LMFDB - Embedded newform 1386.2.k.q.991.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(793,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1386.793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.k (of order $3$, degree $2$, 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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			991.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1386.991
Dual form			1386.2.k.q.793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.207107 + 0.358719i) q^{5} +(-2.62132 + 0.358719i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.207107 + 0.358719i) q^{5} +(-2.62132 + 0.358719i) q^{7} +1.00000 q^{8} +(0.207107 - 0.358719i) q^{10} +(-0.500000 + 0.866025i) q^{11} -1.17157 q^{13} +(1.62132 + 2.09077i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.08579 - 1.88064i) q^{17} +(-0.414214 - 0.717439i) q^{19} -0.414214 q^{20} +1.00000 q^{22} +(1.62132 + 2.80821i) q^{23} +(2.41421 - 4.18154i) q^{25} +(0.585786 + 1.01461i) q^{26} +(1.00000 - 2.44949i) q^{28} +2.82843 q^{29} +(3.24264 - 5.61642i) q^{31} +(-0.500000 + 0.866025i) q^{32} -2.17157 q^{34} +(-0.671573 - 0.866025i) q^{35} +(-4.82843 - 8.36308i) q^{37} +(-0.414214 + 0.717439i) q^{38} +(0.207107 + 0.358719i) q^{40} +4.65685 q^{41} -2.82843 q^{43} +(-0.500000 - 0.866025i) q^{44} +(1.62132 - 2.80821i) q^{46} +(4.62132 + 8.00436i) q^{47} +(6.74264 - 1.88064i) q^{49} -4.82843 q^{50} +(0.585786 - 1.01461i) q^{52} +(2.58579 - 4.47871i) q^{53} -0.414214 q^{55} +(-2.62132 + 0.358719i) q^{56} +(-1.41421 - 2.44949i) q^{58} +(1.82843 - 3.16693i) q^{59} +(0.792893 + 1.37333i) q^{61} -6.48528 q^{62} +1.00000 q^{64} +(-0.242641 - 0.420266i) q^{65} +(6.74264 - 11.6786i) q^{67} +(1.08579 + 1.88064i) q^{68} +(-0.414214 + 1.01461i) q^{70} +13.3137 q^{71} +(2.41421 - 4.18154i) q^{73} +(-4.82843 + 8.36308i) q^{74} +0.828427 q^{76} +(1.00000 - 2.44949i) q^{77} +(-2.37868 - 4.11999i) q^{79} +(0.207107 - 0.358719i) q^{80} +(-2.32843 - 4.03295i) q^{82} -9.82843 q^{83} +0.899495 q^{85} +(1.41421 + 2.44949i) q^{86} +(-0.500000 + 0.866025i) q^{88} +(-6.24264 - 10.8126i) q^{89} +(3.07107 - 0.420266i) q^{91} -3.24264 q^{92} +(4.62132 - 8.00436i) q^{94} +(0.171573 - 0.297173i) q^{95} -10.1716 q^{97} +(-5.00000 - 4.89898i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^8	$ 4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{8} - 2 q^{10} - 2 q^{11} - 16 q^{13} - 2 q^{14} - 2 q^{16} + 10 q^{17} + 4 q^{19} + 4 q^{20} + 4 q^{22} - 2 q^{23} + 4 q^{25} + 8 q^{26} + 4 q^{28} - 4 q^{31} - 2 q^{32} - 20 q^{34} - 14 q^{35} - 8 q^{37} + 4 q^{38} - 2 q^{40} - 4 q^{41} - 2 q^{44} - 2 q^{46} + 10 q^{47} + 10 q^{49} - 8 q^{50} + 8 q^{52} + 16 q^{53} + 4 q^{55} - 2 q^{56} - 4 q^{59} + 6 q^{61} + 8 q^{62} + 4 q^{64} + 16 q^{65} + 10 q^{67} + 10 q^{68} + 4 q^{70} + 8 q^{71} + 4 q^{73} - 8 q^{74} - 8 q^{76} + 4 q^{77} - 18 q^{79} - 2 q^{80} + 2 q^{82} - 28 q^{83} - 36 q^{85} - 2 q^{88} - 8 q^{89} - 16 q^{91} + 4 q^{92} + 10 q^{94} + 12 q^{95} - 52 q^{97} - 20 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^8 - 2 * q^10 - 2 * q^11 - 16 * q^13 - 2 * q^14 - 2 * q^16 + 10 * q^17 + 4 * q^19 + 4 * q^20 + 4 * q^22 - 2 * q^23 + 4 * q^25 + 8 * q^26 + 4 * q^28 - 4 * q^31 - 2 * q^32 - 20 * q^34 - 14 * q^35 - 8 * q^37 + 4 * q^38 - 2 * q^40 - 4 * q^41 - 2 * q^44 - 2 * q^46 + 10 * q^47 + 10 * q^49 - 8 * q^50 + 8 * q^52 + 16 * q^53 + 4 * q^55 - 2 * q^56 - 4 * q^59 + 6 * q^61 + 8 * q^62 + 4 * q^64 + 16 * q^65 + 10 * q^67 + 10 * q^68 + 4 * q^70 + 8 * q^71 + 4 * q^73 - 8 * q^74 - 8 * q^76 + 4 * q^77 - 18 * q^79 - 2 * q^80 + 2 * q^82 - 28 * q^83 - 36 * q^85 - 2 * q^88 - 8 * q^89 - 16 * q^91 + 4 * q^92 + 10 * q^94 + 12 * q^95 - 52 * q^97 - 20 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	0.207107	+	0.358719i	0.0926210	+	0.160424i	0.908613	−	0.417639i	$-0.137142\pi$
$5$	0.207107	+	0.358719i	0.0926210	+	0.160424i	−0.815992	+	0.578063i	$0.803809\pi$
$6$		0			0
$7$	−2.62132	+	0.358719i	−0.990766	+	0.135583i
$7$	−2.62132	+	0.358719i	−0.990766	+	0.135583i
$8$	1.00000			0.353553
$9$		0			0
$10$	0.207107	−	0.358719i	0.0654929	−	0.113437i
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$12$		0			0
$13$	−1.17157			−0.324936			−0.162468	−	0.986714i	$-0.551945\pi$
$13$	−1.17157			−0.324936			−0.162468	+	0.986714i	$0.551945\pi$
$14$	1.62132	+	2.09077i	0.433316	+	0.558782i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	1.08579	−	1.88064i	0.263342	−	0.456122i	−0.703786	−	0.710412i	$-0.748509\pi$
$17$	1.08579	−	1.88064i	0.263342	−	0.456122i	0.967128	+	0.254291i	$0.0818419\pi$
$18$		0			0
$19$	−0.414214	−	0.717439i	−0.0950271	−	0.164592i	0.814593	−	0.580033i	$-0.196960\pi$
$19$	−0.414214	−	0.717439i	−0.0950271	−	0.164592i	−0.909620	+	0.415441i	$0.863627\pi$
$20$	−0.414214			−0.0926210
$21$		0			0
$22$	1.00000			0.213201
$23$	1.62132	+	2.80821i	0.338069	+	0.585552i	0.984069	−	0.177785i	$-0.0568931\pi$
$23$	1.62132	+	2.80821i	0.338069	+	0.585552i	−0.646001	+	0.763337i	$0.723560\pi$
$24$		0			0
$25$	2.41421	−	4.18154i	0.482843	−	0.836308i
$26$	0.585786	+	1.01461i	0.114882	+	0.198982i
$27$		0			0
$28$	1.00000	−	2.44949i	0.188982	−	0.462910i
$29$	2.82843			0.525226			0.262613	−	0.964901i	$-0.415416\pi$
$29$	2.82843			0.525226			0.262613	+	0.964901i	$0.415416\pi$
$30$		0			0
$31$	3.24264	−	5.61642i	0.582395	−	1.00874i	−0.412799	−	0.910822i	$-0.635449\pi$
$31$	3.24264	−	5.61642i	0.582395	−	1.00874i	0.995195	−	0.0979165i	$-0.0312178\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−2.17157			−0.372422
$35$	−0.671573	−	0.866025i	−0.113517	−	0.146385i
$36$		0			0
$37$	−4.82843	−	8.36308i	−0.793789	−	1.37488i	−0.923606	−	0.383344i	$-0.874772\pi$
$37$	−4.82843	−	8.36308i	−0.793789	−	1.37488i	0.129817	−	0.991538i	$-0.458561\pi$
$38$	−0.414214	+	0.717439i	−0.0671943	+	0.116384i
$39$		0			0
$40$	0.207107	+	0.358719i	0.0327465	+	0.0567185i
$41$	4.65685			0.727278			0.363639	−	0.931540i	$-0.381534\pi$
$41$	4.65685			0.727278			0.363639	+	0.931540i	$0.381534\pi$
$42$		0			0
$43$	−2.82843			−0.431331			−0.215666	−	0.976467i	$-0.569192\pi$
$43$	−2.82843			−0.431331			−0.215666	+	0.976467i	$0.569192\pi$
$44$	−0.500000	−	0.866025i	−0.0753778	−	0.130558i
$45$		0			0
$46$	1.62132	−	2.80821i	0.239051	−	0.414048i
$47$	4.62132	+	8.00436i	0.674089	+	1.16756i	0.976734	+	0.214453i	$0.0687969\pi$
$47$	4.62132	+	8.00436i	0.674089	+	1.16756i	−0.302645	+	0.953103i	$0.597870\pi$
$48$		0			0
$49$	6.74264	−	1.88064i	0.963234	−	0.268662i
$50$	−4.82843			−0.682843
$51$		0			0
$52$	0.585786	−	1.01461i	0.0812340	−	0.140701i
$53$	2.58579	−	4.47871i	0.355185	−	0.615199i	−0.631965	−	0.774997i	$-0.717751\pi$
$53$	2.58579	−	4.47871i	0.355185	−	0.615199i	0.987150	+	0.159799i	$0.0510845\pi$
$54$		0			0
$55$	−0.414214			−0.0558525
$56$	−2.62132	+	0.358719i	−0.350289	+	0.0479359i
$57$		0			0
$58$	−1.41421	−	2.44949i	−0.185695	−	0.321634i
$59$	1.82843	−	3.16693i	0.238041	−	0.412299i	−0.722111	−	0.691777i	$-0.756828\pi$
$59$	1.82843	−	3.16693i	0.238041	−	0.412299i	0.960152	+	0.279478i	$0.0901614\pi$
$60$		0			0
$61$	0.792893	+	1.37333i	0.101520	+	0.175837i	0.912311	−	0.409498i	$-0.134296\pi$
$61$	0.792893	+	1.37333i	0.101520	+	0.175837i	−0.810791	+	0.585335i	$0.800963\pi$
$62$	−6.48528			−0.823632
$63$		0			0
$64$	1.00000			0.125000
$65$	−0.242641	−	0.420266i	−0.0300959	−	0.0521276i
$66$		0			0
$67$	6.74264	−	11.6786i	0.823745	−	1.42677i	−0.0791303	−	0.996864i	$-0.525214\pi$
$67$	6.74264	−	11.6786i	0.823745	−	1.42677i	0.902875	−	0.429903i	$-0.141452\pi$
$68$	1.08579	+	1.88064i	0.131671	+	0.228061i
$69$		0			0
$70$	−0.414214	+	1.01461i	−0.0495080	+	0.121269i
$71$	13.3137			1.58005			0.790023	−	0.613077i	$-0.210068\pi$
$71$	13.3137			1.58005			0.790023	+	0.613077i	$0.210068\pi$
$72$		0			0
$73$	2.41421	−	4.18154i	0.282562	−	0.489412i	−0.689453	−	0.724331i	$-0.742149\pi$
$73$	2.41421	−	4.18154i	0.282562	−	0.489412i	0.972015	+	0.234918i	$0.0754823\pi$
$74$	−4.82843	+	8.36308i	−0.561293	+	0.972188i
$75$		0			0
$76$	0.828427			0.0950271
$77$	1.00000	−	2.44949i	0.113961	−	0.279145i
$78$		0			0
$79$	−2.37868	−	4.11999i	−0.267622	−	0.463536i	0.700625	−	0.713530i	$-0.252905\pi$
$79$	−2.37868	−	4.11999i	−0.267622	−	0.463536i	−0.968247	+	0.249994i	$0.919571\pi$
$80$	0.207107	−	0.358719i	0.0231552	−	0.0401061i
$81$		0			0
$82$	−2.32843	−	4.03295i	−0.257132	−	0.445365i
$83$	−9.82843			−1.07881			−0.539405	−	0.842046i	$-0.681351\pi$
$83$	−9.82843			−1.07881			−0.539405	+	0.842046i	$0.681351\pi$
$84$		0			0
$85$	0.899495			0.0975639
$86$	1.41421	+	2.44949i	0.152499	+	0.264135i
$87$		0			0
$88$	−0.500000	+	0.866025i	−0.0533002	+	0.0923186i
$89$	−6.24264	−	10.8126i	−0.661719	−	1.14613i	−0.980164	−	0.198189i	$-0.936494\pi$
$89$	−6.24264	−	10.8126i	−0.661719	−	1.14613i	0.318445	−	0.947941i	$-0.396839\pi$
$90$		0			0
$91$	3.07107	−	0.420266i	0.321935	−	0.0440558i
$92$	−3.24264			−0.338069
$93$		0			0
$94$	4.62132	−	8.00436i	0.476653	−	0.825587i
$95$	0.171573	−	0.297173i	0.0176030	−	0.0304893i
$96$		0			0
$97$	−10.1716			−1.03277			−0.516383	−	0.856358i	$-0.672722\pi$
$97$	−10.1716			−1.03277			−0.516383	+	0.856358i	$0.672722\pi$
$98$	−5.00000	−	4.89898i	−0.505076	−	0.494872i
$99$		0			0
$100$	2.41421	+	4.18154i	0.241421	+	0.418154i
$101$	7.07107	−	12.2474i	0.703598	−	1.21867i	−0.263598	−	0.964633i	$-0.584909\pi$
$101$	7.07107	−	12.2474i	0.703598	−	1.21867i	0.967195	−	0.254034i	$-0.0817575\pi$
$102$		0			0
$103$	4.58579	+	7.94282i	0.451851	+	0.782629i	0.998501	−	0.0547323i	$-0.0174305\pi$
$103$	4.58579	+	7.94282i	0.451851	+	0.782629i	−0.546650	+	0.837361i	$0.684097\pi$
$104$	−1.17157			−0.114882
$105$		0			0
$106$	−5.17157			−0.502308
$107$	8.32843	+	14.4253i	0.805139	+	1.39454i	0.916197	+	0.400728i	$0.131243\pi$
$107$	8.32843	+	14.4253i	0.805139	+	1.39454i	−0.111057	+	0.993814i	$0.535424\pi$
$108$		0			0
$109$	−1.62132	+	2.80821i	−0.155294	+	0.268978i	−0.933166	−	0.359445i	$-0.882966\pi$
$109$	−1.62132	+	2.80821i	−0.155294	+	0.268978i	0.777872	+	0.628423i	$0.216299\pi$
$110$	0.207107	+	0.358719i	0.0197469	+	0.0342026i
$111$		0			0
$112$	1.62132	+	2.09077i	0.153200	+	0.197559i
$113$	3.65685			0.344008			0.172004	−	0.985096i	$-0.444976\pi$
$113$	3.65685			0.344008			0.172004	+	0.985096i	$0.444976\pi$
$114$		0			0
$115$	−0.671573	+	1.16320i	−0.0626245	+	0.108469i
$116$	−1.41421	+	2.44949i	−0.131306	+	0.227429i
$117$		0			0
$118$	−3.65685			−0.336641
$119$	−2.17157	+	5.31925i	−0.199068	+	0.487614i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$	0.792893	−	1.37333i	0.0717852	−	0.124336i
$123$		0			0
$124$	3.24264	+	5.61642i	0.291198	+	0.504369i
$125$	4.07107			0.364127
$126$		0			0
$127$	1.24264			0.110267			0.0551333	−	0.998479i	$-0.482442\pi$
$127$	1.24264			0.110267			0.0551333	+	0.998479i	$0.482442\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−0.242641	+	0.420266i	−0.0212810	+	0.0368598i
$131$	−7.65685	−	13.2621i	−0.668982	−	1.15871i	−0.978189	−	0.207717i	$-0.933397\pi$
$131$	−7.65685	−	13.2621i	−0.668982	−	1.15871i	0.309207	−	0.950995i	$-0.399937\pi$
$132$		0			0
$133$	1.34315	+	1.73205i	0.116466	+	0.150188i
$134$	−13.4853			−1.16495
$135$		0			0
$136$	1.08579	−	1.88064i	0.0931054	−	0.161263i
$137$	2.41421	−	4.18154i	0.206260	−	0.357253i	−0.744273	−	0.667875i	$-0.767204\pi$
$137$	2.41421	−	4.18154i	0.206260	−	0.357253i	0.950534	+	0.310622i	$0.100537\pi$
$138$		0			0
$139$	−6.00000			−0.508913			−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000			−0.508913			−0.254457	+	0.967084i	$0.581897\pi$
$140$	1.08579	−	0.148586i	0.0917657	−	0.0125578i
$141$		0			0
$142$	−6.65685	−	11.5300i	−0.558631	−	0.967577i
$143$	0.585786	−	1.01461i	0.0489859	−	0.0848461i
$144$		0			0
$145$	0.585786	+	1.01461i	0.0486469	+	0.0842589i
$146$	−4.82843			−0.399603
$147$		0			0
$148$	9.65685			0.793789
$149$	−5.65685	−	9.79796i	−0.463428	−	0.802680i	0.535701	−	0.844407i	$-0.320047\pi$
$149$	−5.65685	−	9.79796i	−0.463428	−	0.802680i	−0.999129	+	0.0417274i	$0.986714\pi$
$150$		0			0
$151$	−0.621320	+	1.07616i	−0.0505623	+	0.0875765i	−0.890199	−	0.455572i	$-0.849435\pi$
$151$	−0.621320	+	1.07616i	−0.0505623	+	0.0875765i	0.839637	+	0.543149i	$0.182768\pi$
$152$	−0.414214	−	0.717439i	−0.0335972	−	0.0581920i
$153$		0			0
$154$	−2.62132	+	0.358719i	−0.211232	+	0.0289064i
$155$	2.68629			0.215768
$156$		0			0
$157$	4.58579	−	7.94282i	0.365986	−	0.633906i	−0.622948	−	0.782263i	$-0.714065\pi$
$157$	4.58579	−	7.94282i	0.365986	−	0.633906i	0.988934	+	0.148357i	$0.0473986\pi$
$158$	−2.37868	+	4.11999i	−0.189238	+	0.327769i
$159$		0			0
$160$	−0.414214			−0.0327465
$161$	−5.25736	−	6.77962i	−0.414338	−	0.534309i
$162$		0			0
$163$	−1.50000	−	2.59808i	−0.117489	−	0.203497i	0.801283	−	0.598286i	$-0.204151\pi$
$163$	−1.50000	−	2.59808i	−0.117489	−	0.203497i	−0.918772	+	0.394789i	$0.870818\pi$
$164$	−2.32843	+	4.03295i	−0.181820	+	0.314921i
$165$		0			0
$166$	4.91421	+	8.51167i	0.381417	+	0.660634i
$167$	−16.4853			−1.27567			−0.637835	−	0.770173i	$-0.720170\pi$
$167$	−16.4853			−1.27567			−0.637835	+	0.770173i	$0.720170\pi$
$168$		0			0
$169$	−11.6274			−0.894417
$170$	−0.449747	−	0.778985i	−0.0344941	−	0.0597455i
$171$		0			0
$172$	1.41421	−	2.44949i	0.107833	−	0.186772i
$173$	6.07107	+	10.5154i	0.461575	+	0.799471i	0.999040	−	0.0438152i	$-0.0139513\pi$
$173$	6.07107	+	10.5154i	0.461575	+	0.799471i	−0.537465	+	0.843286i	$0.680618\pi$
$174$		0			0
$175$	−4.82843	+	11.8272i	−0.364995	+	0.894051i
$176$	1.00000			0.0753778
$177$		0			0
$178$	−6.24264	+	10.8126i	−0.467906	+	0.810436i
$179$	1.58579	−	2.74666i	0.118527	−	0.205295i	−0.800657	−	0.599123i	$-0.795516\pi$
$179$	1.58579	−	2.74666i	0.118527	−	0.205295i	0.919184	+	0.393828i	$0.128849\pi$
$180$		0			0
$181$	−2.34315			−0.174165			−0.0870823	−	0.996201i	$-0.527754\pi$
$181$	−2.34315			−0.174165			−0.0870823	+	0.996201i	$0.527754\pi$
$182$	−1.89949	−	2.44949i	−0.140800	−	0.181568i
$183$		0			0
$184$	1.62132	+	2.80821i	0.119525	+	0.207024i
$185$	2.00000	−	3.46410i	0.147043	−	0.254686i
$186$		0			0
$187$	1.08579	+	1.88064i	0.0794006	+	0.137526i
$188$	−9.24264			−0.674089
$189$		0			0
$190$	−0.343146			−0.0248944
$191$	4.65685	+	8.06591i	0.336958	+	0.583629i	0.983859	−	0.178945i	$-0.0572686\pi$
$191$	4.65685	+	8.06591i	0.336958	+	0.583629i	−0.646901	+	0.762574i	$0.723935\pi$
$192$		0			0
$193$	4.58579	−	7.94282i	0.330092	−	0.571736i	−0.652438	−	0.757843i	$-0.726254\pi$
$193$	4.58579	−	7.94282i	0.330092	−	0.571736i	0.982530	+	0.186106i	$0.0595868\pi$
$194$	5.08579	+	8.80884i	0.365138	+	0.632438i
$195$		0			0
$196$	−1.74264	+	6.77962i	−0.124474	+	0.484258i
$197$	−3.51472			−0.250413			−0.125207	−	0.992131i	$-0.539959\pi$
$197$	−3.51472			−0.250413			−0.125207	+	0.992131i	$0.539959\pi$
$198$		0			0
$199$	4.17157	−	7.22538i	0.295715	−	0.512193i	−0.679436	−	0.733735i	$-0.737775\pi$
$199$	4.17157	−	7.22538i	0.295715	−	0.512193i	0.975151	+	0.221541i	$0.0711088\pi$
$200$	2.41421	−	4.18154i	0.170711	−	0.295680i
$201$		0			0
$202$	−14.1421			−0.995037
$203$	−7.41421	+	1.01461i	−0.520376	+	0.0712118i
$204$		0			0
$205$	0.964466	+	1.67050i	0.0673612	+	0.116673i
$206$	4.58579	−	7.94282i	0.319507	−	0.553402i
$207$		0			0
$208$	0.585786	+	1.01461i	0.0406170	+	0.0703507i
$209$	0.828427			0.0573035
$210$		0			0
$211$	14.8284			1.02083			0.510416	−	0.859928i	$-0.329492\pi$
$211$	14.8284			1.02083			0.510416	+	0.859928i	$0.329492\pi$
$212$	2.58579	+	4.47871i	0.177593	+	0.307599i
$213$		0			0
$214$	8.32843	−	14.4253i	0.569320	−	0.986090i
$215$	−0.585786	−	1.01461i	−0.0399503	−	0.0691960i
$216$		0			0
$217$	−6.48528	+	15.8856i	−0.440250	+	1.07839i
$218$	3.24264			0.219619
$219$		0			0
$220$	0.207107	−	0.358719i	0.0139631	−	0.0241849i
$221$	−1.27208	+	2.20330i	−0.0855692	+	0.148210i
$222$		0			0
$223$	−13.3137			−0.891552			−0.445776	−	0.895145i	$-0.647072\pi$
$223$	−13.3137			−0.891552			−0.445776	+	0.895145i	$0.647072\pi$
$224$	1.00000	−	2.44949i	0.0668153	−	0.163663i
$225$		0			0
$226$	−1.82843	−	3.16693i	−0.121625	−	0.210661i
$227$	−6.15685	+	10.6640i	−0.408645	+	0.707794i	−0.994738	−	0.102450i	$-0.967332\pi$
$227$	−6.15685	+	10.6640i	−0.408645	+	0.707794i	0.586093	+	0.810244i	$0.300665\pi$
$228$		0			0
$229$	5.65685	+	9.79796i	0.373815	+	0.647467i	0.990149	−	0.140018i	$-0.0447160\pi$
$229$	5.65685	+	9.79796i	0.373815	+	0.647467i	−0.616334	+	0.787485i	$0.711383\pi$
$230$	1.34315			0.0885644
$231$		0			0
$232$	2.82843			0.185695
$233$	−4.15685	−	7.19988i	−0.272325	−	0.471680i	0.697132	−	0.716943i	$-0.254459\pi$
$233$	−4.15685	−	7.19988i	−0.272325	−	0.471680i	−0.969457	+	0.245263i	$0.921126\pi$
$234$		0			0
$235$	−1.91421	+	3.31552i	−0.124870	+	0.216280i
$236$	1.82843	+	3.16693i	0.119020	+	0.206149i
$237$		0			0
$238$	5.69239	−	0.778985i	0.368983	−	0.0504941i
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$	−12.4853	+	21.6251i	−0.804248	+	1.39300i	0.112550	+	0.993646i	$0.464098\pi$
$241$	−12.4853	+	21.6251i	−0.804248	+	1.39300i	−0.916798	+	0.399352i	$0.869235\pi$
$242$	−0.500000	+	0.866025i	−0.0321412	+	0.0556702i
$243$		0			0
$244$	−1.58579			−0.101520
$245$	2.07107	+	2.02922i	0.132316	+	0.129642i
$246$		0			0
$247$	0.485281	+	0.840532i	0.0308777	+	0.0534818i
$248$	3.24264	−	5.61642i	0.205908	−	0.356643i
$249$		0			0
$250$	−2.03553	−	3.52565i	−0.128738	−	0.222982i
$251$	26.1421			1.65008			0.825038	−	0.565077i	$-0.191153\pi$
$251$	26.1421			1.65008			0.825038	+	0.565077i	$0.191153\pi$
$252$		0			0
$253$	−3.24264			−0.203863
$254$	−0.621320	−	1.07616i	−0.0389851	−	0.0675242i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−12.5858	−	21.7992i	−0.785080	−	1.35980i	−0.928951	−	0.370202i	$-0.879289\pi$
$257$	−12.5858	−	21.7992i	−0.785080	−	1.35980i	0.143872	−	0.989596i	$-0.454045\pi$
$258$		0			0
$259$	15.6569	+	20.1903i	0.972870	+	1.25456i
$260$	0.485281			0.0300959
$261$		0			0
$262$	−7.65685	+	13.2621i	−0.473042	+	0.819333i
$263$	−4.65685	+	8.06591i	−0.287154	+	0.497365i	−0.973129	−	0.230260i	$-0.926042\pi$
$263$	−4.65685	+	8.06591i	−0.287154	+	0.497365i	0.685975	+	0.727625i	$0.259376\pi$
$264$		0			0
$265$	2.14214			0.131590
$266$	0.828427	−	2.02922i	0.0507941	−	0.124420i
$267$		0			0
$268$	6.74264	+	11.6786i	0.411872	+	0.713384i
$269$	4.03553	−	6.98975i	0.246051	−	0.426173i	−0.716376	−	0.697715i	$-0.754200\pi$
$269$	4.03553	−	6.98975i	0.246051	−	0.426173i	0.962427	+	0.271542i	$0.0875336\pi$
$270$		0			0
$271$	6.65685	+	11.5300i	0.404375	+	0.700398i	0.994249	−	0.107098i	$-0.0341557\pi$
$271$	6.65685	+	11.5300i	0.404375	+	0.700398i	−0.589873	+	0.807496i	$0.700822\pi$
$272$	−2.17157			−0.131671
$273$		0			0
$274$	−4.82843			−0.291696
$275$	2.41421	+	4.18154i	0.145583	+	0.252156i
$276$		0			0
$277$	3.41421	−	5.91359i	0.205140	−	0.355313i	−0.745037	−	0.667023i	$-0.767568\pi$
$277$	3.41421	−	5.91359i	0.205140	−	0.355313i	0.950177	+	0.311710i	$0.100902\pi$
$278$	3.00000	+	5.19615i	0.179928	+	0.311645i
$279$		0			0
$280$	−0.671573	−	0.866025i	−0.0401342	−	0.0517549i
$281$	2.51472			0.150016			0.0750078	−	0.997183i	$-0.476102\pi$
$281$	2.51472			0.150016			0.0750078	+	0.997183i	$0.476102\pi$
$282$		0			0
$283$	0.928932	−	1.60896i	0.0552193	−	0.0956426i	−0.837094	−	0.547058i	$-0.815748\pi$
$283$	0.928932	−	1.60896i	0.0552193	−	0.0956426i	0.892314	+	0.451416i	$0.149081\pi$
$284$	−6.65685	+	11.5300i	−0.395012	+	0.684180i
$285$		0			0
$286$	−1.17157			−0.0692766
$287$	−12.2071	+	1.67050i	−0.720563	+	0.0986067i
$288$		0			0
$289$	6.14214	+	10.6385i	0.361302	+	0.625794i
$290$	0.585786	−	1.01461i	0.0343986	−	0.0595801i
$291$		0			0
$292$	2.41421	+	4.18154i	0.141281	+	0.244706i
$293$	12.8284			0.749445			0.374722	−	0.927137i	$-0.377738\pi$
$293$	12.8284			0.749445			0.374722	+	0.927137i	$0.377738\pi$
$294$		0			0
$295$	1.51472			0.0881903
$296$	−4.82843	−	8.36308i	−0.280647	−	0.486094i
$297$		0			0
$298$	−5.65685	+	9.79796i	−0.327693	+	0.567581i
$299$	−1.89949	−	3.29002i	−0.109851	−	0.190267i
$300$		0			0
$301$	7.41421	−	1.01461i	0.427348	−	0.0584813i
$302$	1.24264			0.0715059
$303$		0			0
$304$	−0.414214	+	0.717439i	−0.0237568	+	0.0411479i
$305$	−0.328427	+	0.568852i	−0.0188057	+	0.0325724i
$306$		0			0
$307$	−34.6274			−1.97629			−0.988146	−	0.153520i	$-0.950939\pi$
$307$	−34.6274			−1.97629			−0.988146	+	0.153520i	$0.950939\pi$
$308$	1.62132	+	2.09077i	0.0923833	+	0.119133i
$309$		0			0
$310$	−1.34315	−	2.32640i	−0.0762856	−	0.132130i
$311$	−5.10660	+	8.84489i	−0.289569	+	0.501548i	−0.973707	−	0.227805i	$-0.926845\pi$
$311$	−5.10660	+	8.84489i	−0.289569	+	0.501548i	0.684138	+	0.729353i	$0.260179\pi$
$312$		0			0
$313$	14.6569	+	25.3864i	0.828454	+	1.43493i	0.899250	+	0.437434i	$0.144113\pi$
$313$	14.6569	+	25.3864i	0.828454	+	1.43493i	−0.0707960	+	0.997491i	$0.522554\pi$
$314$	−9.17157			−0.517582
$315$		0			0
$316$	4.75736			0.267622
$317$	16.9350	+	29.3323i	0.951166	+	1.64747i	0.742908	+	0.669394i	$0.233446\pi$
$317$	16.9350	+	29.3323i	0.951166	+	1.64747i	0.208258	+	0.978074i	$0.433221\pi$
$318$		0			0
$319$	−1.41421	+	2.44949i	−0.0791808	+	0.137145i
$320$	0.207107	+	0.358719i	0.0115776	+	0.0200530i
$321$		0			0
$322$	−3.24264	+	7.94282i	−0.180705	+	0.442636i
$323$	−1.79899			−0.100098
$324$		0			0
$325$	−2.82843	+	4.89898i	−0.156893	+	0.271746i
$326$	−1.50000	+	2.59808i	−0.0830773	+	0.143894i
$327$		0			0
$328$	4.65685			0.257132
$329$	−14.9853	−	19.3242i	−0.826165	−	1.06538i
$330$		0			0
$331$	−3.15685	−	5.46783i	−0.173516	−	0.300539i	0.766130	−	0.642685i	$-0.222180\pi$
$331$	−3.15685	−	5.46783i	−0.173516	−	0.300539i	−0.939647	+	0.342146i	$0.888846\pi$
$332$	4.91421	−	8.51167i	0.269703	−	0.467138i
$333$		0			0
$334$	8.24264	+	14.2767i	0.451017	+	0.781185i
$335$	5.58579			0.305184
$336$		0			0
$337$	9.51472			0.518300			0.259150	−	0.965837i	$-0.416558\pi$
$337$	9.51472			0.518300			0.259150	+	0.965837i	$0.416558\pi$
$338$	5.81371	+	10.0696i	0.316224	+	0.547716i
$339$		0			0
$340$	−0.449747	+	0.778985i	−0.0243910	+	0.0422464i
$341$	3.24264	+	5.61642i	0.175599	+	0.304146i
$342$		0			0
$343$	−17.0000	+	7.34847i	−0.917914	+	0.396780i
$344$	−2.82843			−0.152499
$345$		0			0
$346$	6.07107	−	10.5154i	0.326383	−	0.565311i
$347$	−2.42893	+	4.20703i	−0.130392	+	0.225845i	−0.923828	−	0.382809i	$-0.874957\pi$
$347$	−2.42893	+	4.20703i	−0.130392	+	0.225845i	0.793436	+	0.608654i	$0.208290\pi$
$348$		0			0
$349$	−15.7279			−0.841896			−0.420948	−	0.907085i	$-0.638303\pi$
$349$	−15.7279			−0.841896			−0.420948	+	0.907085i	$0.638303\pi$
$350$	12.6569	−	1.73205i	0.676537	−	0.0925820i
$351$		0			0
$352$	−0.500000	−	0.866025i	−0.0266501	−	0.0461593i
$353$	8.41421	−	14.5738i	0.447843	−	0.775688i	−0.550402	−	0.834900i	$-0.685526\pi$
$353$	8.41421	−	14.5738i	0.447843	−	0.775688i	0.998245	+	0.0592122i	$0.0188589\pi$
$354$		0			0
$355$	2.75736	+	4.77589i	0.146345	+	0.253478i
$356$	12.4853			0.661719
$357$		0			0
$358$	−3.17157			−0.167623
$359$	4.24264	+	7.34847i	0.223918	+	0.387837i	0.955994	−	0.293385i	$-0.0947819\pi$
$359$	4.24264	+	7.34847i	0.223918	+	0.387837i	−0.732076	+	0.681223i	$0.761449\pi$
$360$		0			0
$361$	9.15685	−	15.8601i	0.481940	−	0.834744i
$362$	1.17157	+	2.02922i	0.0615765	+	0.106654i
$363$		0			0
$364$	−1.17157	+	2.86976i	−0.0614071	+	0.150416i
$365$	2.00000			0.104685
$366$		0			0
$367$	−2.75736	+	4.77589i	−0.143933	+	0.249299i	−0.928974	−	0.370144i	$-0.879308\pi$
$367$	−2.75736	+	4.77589i	−0.143933	+	0.249299i	0.785041	+	0.619443i	$0.212642\pi$
$368$	1.62132	−	2.80821i	0.0845172	−	0.146388i
$369$		0			0
$370$	−4.00000			−0.207950
$371$	−5.17157	+	12.6677i	−0.268495	+	0.657675i
$372$		0			0
$373$	7.86396	+	13.6208i	0.407180	+	0.705257i	0.994573	−	0.104045i	$-0.0331787\pi$
$373$	7.86396	+	13.6208i	0.407180	+	0.705257i	−0.587392	+	0.809302i	$0.699845\pi$
$374$	1.08579	−	1.88064i	0.0561447	−	0.0972454i
$375$		0			0
$376$	4.62132	+	8.00436i	0.238326	+	0.412793i
$377$	−3.31371			−0.170665
$378$		0			0
$379$	19.3431			0.993591			0.496795	−	0.867868i	$-0.334510\pi$
$379$	19.3431			0.993591			0.496795	+	0.867868i	$0.334510\pi$
$380$	0.171573	+	0.297173i	0.00880150	+	0.0152447i
$381$		0			0
$382$	4.65685	−	8.06591i	0.238265	−	0.412688i
$383$	17.1421	+	29.6910i	0.875922	+	1.51714i	0.855777	+	0.517345i	$0.173079\pi$
$383$	17.1421	+	29.6910i	0.875922	+	1.51714i	0.0201451	+	0.999797i	$0.493587\pi$
$384$		0			0
$385$	1.08579	−	0.148586i	0.0553368	−	0.00757267i
$386$	−9.17157			−0.466821
$387$		0			0
$388$	5.08579	−	8.80884i	0.258192	−	0.447201i
$389$	0.863961	−	1.49642i	0.0438046	−	0.0758717i	−0.843292	−	0.537456i	$-0.819385\pi$
$389$	0.863961	−	1.49642i	0.0438046	−	0.0758717i	0.887096	+	0.461584i	$0.152719\pi$
$390$		0			0
$391$	7.04163			0.356111
$392$	6.74264	−	1.88064i	0.340555	−	0.0949865i
$393$		0			0
$394$	1.75736	+	3.04384i	0.0885345	+	0.153346i
$395$	0.985281	−	1.70656i	0.0495749	−	0.0858662i
$396$		0			0
$397$	−6.24264	−	10.8126i	−0.313309	−	0.542667i	0.665767	−	0.746159i	$-0.268104\pi$
$397$	−6.24264	−	10.8126i	−0.313309	−	0.542667i	−0.979077	+	0.203492i	$0.934771\pi$
$398$	−8.34315			−0.418204
$399$		0			0
$400$	−4.82843			−0.241421
$401$	7.89949	+	13.6823i	0.394482	+	0.683263i	0.993035	−	0.117820i	$-0.0375907\pi$
$401$	7.89949	+	13.6823i	0.394482	+	0.683263i	−0.598553	+	0.801083i	$0.704257\pi$
$402$		0			0
$403$	−3.79899	+	6.58004i	−0.189241	+	0.327775i
$404$	7.07107	+	12.2474i	0.351799	+	0.609333i
$405$		0			0
$406$	4.58579	+	5.91359i	0.227589	+	0.293487i
$407$	9.65685			0.478672
$408$		0			0
$409$	5.72792	−	9.92105i	0.283228	−	0.490564i	−0.688950	−	0.724809i	$-0.741928\pi$
$409$	5.72792	−	9.92105i	0.283228	−	0.490564i	0.972178	+	0.234244i	$0.0752615\pi$
$410$	0.964466	−	1.67050i	0.0476316	−	0.0825003i
$411$		0			0
$412$	−9.17157			−0.451851
$413$	−3.65685	+	8.95743i	−0.179942	+	0.440766i
$414$		0			0
$415$	−2.03553	−	3.52565i	−0.0999204	−	0.173067i
$416$	0.585786	−	1.01461i	0.0287205	−	0.0497454i
$417$		0			0
$418$	−0.414214	−	0.717439i	−0.0202598	−	0.0350911i
$419$	−25.7990			−1.26036			−0.630182	−	0.776448i	$-0.717020\pi$
$419$	−25.7990			−1.26036			−0.630182	+	0.776448i	$0.717020\pi$
$420$		0			0
$421$	14.8284			0.722693			0.361347	−	0.932432i	$-0.382317\pi$
$421$	14.8284			0.722693			0.361347	+	0.932432i	$0.382317\pi$
$422$	−7.41421	−	12.8418i	−0.360918	−	0.625129i
$423$		0			0
$424$	2.58579	−	4.47871i	0.125577	−	0.217506i
$425$	−5.24264	−	9.08052i	−0.254305	−	0.440470i
$426$		0			0
$427$	−2.57107	−	3.31552i	−0.124423	−	0.160449i
$428$	−16.6569			−0.805139
$429$		0			0
$430$	−0.585786	+	1.01461i	−0.0282491	+	0.0489289i
$431$	9.58579	−	16.6031i	0.461731	−	0.799742i	−0.537316	−	0.843381i	$-0.680562\pi$
$431$	9.58579	−	16.6031i	0.461731	−	0.799742i	0.999047	+	0.0436391i	$0.0138952\pi$
$432$		0			0
$433$	−26.6569			−1.28105			−0.640523	−	0.767939i	$-0.721283\pi$
$433$	−26.6569			−1.28105			−0.640523	+	0.767939i	$0.721283\pi$
$434$	17.0000	−	2.32640i	0.816026	−	0.111671i
$435$		0			0
$436$	−1.62132	−	2.80821i	−0.0776472	−	0.134489i
$437$	1.34315	−	2.32640i	0.0642514	−	0.111287i
$438$		0			0
$439$	−1.69239	−	2.93130i	−0.0807733	−	0.139903i	0.822809	−	0.568318i	$-0.192406\pi$
$439$	−1.69239	−	2.93130i	−0.0807733	−	0.139903i	−0.903582	+	0.428415i	$0.859072\pi$
$440$	−0.414214			−0.0197469
$441$		0			0
$442$	2.54416			0.121013
$443$	−7.17157	−	12.4215i	−0.340732	−	0.590165i	0.643837	−	0.765163i	$-0.277341\pi$
$443$	−7.17157	−	12.4215i	−0.340732	−	0.590165i	−0.984569	+	0.174998i	$0.944008\pi$
$444$		0			0
$445$	2.58579	−	4.47871i	0.122578	−	0.212311i
$446$	6.65685	+	11.5300i	0.315211	+	0.545962i
$447$		0			0
$448$	−2.62132	+	0.358719i	−0.123846	+	0.0169479i
$449$	23.1127			1.09076			0.545378	−	0.838190i	$-0.316386\pi$
$449$	23.1127			1.09076			0.545378	+	0.838190i	$0.316386\pi$
$450$		0			0
$451$	−2.32843	+	4.03295i	−0.109641	+	0.189904i
$452$	−1.82843	+	3.16693i	−0.0860020	+	0.148960i
$453$		0			0
$454$	12.3137			0.577911
$455$	0.786797	+	1.01461i	0.0368856	+	0.0475657i
$456$		0			0
$457$	−17.3137	−	29.9882i	−0.809901	−	1.40279i	−0.912932	−	0.408112i	$-0.866187\pi$
$457$	−17.3137	−	29.9882i	−0.809901	−	1.40279i	0.103031	−	0.994678i	$-0.467146\pi$
$458$	5.65685	−	9.79796i	0.264327	−	0.457829i
$459$		0			0
$460$	−0.671573	−	1.16320i	−0.0313122	−	0.0542344i
$461$	−23.6569			−1.10181			−0.550905	−	0.834568i	$-0.685717\pi$
$461$	−23.6569			−1.10181			−0.550905	+	0.834568i	$0.685717\pi$
$462$		0			0
$463$	23.1127			1.07414			0.537069	−	0.843538i	$-0.319531\pi$
$463$	23.1127			1.07414			0.537069	+	0.843538i	$0.319531\pi$
$464$	−1.41421	−	2.44949i	−0.0656532	−	0.113715i
$465$		0			0
$466$	−4.15685	+	7.19988i	−0.192563	+	0.333528i
$467$	1.48528	+	2.57258i	0.0687306	+	0.119045i	0.898343	−	0.439295i	$-0.144772\pi$
$467$	1.48528	+	2.57258i	0.0687306	+	0.119045i	−0.829612	+	0.558340i	$0.811438\pi$
$468$		0			0
$469$	−13.4853	+	33.0321i	−0.622692	+	1.52528i
$470$	3.82843			0.176592
$471$		0			0
$472$	1.82843	−	3.16693i	0.0841602	−	0.145770i
$473$	1.41421	−	2.44949i	0.0650256	−	0.112628i
$474$		0			0
$475$	−4.00000			−0.183533
$476$	−3.52082	−	4.54026i	−0.161376	−	0.208102i
$477$		0			0
$478$	−10.0000	−	17.3205i	−0.457389	−	0.792222i
$479$	−2.58579	+	4.47871i	−0.118148	+	0.204638i	−0.919034	−	0.394179i	$-0.871029\pi$
$479$	−2.58579	+	4.47871i	−0.118148	+	0.204638i	0.800886	+	0.598817i	$0.204362\pi$
$480$		0			0
$481$	5.65685	+	9.79796i	0.257930	+	0.446748i
$482$	24.9706			1.13738
$483$		0			0
$484$	1.00000			0.0454545
$485$	−2.10660	−	3.64874i	−0.0956559	−	0.165681i
$486$		0			0
$487$	10.6569	−	18.4582i	0.482908	−	0.836421i	−0.516899	−	0.856046i	$-0.672914\pi$
$487$	10.6569	−	18.4582i	0.482908	−	0.836421i	0.999807	+	0.0196248i	$0.00624716\pi$
$488$	0.792893	+	1.37333i	0.0358926	+	0.0621678i
$489$		0			0
$490$	0.721825	−	2.80821i	0.0326087	−	0.126862i
$491$	−15.6274			−0.705255			−0.352628	−	0.935764i	$-0.614712\pi$
$491$	−15.6274			−0.705255			−0.352628	+	0.935764i	$0.614712\pi$
$492$		0			0
$493$	3.07107	−	5.31925i	0.138314	−	0.239567i
$494$	0.485281	−	0.840532i	0.0218338	−	0.0378173i
$495$		0			0
$496$	−6.48528			−0.291198
$497$	−34.8995	+	4.77589i	−1.56546	+	0.214228i
$498$		0			0
$499$	−7.17157	−	12.4215i	−0.321044	−	0.556064i	0.659660	−	0.751564i	$-0.270700\pi$
$499$	−7.17157	−	12.4215i	−0.321044	−	0.556064i	−0.980704	+	0.195500i	$0.937367\pi$
$500$	−2.03553	+	3.52565i	−0.0910318	+	0.157672i
$501$		0			0
$502$	−13.0711	−	22.6398i	−0.583390	−	1.01046i
$503$	8.97056			0.399978			0.199989	−	0.979798i	$-0.435909\pi$
$503$	8.97056			0.399978			0.199989	+	0.979798i	$0.435909\pi$
$504$		0			0
$505$	5.85786			0.260672
$506$	1.62132	+	2.80821i	0.0720765	+	0.124840i
$507$		0			0
$508$	−0.621320	+	1.07616i	−0.0275666	+	0.0477468i
$509$	−12.5858	−	21.7992i	−0.557855	−	0.966234i	−0.997675	−	0.0681476i	$-0.978291\pi$
$509$	−12.5858	−	21.7992i	−0.557855	−	0.966234i	0.439820	−	0.898086i	$-0.355042\pi$
$510$		0			0
$511$	−4.82843	+	11.8272i	−0.213597	+	0.523204i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−12.5858	+	21.7992i	−0.555135	+	0.961522i
$515$	−1.89949	+	3.29002i	−0.0837017	+	0.144976i
$516$		0			0
$517$	−9.24264			−0.406491
$518$	9.65685	−	23.6544i	0.424298	−	1.03931i
$519$		0			0
$520$	−0.242641	−	0.420266i	−0.0106405	−	0.0184299i
$521$	−14.6569	+	25.3864i	−0.642128	+	1.11220i	0.342828	+	0.939398i	$0.388615\pi$
$521$	−14.6569	+	25.3864i	−0.642128	+	1.11220i	−0.984957	+	0.172801i	$0.944718\pi$
$522$		0			0
$523$	15.7279	+	27.2416i	0.687734	+	1.19119i	0.972569	+	0.232614i	$0.0747278\pi$
$523$	15.7279	+	27.2416i	0.687734	+	1.19119i	−0.284835	+	0.958577i	$0.591939\pi$
$524$	15.3137			0.668982
$525$		0			0
$526$	9.31371			0.406097
$527$	−7.04163	−	12.1965i	−0.306738	−	0.531286i
$528$		0			0
$529$	6.24264	−	10.8126i	0.271419	−	0.470112i
$530$	−1.07107	−	1.85514i	−0.0465242	−	0.0805823i
$531$		0			0
$532$	−2.17157	+	0.297173i	−0.0941496	+	0.0128841i
$533$	−5.45584			−0.236319
$534$		0			0
$535$	−3.44975	+	5.97514i	−0.149146	+	0.258328i
$536$	6.74264	−	11.6786i	0.291238	−	0.504439i
$537$		0			0
$538$	−8.07107			−0.347968
$539$	−1.74264	+	6.77962i	−0.0750608	+	0.292019i
$540$		0			0
$541$	2.72183	+	4.71434i	0.117020	+	0.202685i	0.918586	−	0.395222i	$-0.129332\pi$
$541$	2.72183	+	4.71434i	0.117020	+	0.202685i	−0.801565	+	0.597907i	$0.795999\pi$
$542$	6.65685	−	11.5300i	0.285936	−	0.495256i
$543$		0			0
$544$	1.08579	+	1.88064i	0.0465527	+	0.0806317i
$545$	−1.34315			−0.0575340
$546$		0			0
$547$	−36.1421			−1.54533			−0.772663	−	0.634816i	$-0.781076\pi$
$547$	−36.1421			−1.54533			−0.772663	+	0.634816i	$0.781076\pi$
$548$	2.41421	+	4.18154i	0.103130	+	0.178627i
$549$		0			0
$550$	2.41421	−	4.18154i	0.102942	−	0.178301i
$551$	−1.17157	−	2.02922i	−0.0499107	−	0.0864478i
$552$		0			0
$553$	7.71320	+	9.94655i	0.327999	+	0.422970i
$554$	−6.82843			−0.290112
$555$		0			0
$556$	3.00000	−	5.19615i	0.127228	−	0.220366i
$557$	2.24264	−	3.88437i	0.0950237	−	0.164586i	−0.814595	−	0.580030i	$-0.803041\pi$
$557$	2.24264	−	3.88437i	0.0950237	−	0.164586i	0.909619	+	0.415445i	$0.136374\pi$
$558$		0			0
$559$	3.31371			0.140155
$560$	−0.414214	+	1.01461i	−0.0175037	+	0.0428752i
$561$		0			0
$562$	−1.25736	−	2.17781i	−0.0530385	−	0.0918654i
$563$	−12.8284	+	22.2195i	−0.540654	+	0.936440i	0.458213	+	0.888842i	$0.348490\pi$
$563$	−12.8284	+	22.2195i	−0.540654	+	0.936440i	−0.998867	+	0.0475973i	$0.984844\pi$
$564$		0			0
$565$	0.757359	+	1.31178i	0.0318623	+	0.0551872i
$566$	−1.85786			−0.0780919
$567$		0			0
$568$	13.3137			0.558631
$569$	11.0000	+	19.0526i	0.461144	+	0.798725i	0.999018	−	0.0443003i	$-0.0141058\pi$
$569$	11.0000	+	19.0526i	0.461144	+	0.798725i	−0.537874	+	0.843025i	$0.680772\pi$
$570$		0			0
$571$	10.8284	−	18.7554i	0.453156	−	0.784888i	−0.545424	−	0.838160i	$-0.683632\pi$
$571$	10.8284	−	18.7554i	0.453156	−	0.784888i	0.998580	+	0.0532715i	$0.0169649\pi$
$572$	0.585786	+	1.01461i	0.0244930	+	0.0424231i
$573$		0			0
$574$	7.55025	+	9.73641i	0.315141	+	0.406390i
$575$	15.6569			0.652936
$576$		0			0
$577$	−2.57107	+	4.45322i	−0.107035	+	0.185390i	−0.914568	−	0.404433i	$-0.867469\pi$
$577$	−2.57107	+	4.45322i	−0.107035	+	0.185390i	0.807533	+	0.589823i	$0.200802\pi$
$578$	6.14214	−	10.6385i	0.255479	−	0.442503i
$579$		0			0
$580$	−1.17157			−0.0486469
$581$	25.7635	−	3.52565i	1.06885	−	0.146269i
$582$		0			0
$583$	2.58579	+	4.47871i	0.107092	+	0.185489i
$584$	2.41421	−	4.18154i	0.0999009	−	0.173033i
$585$		0			0
$586$	−6.41421	−	11.1097i	−0.264969	−	0.458939i
$587$	−28.1421			−1.16155			−0.580775	−	0.814064i	$-0.697250\pi$
$587$	−28.1421			−1.16155			−0.580775	+	0.814064i	$0.697250\pi$
$588$		0			0
$589$	−5.37258			−0.221373
$590$	−0.757359	−	1.31178i	−0.0311800	−	0.0540053i
$591$		0			0
$592$	−4.82843	+	8.36308i	−0.198447	+	0.343721i
$593$	−19.9706	−	34.5900i	−0.820093	−	1.42044i	−0.905613	−	0.424106i	$-0.860588\pi$
$593$	−19.9706	−	34.5900i	−0.820093	−	1.42044i	0.0855199	−	0.996336i	$-0.472745\pi$
$594$		0			0
$595$	−2.35786	+	0.322666i	−0.0966630	+	0.0132280i
$596$	11.3137			0.463428
$597$		0			0
$598$	−1.89949	+	3.29002i	−0.0776761	+	0.134539i
$599$	−14.5208	+	25.1508i	−0.593304	+	1.02763i	0.400479	+	0.916306i	$0.368844\pi$
$599$	−14.5208	+	25.1508i	−0.593304	+	1.02763i	−0.993784	+	0.111328i	$0.964490\pi$
$600$		0			0
$601$	−20.0000			−0.815817			−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000			−0.815817			−0.407909	+	0.913023i	$0.633742\pi$
$602$	−4.58579	−	5.91359i	−0.186903	−	0.241020i
$603$		0			0
$604$	−0.621320	−	1.07616i	−0.0252812	−	0.0437883i
$605$	0.207107	−	0.358719i	0.00842009	−	0.0145840i
$606$		0			0
$607$	−23.1066	−	40.0218i	−0.937868	−	1.62444i	−0.769438	−	0.638721i	$-0.779464\pi$
$607$	−23.1066	−	40.0218i	−0.937868	−	1.62444i	−0.168430	−	0.985714i	$-0.553870\pi$
$608$	0.828427			0.0335972
$609$		0			0
$610$	0.656854			0.0265953
$611$	−5.41421	−	9.37769i	−0.219036	−	0.379381i
$612$		0			0
$613$	15.4497	−	26.7597i	0.624009	−	1.08082i	−0.364722	−	0.931116i	$-0.618836\pi$
$613$	15.4497	−	26.7597i	0.624009	−	1.08082i	0.988732	−	0.149700i	$-0.0478307\pi$
$614$	17.3137	+	29.9882i	0.698724	+	1.21023i
$615$		0			0
$616$	1.00000	−	2.44949i	0.0402911	−	0.0986928i
$617$	−24.4853			−0.985740			−0.492870	−	0.870103i	$-0.664052\pi$
$617$	−24.4853			−0.985740			−0.492870	+	0.870103i	$0.664052\pi$
$618$		0			0
$619$	−0.257359	+	0.445759i	−0.0103441	+	0.0179166i	−0.871151	−	0.491015i	$-0.836626\pi$
$619$	−0.257359	+	0.445759i	−0.0103441	+	0.0179166i	0.860807	+	0.508932i	$0.169959\pi$
$620$	−1.34315	+	2.32640i	−0.0539420	+	0.0934303i
$621$		0			0
$622$	10.2132			0.409512
$623$	20.2426	+	26.1039i	0.811004	+	1.04583i
$624$		0			0
$625$	−11.2279	−	19.4473i	−0.449117	−	0.777893i
$626$	14.6569	−	25.3864i	0.585806	−	1.01465i
$627$		0			0
$628$	4.58579	+	7.94282i	0.182993	+	0.316953i
$629$	−20.9706			−0.836151
$630$		0			0
$631$	−1.02944			−0.0409812			−0.0204906	−	0.999790i	$-0.506523\pi$
$631$	−1.02944			−0.0409812			−0.0204906	+	0.999790i	$0.506523\pi$
$632$	−2.37868	−	4.11999i	−0.0946188	−	0.163885i
$633$		0			0
$634$	16.9350	−	29.3323i	0.672576	−	1.16494i
$635$	0.257359	+	0.445759i	0.0102130	+	0.0176894i
$636$		0			0
$637$	−7.89949	+	2.20330i	−0.312989	+	0.0872981i
$638$	2.82843			0.111979
$639$		0			0
$640$	0.207107	−	0.358719i	0.00818661	−	0.0141796i
$641$	13.6569	−	23.6544i	0.539413	−	0.934291i	−0.459522	−	0.888166i	$-0.651979\pi$
$641$	13.6569	−	23.6544i	0.539413	−	0.934291i	0.998936	−	0.0461251i	$-0.0146873\pi$
$642$		0			0
$643$	17.6569			0.696318			0.348159	−	0.937435i	$-0.386807\pi$
$643$	17.6569			0.696318			0.348159	+	0.937435i	$0.386807\pi$
$644$	8.50000	−	1.16320i	0.334947	−	0.0458364i
$645$		0			0
$646$	0.899495	+	1.55797i	0.0353902	+	0.0612975i
$647$	−21.3492	+	36.9780i	−0.839325	+	1.45375i	0.0511343	+	0.998692i	$0.483716\pi$
$647$	−21.3492	+	36.9780i	−0.839325	+	1.45375i	−0.890460	+	0.455062i	$0.849617\pi$
$648$		0			0
$649$	1.82843	+	3.16693i	0.0717720	+	0.124313i
$650$	5.65685			0.221880
$651$		0			0
$652$	3.00000			0.117489
$653$	−22.1777	−	38.4129i	−0.867879	−	1.50321i	−0.864160	−	0.503218i	$-0.832149\pi$
$653$	−22.1777	−	38.4129i	−0.867879	−	1.50321i	−0.00371972	−	0.999993i	$-0.501184\pi$
$654$		0			0
$655$	3.17157	−	5.49333i	0.123924	−	0.214642i
$656$	−2.32843	−	4.03295i	−0.0909098	−	0.157460i
$657$		0			0
$658$	−9.24264	+	22.6398i	−0.360316	+	0.882589i
$659$	−36.9411			−1.43902			−0.719511	−	0.694481i	$-0.755634\pi$
$659$	−36.9411			−1.43902			−0.719511	+	0.694481i	$0.755634\pi$
$660$		0			0
$661$	19.4853	−	33.7495i	0.757890	−	1.31270i	−0.186035	−	0.982543i	$-0.559564\pi$
$661$	19.4853	−	33.7495i	0.757890	−	1.31270i	0.943925	−	0.330160i	$-0.107103\pi$
$662$	−3.15685	+	5.46783i	−0.122695	+	0.212513i
$663$		0			0
$664$	−9.82843			−0.381417
$665$	−0.343146	+	0.840532i	−0.0133066	+	0.0325944i
$666$		0			0
$667$	4.58579	+	7.94282i	0.177562	+	0.307547i
$668$	8.24264	−	14.2767i	0.318917	−	0.552381i
$669$		0			0
$670$	−2.79289	−	4.83743i	−0.107899	−	0.186886i
$671$	−1.58579			−0.0612186
$672$		0			0
$673$	24.6274			0.949317			0.474659	−	0.880170i	$-0.342572\pi$
$673$	24.6274			0.949317			0.474659	+	0.880170i	$0.342572\pi$
$674$	−4.75736	−	8.23999i	−0.183247	−	0.317392i
$675$		0			0
$676$	5.81371	−	10.0696i	0.223604	−	0.387294i
$677$	−21.3848	−	37.0395i	−0.821884	−	1.42354i	−0.904278	−	0.426945i	$-0.859590\pi$
$677$	−21.3848	−	37.0395i	−0.821884	−	1.42354i	0.0823941	−	0.996600i	$-0.473743\pi$
$678$		0			0
$679$	26.6630	−	3.64874i	1.02323	−	0.140026i
$680$	0.899495			0.0344941
$681$		0			0
$682$	3.24264	−	5.61642i	0.124167	−	0.215064i
$683$	−8.82843	+	15.2913i	−0.337810	+	0.585105i	−0.984021	−	0.178055i	$-0.943020\pi$
$683$	−8.82843	+	15.2913i	−0.337810	+	0.585105i	0.646210	+	0.763159i	$0.276353\pi$
$684$		0			0
$685$	2.00000			0.0764161
$686$	14.8640	+	11.0482i	0.567509	+	0.421822i
$687$		0			0
$688$	1.41421	+	2.44949i	0.0539164	+	0.0933859i
$689$	−3.02944	+	5.24714i	−0.115412	+	0.199900i
$690$		0			0
$691$	−14.5711	−	25.2378i	−0.554310	−	0.960092i	−0.997957	−	0.0638908i	$-0.979649\pi$
$691$	−14.5711	−	25.2378i	−0.554310	−	0.960092i	0.443647	−	0.896201i	$-0.353684\pi$
$692$	−12.1421			−0.461575
$693$		0			0
$694$	4.85786			0.184402
$695$	−1.24264	−	2.15232i	−0.0471360	−	0.0816420i
$696$		0			0
$697$	5.05635	−	8.75785i	0.191523	−	0.331727i
$698$	7.86396	+	13.6208i	0.297655	+	0.515554i
$699$		0			0
$700$	−7.82843	−	10.0951i	−0.295887	−	0.381560i
$701$	28.1421			1.06291			0.531457	−	0.847085i	$-0.321645\pi$
$701$	28.1421			1.06291			0.531457	+	0.847085i	$0.321645\pi$
$702$		0			0
$703$	−4.00000	+	6.92820i	−0.150863	+	0.261302i
$704$	−0.500000	+	0.866025i	−0.0188445	+	0.0326396i
$705$		0			0
$706$	−16.8284			−0.633346
$707$	−14.1421	+	34.6410i	−0.531870	+	1.30281i
$708$		0			0
$709$	−1.92893	−	3.34101i	−0.0724426	−	0.125474i	0.827529	−	0.561423i	$-0.189746\pi$
$709$	−1.92893	−	3.34101i	−0.0724426	−	0.125474i	−0.899971	+	0.435949i	$0.856413\pi$
$710$	2.75736	−	4.77589i	0.103482	−	0.179236i
$711$		0			0
$712$	−6.24264	−	10.8126i	−0.233953	−	0.405218i
$713$	21.0294			0.787559
$714$		0			0
$715$	0.485281			0.0181485
$716$	1.58579	+	2.74666i	0.0592636	+	0.102648i
$717$		0			0
$718$	4.24264	−	7.34847i	0.158334	−	0.274242i
$719$	8.10660	+	14.0410i	0.302325	+	0.523643i	0.976662	−	0.214781i	$-0.0689038\pi$
$719$	8.10660	+	14.0410i	0.302325	+	0.523643i	−0.674337	+	0.738424i	$0.735570\pi$
$720$		0			0
$721$	−14.8701	−	19.1757i	−0.553790	−	0.714139i
$722$	−18.3137			−0.681566
$723$		0			0
$724$	1.17157	−	2.02922i	0.0435412	−	0.0754155i
$725$	6.82843	−	11.8272i	0.253601	−	0.439251i
$726$		0			0
$727$	−32.4853			−1.20481			−0.602406	−	0.798190i	$-0.705791\pi$
$727$	−32.4853			−1.20481			−0.602406	+	0.798190i	$0.705791\pi$
$728$	3.07107	−	0.420266i	0.113821	−	0.0155761i
$729$		0			0
$730$	−1.00000	−	1.73205i	−0.0370117	−	0.0641061i
$731$	−3.07107	+	5.31925i	−0.113588	+	0.196739i
$732$		0			0
$733$	7.37868	+	12.7802i	0.272538	+	0.472049i	0.969511	−	0.245048i	$-0.0788037\pi$
$733$	7.37868	+	12.7802i	0.272538	+	0.472049i	−0.696973	+	0.717097i	$0.745470\pi$
$734$	5.51472			0.203552
$735$		0			0
$736$	−3.24264			−0.119525
$737$	6.74264	+	11.6786i	0.248368	+	0.430187i
$738$		0			0
$739$	17.0000	−	29.4449i	0.625355	−	1.08315i	−0.363117	−	0.931744i	$-0.618287\pi$
$739$	17.0000	−	29.4449i	0.625355	−	1.08315i	0.988472	−	0.151403i	$-0.0483792\pi$
$740$	2.00000	+	3.46410i	0.0735215	+	0.127343i
$741$		0			0
$742$	13.5563	−	1.85514i	0.497669	−	0.0681045i
$743$	−13.6569			−0.501021			−0.250511	−	0.968114i	$-0.580599\pi$
$743$	−13.6569			−0.501021			−0.250511	+	0.968114i	$0.580599\pi$
$744$		0			0
$745$	2.34315	−	4.05845i	0.0858462	−	0.148690i
$746$	7.86396	−	13.6208i	0.287920	−	0.498692i
$747$		0			0
$748$	−2.17157			−0.0794006
$749$	−27.0061	−	34.8257i	−0.986781	−	1.27250i
$750$		0			0
$751$	−14.8995	−	25.8067i	−0.543690	−	0.941699i	−0.998688	−	0.0512066i	$-0.983693\pi$
$751$	−14.8995	−	25.8067i	−0.543690	−	0.941699i	0.454998	−	0.890493i	$-0.349640\pi$
$752$	4.62132	−	8.00436i	0.168522	−	0.291889i
$753$		0			0
$754$	1.65685	+	2.86976i	0.0603391	+	0.104510i
$755$	−0.514719			−0.0187325
$756$		0			0
$757$	19.6569			0.714441			0.357220	−	0.934020i	$-0.383725\pi$
$757$	19.6569			0.714441			0.357220	+	0.934020i	$0.383725\pi$
$758$	−9.67157	−	16.7517i	−0.351287	−	0.608448i
$759$		0			0
$760$	0.171573	−	0.297173i	0.00622360	−	0.0107796i
$761$	−12.2279	−	21.1794i	−0.443262	−	0.767752i	0.554667	−	0.832072i	$-0.312845\pi$
$761$	−12.2279	−	21.1794i	−0.443262	−	0.767752i	−0.997929	+	0.0643201i	$0.979512\pi$
$762$		0			0
$763$	3.24264	−	7.94282i	0.117391	−	0.287549i
$764$	−9.31371			−0.336958
$765$		0			0
$766$	17.1421	−	29.6910i	0.619371	−	1.07278i
$767$	−2.14214	+	3.71029i	−0.0773480	+	0.133971i
$768$		0			0
$769$	26.4853			0.955084			0.477542	−	0.878609i	$-0.341528\pi$
$769$	26.4853			0.955084			0.477542	+	0.878609i	$0.341528\pi$
$770$	−0.671573	−	0.866025i	−0.0242018	−	0.0312094i
$771$		0			0
$772$	4.58579	+	7.94282i	0.165046	+	0.285868i
$773$	−15.6213	+	27.0569i	−0.561860	+	0.973170i	0.435474	+	0.900201i	$0.356581\pi$
$773$	−15.6213	+	27.0569i	−0.561860	+	0.973170i	−0.997334	+	0.0729687i	$0.976753\pi$
$774$		0			0
$775$	−15.6569	−	27.1185i	−0.562411	−	0.974124i
$776$	−10.1716			−0.365138
$777$		0			0
$778$	−1.72792			−0.0619490
$779$	−1.92893	−	3.34101i	−0.0691112	−	0.119704i
$780$		0			0
$781$	−6.65685	+	11.5300i	−0.238201	+	0.412576i
$782$	−3.52082	−	6.09823i	−0.125904	−	0.218072i

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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.207107 + 0.358719i) q^{5} +(-2.62132 + 0.358719i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.207107 + 0.358719i) q^{5} +(-2.62132 + 0.358719i) q^{7} +1.00000 q^{8} +(0.207107 - 0.358719i) q^{10} +(-0.500000 + 0.866025i) q^{11} -1.17157 q^{13} +(1.62132 + 2.09077i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.08579 - 1.88064i) q^{17} +(-0.414214 - 0.717439i) q^{19} -0.414214 q^{20} +1.00000 q^{22} +(1.62132 + 2.80821i) q^{23} +(2.41421 - 4.18154i) q^{25} +(0.585786 + 1.01461i) q^{26} +(1.00000 - 2.44949i) q^{28} +2.82843 q^{29} +(3.24264 - 5.61642i) q^{31} +(-0.500000 + 0.866025i) q^{32} -2.17157 q^{34} +(-0.671573 - 0.866025i) q^{35} +(-4.82843 - 8.36308i) q^{37} +(-0.414214 + 0.717439i) q^{38} +(0.207107 + 0.358719i) q^{40} +4.65685 q^{41} -2.82843 q^{43} +(-0.500000 - 0.866025i) q^{44} +(1.62132 - 2.80821i) q^{46} +(4.62132 + 8.00436i) q^{47} +(6.74264 - 1.88064i) q^{49} -4.82843 q^{50} +(0.585786 - 1.01461i) q^{52} +(2.58579 - 4.47871i) q^{53} -0.414214 q^{55} +(-2.62132 + 0.358719i) q^{56} +(-1.41421 - 2.44949i) q^{58} +(1.82843 - 3.16693i) q^{59} +(0.792893 + 1.37333i) q^{61} -6.48528 q^{62} +1.00000 q^{64} +(-0.242641 - 0.420266i) q^{65} +(6.74264 - 11.6786i) q^{67} +(1.08579 + 1.88064i) q^{68} +(-0.414214 + 1.01461i) q^{70} +13.3137 q^{71} +(2.41421 - 4.18154i) q^{73} +(-4.82843 + 8.36308i) q^{74} +0.828427 q^{76} +(1.00000 - 2.44949i) q^{77} +(-2.37868 - 4.11999i) q^{79} +(0.207107 - 0.358719i) q^{80} +(-2.32843 - 4.03295i) q^{82} -9.82843 q^{83} +0.899495 q^{85} +(1.41421 + 2.44949i) q^{86} +(-0.500000 + 0.866025i) q^{88} +(-6.24264 - 10.8126i) q^{89} +(3.07107 - 0.420266i) q^{91} -3.24264 q^{92} +(4.62132 - 8.00436i) q^{94} +(0.171573 - 0.297173i) q^{95} -10.1716 q^{97} +(-5.00000 - 4.89898i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^8	\( 4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{8} - 2 q^{10} - 2 q^{11} - 16 q^{13} - 2 q^{14} - 2 q^{16} + 10 q^{17} + 4 q^{19} + 4 q^{20} + 4 q^{22} - 2 q^{23} + 4 q^{25} + 8 q^{26} + 4 q^{28} - 4 q^{31} - 2 q^{32} - 20 q^{34} - 14 q^{35} - 8 q^{37} + 4 q^{38} - 2 q^{40} - 4 q^{41} - 2 q^{44} - 2 q^{46} + 10 q^{47} + 10 q^{49} - 8 q^{50} + 8 q^{52} + 16 q^{53} + 4 q^{55} - 2 q^{56} - 4 q^{59} + 6 q^{61} + 8 q^{62} + 4 q^{64} + 16 q^{65} + 10 q^{67} + 10 q^{68} + 4 q^{70} + 8 q^{71} + 4 q^{73} - 8 q^{74} - 8 q^{76} + 4 q^{77} - 18 q^{79} - 2 q^{80} + 2 q^{82} - 28 q^{83} - 36 q^{85} - 2 q^{88} - 8 q^{89} - 16 q^{91} + 4 q^{92} + 10 q^{94} + 12 q^{95} - 52 q^{97} - 20 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^8 - 2 * q^10 - 2 * q^11 - 16 * q^13 - 2 * q^14 - 2 * q^16 + 10 * q^17 + 4 * q^19 + 4 * q^20 + 4 * q^22 - 2 * q^23 + 4 * q^25 + 8 * q^26 + 4 * q^28 - 4 * q^31 - 2 * q^32 - 20 * q^34 - 14 * q^35 - 8 * q^37 + 4 * q^38 - 2 * q^40 - 4 * q^41 - 2 * q^44 - 2 * q^46 + 10 * q^47 + 10 * q^49 - 8 * q^50 + 8 * q^52 + 16 * q^53 + 4 * q^55 - 2 * q^56 - 4 * q^59 + 6 * q^61 + 8 * q^62 + 4 * q^64 + 16 * q^65 + 10 * q^67 + 10 * q^68 + 4 * q^70 + 8 * q^71 + 4 * q^73 - 8 * q^74 - 8 * q^76 + 4 * q^77 - 18 * q^79 - 2 * q^80 + 2 * q^82 - 28 * q^83 - 36 * q^85 - 2 * q^88 - 8 * q^89 - 16 * q^91 + 4 * q^92 + 10 * q^94 + 12 * q^95 - 52 * q^97 - 20 * q^98

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