.414955	−	0.718724i
$173$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$173$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$174$		0			0
$175$	0.580057	−	0.814576i	0.580057	−	0.814576i
$176$	0.0163008	−	0.113375i	0.0163008	−	0.113375i
$177$		0			0
$178$		0			0
$179$	1.91030	+	0.560914i	1.91030	+	0.560914i	0.981929	+	0.189251i	$0.0606061\pi$
$179$	1.91030	+	0.560914i	1.91030	+	0.560914i	0.928368	+	0.371662i	$0.121212\pi$
$180$		0			0
$181$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$181$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$182$		0			0
$183$		0			0
$184$	−0.689030	−	0.0657944i	−0.689030	−	0.0657944i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.473420	+	0.451405i	−0.473420	+	0.451405i	−0.888835	−	0.458227i	$-0.848485\pi$
$191$	−0.473420	+	0.451405i	−0.473420	+	0.451405i	0.415415	+	0.909632i	$0.363636\pi$
$192$		0			0
$193$	−0.0671040	−	0.466718i	−0.0671040	−	0.466718i	−0.995472	−	0.0950560i	$-0.969697\pi$
$193$	−0.0671040	−	0.466718i	−0.0671040	−	0.466718i	0.928368	−	0.371662i	$-0.121212\pi$
$194$		0			0
$195$		0			0
$196$	−0.468468	−	0.540641i	−0.468468	−	0.540641i
$197$	−1.95496	+	0.186677i	−1.95496	+	0.186677i	−0.995472	−	0.0950560i	$-0.969697\pi$
$197$	−1.95496	+	0.186677i	−1.95496	+	0.186677i	−0.959493	+	0.281733i	$0.909091\pi$
$198$	0.119589	−	0.0351146i	0.119589	−	0.0351146i
$199$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$199$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$200$	0.372786			0.372786
$201$		0			0
$202$		0			0
$203$	−0.0845850	−	1.77566i	−0.0845850	−	1.77566i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.607279	−	1.75462i	−0.607279	−	1.75462i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.723734	+	0.690079i	−0.723734	+	0.690079i	−0.959493	−	0.281733i	$-0.909091\pi$
$211$	−0.723734	+	0.690079i	−0.723734	+	0.690079i	0.235759	+	0.971812i	$0.424242\pi$
$212$	−0.338671	+	0.978525i	−0.338671	+	0.978525i
$213$		0			0
$214$	1.19435	+	0.939247i	1.19435	+	0.939247i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−2.04970	−	0.195722i	−2.04970	−	0.195722i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$223$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$224$	0.283763	−	1.16969i	0.283763	−	1.16969i
$225$	0.415415	+	0.909632i	0.415415	+	0.909632i
$226$	−0.186393	+	1.29639i	−0.186393	+	1.29639i
$227$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$227$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$228$		0			0
$229$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$229$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$230$		0			0
$231$		0			0
$232$	0.520910	−	0.409648i	0.520910	−	0.409648i
$233$	−0.419102	−	1.72756i	−0.419102	−	1.72756i	−0.654861	−	0.755750i	$-0.727273\pi$
$233$	−0.419102	−	1.72756i	−0.419102	−	1.72756i	0.235759	−	0.971812i	$-0.424242\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$239$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$240$		0			0
$241$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$241$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$242$	0.752833	−	1.05721i	0.752833	−	1.05721i
$243$		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		−0.995472	−	0.0950560i	$-0.969697\pi$
$251$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$252$	0.702443	−	0.135385i	0.702443	−	0.135385i
$253$	0.148645	+	0.0955280i	0.148645	+	0.0955280i
$254$	−0.452036	+	0.989821i	−0.452036	+	0.989821i
$255$		0			0
$256$	1.21590	−	0.486774i	1.21590	−	0.486774i
$257$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$257$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$258$		0			0
$259$	1.65210	−	1.06174i	1.65210	−	1.06174i
$260$		0			0
$261$	1.58006	+	0.814576i	1.58006	+	0.814576i
$262$		0			0
$263$	1.30379	+	1.50465i	1.30379	+	1.50465i	0.723734	+	0.690079i	$0.242424\pi$
$263$	1.30379	+	1.50465i	1.30379	+	1.50465i	0.580057	+	0.814576i	$0.303030\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.686393	+	0.201543i	−0.686393	+	0.201543i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$271$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$272$		0			0
$273$		0			0
$274$	0.201983	+	0.583592i	0.201983	+	0.583592i
$275$	−0.0845850	−	0.0436066i	−0.0845850	−	0.0436066i
$276$		0			0
$277$	1.41542	−	0.909632i	1.41542	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$277$	1.41542	−	0.909632i	1.41542	−	0.909632i	1.00000			$0$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.223734	+	0.175946i	0.223734	+	0.175946i	0.723734	−	0.690079i	$-0.242424\pi$
$281$	0.223734	+	0.175946i	0.223734	+	0.175946i	−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$283$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$284$	−0.459493	+	0.0885600i	−0.459493	+	0.0885600i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.871098	+	0.830590i	0.871098	+	0.830590i
$289$	0.981929	+	0.189251i	0.981929	+	0.189251i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$293$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$294$		0			0
$295$		0			0
$296$	0.679656	+	0.272093i	0.679656	+	0.272093i
$297$		0			0
$298$	−0.654861	+	1.13425i	−0.654861	+	1.13425i
$299$		0			0
$300$		0			0
$301$	0.0552004	−	1.15880i	0.0552004	−	1.15880i
$302$	−0.104852	+	2.20112i	−0.104852	+	2.20112i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$307$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$308$	−0.0445812	+	0.0514495i	−0.0445812	+	0.0514495i
$309$		0			0
$310$		0			0
$311$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$311$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$312$		0			0
$313$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$313$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$314$		0			0
$315$		0			0
$316$	1.22018	−	0.629047i	1.22018	−	0.629047i
$317$	0.481929	+	0.676774i	0.481929	+	0.676774i	0.981929	−	0.189251i	$-0.0606061\pi$
$317$	0.481929	+	0.676774i	0.481929	+	0.676774i	−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$	−0.166113	+	0.0320156i	−0.166113	+	0.0320156i
$320$		0			0
$321$		0			0
$322$	1.91153	+	1.50324i	1.91153	+	1.50324i
$323$		0			0
$324$	−0.233975	+	0.676026i	−0.233975	+	0.676026i
$325$		0			0
$326$	−0.519522	+	0.333877i	−0.519522	+	0.333877i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.469383	+	0.0448206i	−0.469383	+	0.0448206i	−0.327068	−	0.945001i	$-0.606061\pi$
$331$	−0.469383	+	0.0448206i	−0.469383	+	0.0448206i	−0.142315	+	0.989821i	$0.545455\pi$
$332$		0			0
$333$	0.0934441	+	1.96163i	0.0934441	+	1.96163i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.0947329	−	1.98869i	−0.0947329	−	1.98869i	−0.142315	−	0.989821i	$-0.545455\pi$
$337$	−0.0947329	−	1.98869i	−0.0947329	−	1.98869i	0.0475819	−	0.998867i	$-0.484848\pi$
$338$	1.25667	−	0.368991i	1.25667	−	0.368991i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$344$	0.363820	−	0.233813i	0.363820	−	0.233813i
$345$		0			0
$346$		0			0
$347$	−1.45949	+	0.584293i	−1.45949	+	0.584293i	−0.959493	−	0.281733i	$-0.909091\pi$
$347$	−1.45949	+	0.584293i	−1.45949	+	0.584293i	−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$349$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$350$	−1.10181	−	0.708089i	−1.10181	−	0.708089i
$351$		0			0
$352$	−0.114022	−	0.0108878i	−0.114022	−	0.0108878i
$353$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$353$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.614761	−	2.53408i	0.614761	−	2.53408i
$359$	−0.653077	−	1.43004i	−0.653077	−	1.43004i	−0.888835	−	0.458227i	$-0.848485\pi$
$359$	−0.653077	−	1.43004i	−0.653077	−	1.43004i	0.235759	−	0.971812i	$-0.424242\pi$
$360$		0			0
$361$	0.580057	−	0.814576i	0.580057	−	0.814576i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$367$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$368$	−0.106336	+	2.23227i	−0.106336	+	2.23227i
$369$		0			0
$370$		0			0
$371$	−1.13779	+	0.894765i	−1.13779	+	0.894765i
$372$		0			0
$373$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$373$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0671040	+	0.276606i	−0.0671040	+	0.276606i	−0.995472	−	0.0950560i	$-0.969697\pi$
$379$	−0.0671040	+	0.276606i	−0.0671040	+	0.276606i	0.928368	+	0.371662i	$0.121212\pi$
$380$		0			0
$381$		0			0
$382$	0.620049	+	0.591215i	0.620049	+	0.591215i
$383$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$383$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$384$		0			0
$385$		0			0
$386$	−0.606397	+	0.116873i	−0.606397	+	0.116873i
$387$	0.975950	+	0.627205i	0.975950	+	0.627205i
$388$		0			0
$389$	−1.13779	−	0.894765i	−1.13779	−	0.894765i	−0.142315	−	0.989821i	$-0.545455\pi$
$389$	−1.13779	−	0.894765i	−1.13779	−	0.894765i	−0.995472	+	0.0950560i	$0.969697\pi$
$390$		0			0
$391$		0			0
$392$	0.269798	−	0.257252i	0.269798	−	0.257252i
$393$		0			0
$394$	0.366049	+	2.54593i	0.366049	+	2.54593i
$395$		0			0
$396$	−0.0222659	−	0.0643332i	−0.0222659	−	0.0643332i
$397$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$397$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$398$		0			0
$399$		0			0
$400$	−0.0572703	−	1.20225i	−0.0572703	−	1.20225i
$401$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$401$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−2.31771	+	0.221315i	−2.31771	+	0.221315i
$407$	−0.122386	−	0.141241i	−0.122386	−	0.141241i
$408$		0			0
$409$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$409$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−2.25761	+	0.903811i	−2.25761	+	0.903811i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$419$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$420$		0			0
$421$	1.39734	−	0.720381i	1.39734	−	0.720381i	0.415415	−	0.909632i	$-0.363636\pi$
$421$	1.39734	−	0.720381i	1.39734	−	0.720381i	0.981929	+	0.189251i	$0.0606061\pi$
$422$	0.947890	+	0.903811i	0.947890	+	0.903811i
$423$		0			0
$424$	−0.517738	−	0.152022i	−0.517738	−	0.152022i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.481396	−	0.676026i	0.481396	−	0.676026i
$429$		0			0
$430$		0			0
$431$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$431$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$432$		0			0
$433$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$433$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$434$		0			0
$435$		0			0
$436$	−0.0535124	+	1.12336i	−0.0535124	+	1.12336i
$437$		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
$441$	0.928368	+	0.371662i	0.928368	+	0.371662i
$442$		0			0
$443$	0.481929	−	0.676774i	0.481929	−	0.676774i	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	0.481929	−	0.676774i	0.481929	−	0.676774i	0.981929	+	0.189251i	$0.0606061\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.366049	−	0.0705501i	−0.366049	−	0.0705501i
$449$	0.601300	+	0.573338i	0.601300	+	0.573338i	0.928368	−	0.371662i	$-0.121212\pi$
$449$	0.601300	+	0.573338i	0.601300	+	0.573338i	−0.327068	+	0.945001i	$0.606061\pi$
$450$	1.16413	−	0.600149i	1.16413	−	0.600149i
$451$		0			0
$452$	0.712131	+	0.0680003i	0.712131	+	0.0680003i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.607279	+	0.243118i	−0.607279	+	0.243118i	−0.654861	−	0.755750i	$-0.727273\pi$
$457$	−0.607279	+	0.243118i	−0.607279	+	0.243118i	0.0475819	+	0.998867i	$0.484848\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$461$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$462$		0			0
$463$	0.514186	+	1.48564i	0.514186	+	1.48564i	0.841254	+	0.540641i	$0.181818\pi$
$463$	0.514186	+	1.48564i	0.514186	+	1.48564i	−0.327068	+	0.945001i	$0.606061\pi$
$464$	−1.40116	−	1.61703i	−1.40116	−	1.61703i
$465$		0			0
$466$	−2.23394	+	0.655945i	−2.23394	+	0.655945i
$467$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$467$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$468$		0			0
$469$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.109901	+	0.0104943i	−0.109901	+	0.0104943i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.205996	−	1.43273i	−0.205996	−	1.43273i
$478$	−2.04577	+	1.31473i	−2.04577	+	1.31473i
$479$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$479$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.596358	−	0.383256i	−0.596358	−	0.383256i
$485$		0			0
$486$		0			0
$487$	1.13915	+	1.59971i	1.13915	+	1.59971i	0.723734	+	0.690079i	$0.242424\pi$
$487$	1.13915	+	1.59971i	1.13915	+	1.59971i	0.415415	+	0.909632i	$0.363636\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.84125	+	0.540641i	1.84125	+	0.540641i	1.00000			$0$
$491$	1.84125	+	0.540641i	1.84125	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.607279	−	0.243118i	−0.607279	−	0.243118i
$498$		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		0.0475819	−	0.998867i	$-0.484848\pi$
$503$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$504$	0.0878875	+	0.362277i	0.0878875	+	0.362277i
$505$		0			0
$506$	0.115710	−	0.200416i	0.115710	−	0.200416i
$507$		0			0
$508$	0.551777	+	0.220898i	0.551777	+	0.220898i
$509$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$509$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$510$		0			0
$511$		0			0
$512$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−1.49197	−	2.09518i	−1.49197	−	2.09518i
$519$		0			0
$520$		0			0
$521$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$521$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$522$	0.967192	−	2.11785i	0.967192	−	2.11785i
$523$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$523$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$524$		0			0
$525$		0			0
$526$	1.88720	−	1.79944i	1.88720	−	1.79944i
$527$		0			0
$528$		0			0
$529$	−2.17540	−	1.12149i	−2.17540	−	1.12149i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.121926	−	0.352283i	−0.121926	−	0.352283i
$537$		0			0
$538$		0			0
$539$	−0.0913090	+	0.0268107i	−0.0913090	+	0.0268107i
$540$		0			0
$541$	1.16413	+	1.34347i	1.16413	+	1.34347i	0.928368	+	0.371662i	$0.121212\pi$
$541$	1.16413	+	1.34347i	1.16413	+	1.34347i	0.235759	+	0.971812i	$0.424242\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.607279	+	1.75462i	−0.607279	+	1.75462i	0.0475819	+	0.998867i	$0.484848\pi$
$547$	−0.607279	+	1.75462i	−0.607279	+	1.75462i	−0.654861	+	0.755750i	$0.727273\pi$
$548$	0.313148	−	0.125365i	0.313148	−	0.125365i
$549$		0			0
$550$	−0.0517765	+	0.113375i	−0.0517765	+	0.113375i
$551$		0			0
$552$		0			0
$553$	1.91030	+	0.182411i	1.91030	+	0.182411i
$554$	−1.27822	−	1.79501i	−1.27822	−	1.79501i
$555$		0			0
$556$		0			0
$557$	−1.88431	−	0.363170i	−1.88431	−	0.363170i	−0.888835	−	0.458227i	$-0.848485\pi$
$557$	−1.88431	−	0.363170i	−1.88431	−	0.363170i	−0.995472	+	0.0950560i	$0.969697\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.216237	−	0.303662i	0.216237	−	0.303662i
$563$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$563$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.786053	+	0.618159i	−0.786053	+	0.618159i
$568$	−0.0574904	−	0.236979i	−0.0574904	−	0.236979i
$569$	−0.0623191	+	1.30824i	−0.0623191	+	1.30824i	0.723734	+	0.690079i	$0.242424\pi$
$569$	−0.0623191	+	1.30824i	−0.0623191	+	1.30824i	−0.786053	+	0.618159i	$0.787879\pi$
$570$		0			0
$571$	0.0224357	+	0.0924813i	0.0224357	+	0.0924813i	0.981929	−	0.189251i	$-0.0606061\pi$
$571$	0.0224357	+	0.0924813i	0.0224357	+	0.0924813i	−0.959493	+	0.281733i	$0.909091\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.72373	+	0.690079i	1.72373	+	0.690079i
$576$	0.244123	−	0.281733i	0.244123	−	0.281733i
$577$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$577$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$578$	0.186393	−	1.29639i	0.186393	−	1.29639i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.0996919	+	0.0950560i	0.0996919	+	0.0950560i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$587$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.773100	−	2.23373i	0.773100	−	2.23373i
$593$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$593$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$594$		0			0
$595$		0			0
$596$	0.635847	+	0.327802i	0.635847	+	0.327802i
$597$		0			0
$598$		0			0
$599$	−1.44091	+	0.137591i	−1.44091	+	0.137591i	−0.786053	−	0.618159i	$-0.787879\pi$
$599$	−1.44091	+	0.137591i	−1.44091	+	0.137591i	−0.654861	+	0.755750i	$0.727273\pi$
$600$		0			0
$601$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$601$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$602$	−1.51943			−1.51943
$603$	0.723734	−	0.690079i	0.723734	−	0.690079i
$604$	1.20362			1.20362
$605$		0			0
$606$		0			0
$607$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$607$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.473420	+	0.451405i	−0.473420	+	0.451405i	−0.888835	−	0.458227i	$-0.848485\pi$
$613$	−0.473420	+	0.451405i	−0.473420	+	0.451405i	0.415415	+	0.909632i	$0.363636\pi$
$614$		0			0
$615$		0			0
$616$	−0.0278858	−	0.0219296i	−0.0278858	−	0.0219296i
$617$	0.481929	−	1.05528i	0.481929	−	1.05528i	−0.500000	−	0.866025i	$-0.666667\pi$
$617$	0.481929	−	1.05528i	0.481929	−	1.05528i	0.981929	−	0.189251i	$-0.0606061\pi$
$618$		0			0
$619$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$619$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.65033	−	0.660694i	−1.65033	−	0.660694i	−0.654861	−	0.755750i	$-0.727273\pi$
$631$	−1.65033	−	0.660694i	−1.65033	−	0.660694i	−0.995472	+	0.0950560i	$0.969697\pi$
$632$	0.357685	+	0.619529i	0.357685	+	0.619529i
$633$		0			0
$634$	0.855348	−	0.672653i	0.855348	−	0.672653i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	0.0522361	+	0.215320i	0.0522361	+	0.215320i
$639$	0.514186	−	0.404360i	0.514186	−	0.404360i
$640$		0			0
$641$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$641$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$642$		0			0
$643$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$643$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$644$	0.770463	−	1.08196i	0.770463	−	1.08196i
$645$		0			0
$646$		0			0
$647$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$647$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$648$	−0.357685	−	0.105026i	−0.357685	−	0.105026i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.195659	+	0.274765i	0.195659	+	0.274765i
$653$	1.91030	+	0.182411i	1.91030	+	0.182411i	0.981929	−	0.189251i	$-0.0606061\pi$
$653$	1.91030	+	0.182411i	1.91030	+	0.182411i	0.928368	+	0.371662i	$0.121212\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.514186	−	1.48564i	0.514186	−	1.48564i	−0.327068	−	0.945001i	$-0.606061\pi$
$659$	0.514186	−	1.48564i	0.514186	−	1.48564i	0.841254	−	0.540641i	$-0.181818\pi$
$660$		0			0
$661$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$661$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$662$	0.0878875	+	0.611271i	0.0878875	+	0.611271i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	2.56046	−	0.244494i	2.56046	−	0.244494i
$667$	3.16697	−	0.929905i	3.16697	−	0.929905i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.88431	+	0.553283i	−1.88431	+	0.553283i	−0.888835	+	0.458227i	$0.848485\pi$
$673$	−1.88431	+	0.553283i	−1.88431	+	0.553283i	−0.995472	+	0.0950560i	$0.969697\pi$
$674$	−2.59577	+	0.247866i	−2.59577	+	0.247866i
$675$		0			0
$676$	−0.233975	−	0.676026i	−0.233975	−	0.676026i
$677$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$677$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.514186	+	0.404360i	0.514186	+	0.404360i	0.841254	−	0.540641i	$-0.181818\pi$
$683$	0.514186	+	0.404360i	0.514186	+	0.404360i	−0.327068	+	0.945001i	$0.606061\pi$
$684$		0			0
$685$		0			0
$686$	−1.28605	+	0.247866i	−1.28605	+	0.247866i
$687$		0			0
$688$	−0.809952	−	1.13742i	−0.809952	−	1.13742i
$689$		0			0
$690$		0			0
$691$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$691$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$692$		0			0
$693$	0.0224357	−	0.0924813i	0.0224357	−	0.0924813i
$694$	0.855348	+	1.87295i	0.855348	+	1.87295i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.357685	+	0.619529i	−0.357685	+	0.619529i
$701$	0.223734	−	0.175946i	0.223734	−	0.175946i	−0.500000	−	0.866025i	$-0.666667\pi$
$701$	0.223734	−	0.175946i	0.223734	−	0.175946i	0.723734	+	0.690079i	$0.242424\pi$
$702$		0			0
$703$		0			0
$704$	−0.00168800	+	0.0354355i	−0.00168800	+	0.0354355i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.21590	−	0.486774i	−1.21590	−	0.486774i	−0.327068	−	0.945001i	$-0.606061\pi$
$709$	−1.21590	−	0.486774i	−1.21590	−	0.486774i	−0.888835	+	0.458227i	$0.848485\pi$
$710$		0			0
$711$	−1.11312	+	1.56316i	−1.11312	+	1.56316i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.39852	−	0.269543i	−1.39852	−	0.269543i
$717$		0			0
$718$	−1.83013	+	0.943498i	−1.83013	+	0.943498i
$719$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$719$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$720$		0			0
$721$		0			0
$722$	−1.10181	−	0.708089i	−1.10181	−	0.708089i
$723$		0			0
$724$		0			0
$725$	−1.65033	+	0.660694i	−1.65033	+	0.660694i
$726$		0			0
$727$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$727$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$728$		0			0
$729$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$733$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$734$		0			0
$735$		0			0
$736$	2.23480			2.23480
$737$	−0.0135432	+	0.0941952i	−0.0135432	+	0.0941952i
$738$		0			0
$739$	0.0688733	+	1.44583i	0.0688733	+	1.44583i	0.723734	+	0.690079i	$0.242424\pi$
$739$	0.0688733	+	1.44583i	0.0688733	+	1.44583i	−0.654861	+	0.755750i	$0.727273\pi$
$740$		0			0
$741$		0			0
$742$	1.24147	+	1.43273i	1.24147	+	1.43273i
$743$	0.627639	+	1.81344i	0.627639	+	1.81344i	0.580057	+	0.814576i	$0.303030\pi$
$743$	0.627639	+	1.81344i	0.627639	+	1.81344i	0.0475819	+	0.998867i	$0.484848\pi$
$744$		0			0
$745$		0			0
$746$	2.19364	−	1.40977i	2.19364	−	1.40977i
$747$		0			0
$748$		0			0
$749$	1.07701	−	0.431171i	1.07701	−	0.431171i
$750$		0			0
$751$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$751$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.205996	−	0.196417i	−0.205996	−	0.196417i	0.580057	−	0.814576i	$-0.303030\pi$
$757$	−0.205996	−	0.196417i	−0.205996	−	0.196417i	−0.786053	+	0.618159i	$0.787879\pi$
$758$	0.366049	+	0.0705501i	0.366049	+	0.0705501i
$759$		0			0
$760$		0			0
$761$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$761$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$762$		0			0
$763$	−0.911911	+	1.28060i	−0.911911	+	1.28060i
$764$	0.306442	−	0.353653i	0.306442	−	0.353653i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$769$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$770$		0			0
$771$		0			0
$772$	0.0795238	+	0.327802i	0.0795238	+	0.327802i
$773$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$773$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$774$	0.759713	−	1.31586i	0.759713	−	1.31586i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.09966	+	1.54426i	−1.09966	+	1.54426i
$779$		0			0
$780$		0			0
$781$	−0.0146760	+	0.0604953i	−0.0146760	+	0.0604953i
$782$		0			0
$783$		0			0
$784$	−0.871098	−	0.830590i	−0.871098	−	0.830590i
$785$		0			0
$786$		0			0
$787$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$787$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$788$	1.37950	−	0.265876i	1.37950	−	0.265876i
$789$		0			0
$790$		0			0
$791$	0.786053	+	0.618159i	0.786053	+	0.618159i
$792$	0.0329345	−	0.0131850i	0.0329345	−	0.0131850i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$797$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$798$		0			0
$799$		0			0
$800$	−1.19817	+	0.114411i	−1.19817	+	0.114411i
$801$		0			0
$802$	0.119589	+	2.51049i	0.119589	+	2.51049i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.759713	−	0.876756i	−0.759713	−	0.876756i	0.235759	−	0.971812i	$-0.424242\pi$
$809$	−0.759713	−	0.876756i	−0.759713	−	0.876756i	−0.995472	+	0.0950560i	$0.969697\pi$
$810$		0			0
$811$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$811$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$812$	0.180981	+	1.25875i	0.180981	+	1.25875i
$813$		0			0
$814$	−0.177149	+	0.168912i	−0.177149	+	0.168912i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.469383	−	0.0448206i	−0.469383	−	0.0448206i	−0.142315	−	0.989821i	$-0.545455\pi$
$821$	−0.469383	−	0.0448206i	−0.469383	−	0.0448206i	−0.327068	+	0.945001i	$0.606061\pi$
$822$		0			0
$823$	0.888835	−	0.458227i	0.888835	−	0.458227i	0.0475819	−	0.998867i	$-0.484848\pi$
$823$	0.888835	−	0.458227i	0.888835	−	0.458227i	0.841254	+	0.540641i	$0.181818\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.469383	+	1.93482i	−0.469383	+	1.93482i	−0.142315	+	0.989821i	$0.545455\pi$
$827$	−0.469383	+	1.93482i	−0.469383	+	1.93482i	−0.327068	+	0.945001i	$0.606061\pi$
$828$	0.551777	+	1.20822i	0.551777	+	1.20822i
$829$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$829$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$830$		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		−0.235759	−	0.971812i	$-0.575758\pi$
$839$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$840$		0			0
$841$	−1.08006	+	1.87071i	−1.08006	+	1.87071i
$842$	−1.02951	−	1.78316i	−1.02951	−	1.78316i
$843$		0			0
$844$	0.468468	−	0.540641i	0.468468	−	0.540641i
$845$		0			0
$846$		0			0
$847$	−0.411653	−	0.901394i	−0.411653	−	0.901394i
$848$	−0.410738	+	1.69309i	−0.410738	+	1.69309i
$849$		0			0
$850$		0			0
$851$	2.63900	+	2.51628i	2.63900	+	2.51628i
$852$		0			0
$853$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$853$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$854$		0			0
$855$		0			0
$856$	0.363820	+	0.233813i	0.363820	+	0.233813i
$857$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$857$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$858$		0			0
$859$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$859$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$860$		0			0
$861$		0			0
$862$	−0.104852	+	0.0673845i	−0.104852	+	0.0673845i
$863$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	−0.786053	−	0.618159i	$-0.787879\pi$
$863$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	0.580057	−	0.814576i	$-0.303030\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.00868932	−	0.182411i	−0.00868932	−	0.182411i
$870$		0			0
$871$		0			0
$872$	−0.586059			−0.586059
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.428368	+	1.23769i	0.428368	+	1.23769i	0.928368	+	0.371662i	$0.121212\pi$
$877$	0.428368	+	1.23769i	0.428368	+	1.23769i	−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$881$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$882$	0.428368	−	1.23769i	0.428368	−	1.23769i
$883$	1.34378	−	0.537970i	1.34378	−	0.537970i	0.415415	−	0.909632i	$-0.363636\pi$
$883$	1.34378	−	0.537970i	1.34378	−	0.537970i	0.928368	+	0.371662i	$0.121212\pi$
$884$		0			0
$885$		0			0
$886$	−0.915415	−	0.588302i	−0.915415	−	0.588302i
$887$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$887$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$888$		0			0
$889$	0.481929	+	0.676774i	0.481929	+	0.676774i
$890$		0			0
$891$	0.0688733	+	0.0656706i	0.0688733	+	0.0656706i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.101808	−	0.708089i	0.101808	−	0.708089i
$897$		0			0
$898$	0.712591	−	0.822373i	0.712591	−	0.822373i
$899$		0			0
$900$	−0.357685	−	0.619529i	−0.357685	−	0.619529i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.0177379	+	0.372363i	−0.0177379	+	0.372363i
$905$		0			0
$906$		0			0
$907$	−0.0748038	+	0.0588264i	−0.0748038	+	0.0588264i	−0.654861	−	0.755750i	$-0.727273\pi$
$907$	−0.0748038	+	0.0588264i	−0.0748038	+	0.0588264i	0.580057	+	0.814576i	$0.303030\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.308779	+	0.356349i	−0.308779	+	0.356349i	−0.888835	−	0.458227i	$-0.848485\pi$
$911$	−0.308779	+	0.356349i	−0.308779	+	0.356349i	0.580057	+	0.814576i	$0.303030\pi$
$912$		0			0
$913$		0			0
$914$	0.355901	+	0.779314i	0.355901	+	0.779314i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.76962	−	0.912303i	1.76962	−	0.912303i	0.841254	−	0.540641i	$-0.181818\pi$
$919$	1.76962	−	0.912303i	1.76962	−	0.912303i	0.928368	−	0.371662i	$-0.121212\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.54370	−	1.21398i	−1.54370	−	1.21398i
$926$	1.91153	−	0.765261i	1.91153	−	0.765261i
$927$		0			0
$928$	−1.54852	+	1.47652i	−1.54852	+	1.47652i
$929$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$929$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$930$		0			0
$931$		0			0
$932$	0.415930	+	1.20175i	0.415930	+	1.20175i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−0.308779	+	1.27280i	−0.308779	+	1.27280i
$939$		0			0
$940$		0			0
$941$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$941$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.0205779	+	0.143123i	0.0205779	+	0.143123i
$947$	−1.10181	+	0.708089i	−1.10181	+	0.708089i	−0.959493	−	0.281733i	$-0.909091\pi$
$947$	−1.10181	+	0.708089i	−1.10181	+	0.708089i	−0.142315	+	0.989821i	$0.545455\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.21769	+	0.782560i	1.21769	+	0.782560i	0.981929	−	0.189251i	$-0.0606061\pi$
$953$	1.21769	+	0.782560i	1.21769	+	0.782560i	0.235759	+	0.971812i	$0.424242\pi$
$954$	−1.86152	+	0.358779i	−1.86152	+	0.358779i
$955$		0			0
$956$	0.770463	+	1.08196i	0.770463	+	1.08196i
$957$		0			0
$958$		0			0
$959$	0.462997	+	0.0892353i	0.462997	+	0.0892353i
$960$		0			0
$961$	0.235759	−	0.971812i	0.235759	−	0.971812i
$962$		0			0
$963$	−0.165101	+	1.14831i	−0.165101	+	1.14831i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$967$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$968$	0.184705	−	0.319918i	0.184705	−	0.319918i
$969$		0			0
$970$		0			0
$971$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$971$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$972$		0			0
$973$		0			0
$974$	2.02181	−	1.58997i	2.02181	−	1.58997i
$975$		0			0
$976$		0			0
$977$	−0.928368	−	0.371662i	−0.928368	−	0.371662i	−0.142315	−	0.989821i	$-0.545455\pi$
$977$	−0.928368	−	0.371662i	−0.928368	−	0.371662i	−0.786053	+	0.618159i	$0.787879\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.653077	−	1.43004i	−0.653077	−	1.43004i
$982$	0.592542	−	2.44249i	0.592542	−	2.44249i
$983$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$983$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.11510	−	0.407652i	2.11510	−	0.407652i
$990$		0			0
$991$	0.698939	−	1.53046i	0.698939	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$991$	0.698939	−	1.53046i	0.698939	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$992$		0			0
$993$		0			0
$994$	−0.280211	+	0.809616i	−0.280211	+	0.809616i
$995$		0			0
$996$		0			0
$997$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$997$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$998$	−1.16413	−	0.600149i	−1.16413	−	0.600149i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	469.1.bl.a.167.1	yes	20
7.2	even	3		3283.1.bw.a.2579.1		20
7.3	odd	6		3283.1.cd.a.1440.1		20
7.4	even	3		3283.1.cd.a.1440.1		20
7.5	odd	6		3283.1.bw.a.2579.1		20
7.6	odd	2	CM	469.1.bl.a.167.1	yes	20
67.65	even	33	inner	469.1.bl.a.132.1	✓	20
469.65	even	33		3283.1.cd.a.668.1		20
469.132	odd	66	inner	469.1.bl.a.132.1	✓	20
469.199	odd	66		3283.1.bw.a.2812.1		20
469.333	even	33		3283.1.bw.a.2812.1		20
469.467	odd	66		3283.1.cd.a.668.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
469.1.bl.a.132.1	✓	20	67.65	even	33	inner
469.1.bl.a.132.1	✓	20	469.132	odd	66	inner
469.1.bl.a.167.1	yes	20	1.1	even	1	trivial
469.1.bl.a.167.1	yes	20	7.6	odd	2	CM
3283.1.bw.a.2579.1		20	7.2	even	3
3283.1.bw.a.2579.1		20	7.5	odd	6
3283.1.bw.a.2812.1		20	469.199	odd	66
3283.1.bw.a.2812.1		20	469.333	even	33
3283.1.cd.a.668.1		20	469.65	even	33
3283.1.cd.a.668.1		20	469.467	odd	66
3283.1.cd.a.1440.1		20	7.3	odd	6
3283.1.cd.a.1440.1		20	7.4	even	3

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)

\(f(q)\)	\(=\)	\(q+(-0.0623191 - 1.30824i) q^{2} +(-0.712131 + 0.0680003i) q^{4} +(-0.888835 - 0.458227i) q^{7} +(-0.0530529 - 0.368991i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)	\(q+(-0.0623191 - 1.30824i) q^{2} +(-0.712131 + 0.0680003i) q^{4} +(-0.888835 - 0.458227i) q^{7} +(-0.0530529 - 0.368991i) q^{8} +(0.841254 - 0.540641i) q^{9} +(-0.0311250 + 0.0899299i) q^{11} +(-0.544078 + 1.19136i) q^{14} +(-1.18186 + 0.227786i) q^{16} +(-0.759713 - 1.06687i) q^{18} +(0.119589 + 0.0351146i) q^{22} +(0.437742 - 1.80440i) q^{23} +(-0.142315 + 0.989821i) q^{25} +(0.664127 + 0.265876i) q^{28} +(0.888835 + 1.53951i) q^{29} +(0.283763 + 1.16969i) q^{32} +(-0.562319 + 0.442213i) q^{36} +(-0.981929 + 1.70075i) q^{37} +(0.481929 + 1.05528i) q^{43} +(0.0160499 - 0.0661584i) q^{44} +(-2.38786 - 0.460222i) q^{46} +(0.580057 + 0.814576i) q^{49} +(1.30379 + 0.124497i) q^{50} +(0.601300 - 1.31666i) q^{53} +(-0.121926 + 0.352283i) q^{56} +(1.95865 - 1.25875i) q^{58} +(-0.995472 + 0.0950560i) q^{63} +(0.357685 - 0.105026i) q^{64} +(0.981929 - 0.189251i) q^{67} +(0.651174 - 0.0621796i) q^{71} +(-0.244123 - 0.281733i) q^{72} +(2.28618 + 1.17861i) q^{74} +(0.0688733 - 0.0656706i) q^{77} +(-1.78153 + 0.713215i) q^{79} +(0.415415 - 0.909632i) q^{81} +(1.35052 - 0.696241i) q^{86} +(0.0348346 + 0.00671382i) q^{88} +(-0.189030 + 1.31473i) q^{92} +(1.02951 - 0.809616i) q^{98} +(0.0224357 + 0.0924813i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} + 3 q^{4} + q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \) 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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