LMFDB - Embedded newform 1078.2.e.o.67.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(67,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("1078.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.e (of order $3$, degree $2$, not 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:	$8.60787333789$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			67.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1078.67
Dual form			1078.2.e.o.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-1.41421 + 2.44949i) q^{3} +(-0.500000 + 0.866025i) q^{4} +2.82843 q^{6} +1.00000 q^{8} +(-2.50000 - 4.33013i) q^{9} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-1.41421 + 2.44949i) q^{3} +(-0.500000 + 0.866025i) q^{4} +2.82843 q^{6} +1.00000 q^{8} +(-2.50000 - 4.33013i) q^{9} +(0.500000 - 0.866025i) q^{11} +(-1.41421 - 2.44949i) q^{12} +4.24264 q^{13} +(-0.500000 - 0.866025i) q^{16} +(1.41421 - 2.44949i) q^{17} +(-2.50000 + 4.33013i) q^{18} +(2.12132 + 3.67423i) q^{19} -1.00000 q^{22} +(-3.00000 - 5.19615i) q^{23} +(-1.41421 + 2.44949i) q^{24} +(2.50000 - 4.33013i) q^{25} +(-2.12132 - 3.67423i) q^{26} +5.65685 q^{27} -4.00000 q^{29} +(3.53553 - 6.12372i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(1.41421 + 2.44949i) q^{33} -2.82843 q^{34} +5.00000 q^{36} +(-1.00000 - 1.73205i) q^{37} +(2.12132 - 3.67423i) q^{38} +(-6.00000 + 10.3923i) q^{39} +2.82843 q^{41} +10.0000 q^{43} +(0.500000 + 0.866025i) q^{44} +(-3.00000 + 5.19615i) q^{46} +(6.36396 + 11.0227i) q^{47} +2.82843 q^{48} -5.00000 q^{50} +(4.00000 + 6.92820i) q^{51} +(-2.12132 + 3.67423i) q^{52} +(-1.00000 + 1.73205i) q^{53} +(-2.82843 - 4.89898i) q^{54} -12.0000 q^{57} +(2.00000 + 3.46410i) q^{58} +(-5.65685 + 9.79796i) q^{59} +(4.94975 + 8.57321i) q^{61} -7.07107 q^{62} +1.00000 q^{64} +(1.41421 - 2.44949i) q^{66} +(-4.00000 + 6.92820i) q^{67} +(1.41421 + 2.44949i) q^{68} +16.9706 q^{69} +16.0000 q^{71} +(-2.50000 - 4.33013i) q^{72} +(4.24264 - 7.34847i) q^{73} +(-1.00000 + 1.73205i) q^{74} +(7.07107 + 12.2474i) q^{75} -4.24264 q^{76} +12.0000 q^{78} +(4.00000 + 6.92820i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-1.41421 - 2.44949i) q^{82} -12.7279 q^{83} +(-5.00000 - 8.66025i) q^{86} +(5.65685 - 9.79796i) q^{87} +(0.500000 - 0.866025i) q^{88} +(-3.53553 - 6.12372i) q^{89} +6.00000 q^{92} +(10.0000 + 17.3205i) q^{93} +(6.36396 - 11.0227i) q^{94} +(-1.41421 - 2.44949i) q^{96} -7.07107 q^{97} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 10 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 - 10 * q^9	$ 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 10 q^{9} + 2 q^{11} - 2 q^{16} - 10 q^{18} - 4 q^{22} - 12 q^{23} + 10 q^{25} - 16 q^{29} - 2 q^{32} + 20 q^{36} - 4 q^{37} - 24 q^{39} + 40 q^{43} + 2 q^{44} - 12 q^{46} - 20 q^{50} + 16 q^{51} - 4 q^{53} - 48 q^{57} + 8 q^{58} + 4 q^{64} - 16 q^{67} + 64 q^{71} - 10 q^{72} - 4 q^{74} + 48 q^{78} + 16 q^{79} - 2 q^{81} - 20 q^{86} + 2 q^{88} + 24 q^{92} + 40 q^{93} - 20 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 - 10 * q^9 + 2 * q^11 - 2 * q^16 - 10 * q^18 - 4 * q^22 - 12 * q^23 + 10 * q^25 - 16 * q^29 - 2 * q^32 + 20 * q^36 - 4 * q^37 - 24 * q^39 + 40 * q^43 + 2 * q^44 - 12 * q^46 - 20 * q^50 + 16 * q^51 - 4 * q^53 - 48 * q^57 + 8 * q^58 + 4 * q^64 - 16 * q^67 + 64 * q^71 - 10 * q^72 - 4 * q^74 + 48 * q^78 + 16 * q^79 - 2 * q^81 - 20 * q^86 + 2 * q^88 + 24 * q^92 + 40 * q^93 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$.

$n$	$199$	$981$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$	−1.41421	+	2.44949i	−0.816497	+	1.41421i	0.0917517	+	0.995782i	$0.470753\pi$
$3$	−1.41421	+	2.44949i	−0.816497	+	1.41421i	−0.908248	+	0.418432i	$0.862580\pi$
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$	2.82843			1.15470
$7$		0			0
$7$		0			0
$8$	1.00000			0.353553
$9$	−2.50000	−	4.33013i	−0.833333	−	1.44338i
$10$		0			0
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i
$12$	−1.41421	−	2.44949i	−0.408248	−	0.707107i
$13$	4.24264			1.17670			0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264			1.17670			0.588348	+	0.808608i	$0.299778\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	1.41421	−	2.44949i	0.342997	−	0.594089i	−0.641991	−	0.766712i	$-0.721891\pi$
$17$	1.41421	−	2.44949i	0.342997	−	0.594089i	0.984988	+	0.172624i	$0.0552245\pi$
$18$	−2.50000	+	4.33013i	−0.589256	+	1.02062i
$19$	2.12132	+	3.67423i	0.486664	+	0.842927i	0.999882	−	0.0153309i	$-0.00488018\pi$
$19$	2.12132	+	3.67423i	0.486664	+	0.842927i	−0.513218	+	0.858258i	$0.671547\pi$
$20$		0			0
$21$		0			0
$22$	−1.00000			−0.213201
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$	−1.41421	+	2.44949i	−0.288675	+	0.500000i
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$	−2.12132	−	3.67423i	−0.416025	−	0.720577i
$27$	5.65685			1.08866
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$	3.53553	−	6.12372i	0.635001	−	1.09985i	−0.351514	−	0.936182i	$-0.614333\pi$
$31$	3.53553	−	6.12372i	0.635001	−	1.09985i	0.986515	−	0.163671i	$-0.0523335\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$	1.41421	+	2.44949i	0.246183	+	0.426401i
$34$	−2.82843			−0.485071
$35$		0			0
$36$	5.00000			0.833333
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	0.772043	−	0.635571i	$-0.219235\pi$
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	−0.936442	+	0.350823i	$0.885902\pi$
$38$	2.12132	−	3.67423i	0.344124	−	0.596040i
$39$	−6.00000	+	10.3923i	−0.960769	+	1.66410i
$40$		0			0
$41$	2.82843			0.441726			0.220863	−	0.975305i	$-0.429113\pi$
$41$	2.82843			0.441726			0.220863	+	0.975305i	$0.429113\pi$
$42$		0			0
$43$	10.0000			1.52499			0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000			1.52499			0.762493	+	0.646997i	$0.223975\pi$
$44$	0.500000	+	0.866025i	0.0753778	+	0.130558i
$45$		0			0
$46$	−3.00000	+	5.19615i	−0.442326	+	0.766131i
$47$	6.36396	+	11.0227i	0.928279	+	1.60783i	0.786201	+	0.617971i	$0.212045\pi$
$47$	6.36396	+	11.0227i	0.928279	+	1.60783i	0.142078	+	0.989855i	$0.454621\pi$
$48$	2.82843			0.408248
$49$		0			0
$50$	−5.00000			−0.707107
$51$	4.00000	+	6.92820i	0.560112	+	0.970143i
$52$	−2.12132	+	3.67423i	−0.294174	+	0.509525i
$53$	−1.00000	+	1.73205i	−0.137361	+	0.237915i	−0.926497	−	0.376303i	$-0.877195\pi$
$53$	−1.00000	+	1.73205i	−0.137361	+	0.237915i	0.789136	+	0.614218i	$0.210529\pi$
$54$	−2.82843	−	4.89898i	−0.384900	−	0.666667i
$55$		0			0
$56$		0			0
$57$	−12.0000			−1.58944
$58$	2.00000	+	3.46410i	0.262613	+	0.454859i
$59$	−5.65685	+	9.79796i	−0.736460	+	1.27559i	0.217620	+	0.976034i	$0.430171\pi$
$59$	−5.65685	+	9.79796i	−0.736460	+	1.27559i	−0.954080	+	0.299552i	$0.903163\pi$
$60$		0			0
$61$	4.94975	+	8.57321i	0.633750	+	1.09769i	0.986778	+	0.162075i	$0.0518186\pi$
$61$	4.94975	+	8.57321i	0.633750	+	1.09769i	−0.353028	+	0.935613i	$0.614848\pi$
$62$	−7.07107			−0.898027
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$	1.41421	−	2.44949i	0.174078	−	0.301511i
$67$	−4.00000	+	6.92820i	−0.488678	+	0.846415i	−0.999915	−	0.0130248i	$-0.995854\pi$
$67$	−4.00000	+	6.92820i	−0.488678	+	0.846415i	0.511237	+	0.859440i	$0.329187\pi$
$68$	1.41421	+	2.44949i	0.171499	+	0.297044i
$69$	16.9706			2.04302
$70$		0			0
$71$	16.0000			1.89885			0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000			1.89885			0.949425	+	0.313993i	$0.101667\pi$
$72$	−2.50000	−	4.33013i	−0.294628	−	0.510310i
$73$	4.24264	−	7.34847i	0.496564	−	0.860073i	−0.503429	−	0.864037i	$-0.667928\pi$
$73$	4.24264	−	7.34847i	0.496564	−	0.860073i	0.999992	+	0.00396356i	$0.00126164\pi$
$74$	−1.00000	+	1.73205i	−0.116248	+	0.201347i
$75$	7.07107	+	12.2474i	0.816497	+	1.41421i
$76$	−4.24264			−0.486664
$77$		0			0
$78$	12.0000			1.35873
$79$	4.00000	+	6.92820i	0.450035	+	0.779484i	0.998388	−	0.0567635i	$-0.0180781\pi$
$79$	4.00000	+	6.92820i	0.450035	+	0.779484i	−0.548352	+	0.836247i	$0.684745\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$	−1.41421	−	2.44949i	−0.156174	−	0.270501i
$83$	−12.7279			−1.39707			−0.698535	−	0.715575i	$-0.746165\pi$
$83$	−12.7279			−1.39707			−0.698535	+	0.715575i	$0.746165\pi$
$84$		0			0
$85$		0			0
$86$	−5.00000	−	8.66025i	−0.539164	−	0.933859i
$87$	5.65685	−	9.79796i	0.606478	−	1.05045i
$88$	0.500000	−	0.866025i	0.0533002	−	0.0923186i
$89$	−3.53553	−	6.12372i	−0.374766	−	0.649113i	0.615526	−	0.788116i	$-0.288944\pi$
$89$	−3.53553	−	6.12372i	−0.374766	−	0.649113i	−0.990292	+	0.139003i	$0.955610\pi$
$90$		0			0
$91$		0			0
$92$	6.00000			0.625543
$93$	10.0000	+	17.3205i	1.03695	+	1.79605i
$94$	6.36396	−	11.0227i	0.656392	−	1.13691i
$95$		0			0
$96$	−1.41421	−	2.44949i	−0.144338	−	0.250000i
$97$	−7.07107			−0.717958			−0.358979	−	0.933346i	$-0.616875\pi$
$97$	−7.07107			−0.717958			−0.358979	+	0.933346i	$0.616875\pi$
$98$		0			0
$99$	−5.00000			−0.502519
$100$	2.50000	+	4.33013i	0.250000	+	0.433013i
$101$	0.707107	−	1.22474i	0.0703598	−	0.121867i	−0.828699	−	0.559694i	$-0.810919\pi$
$101$	0.707107	−	1.22474i	0.0703598	−	0.121867i	0.899059	+	0.437828i	$0.144252\pi$
$102$	4.00000	−	6.92820i	0.396059	−	0.685994i
$103$	0.707107	+	1.22474i	0.0696733	+	0.120678i	0.898757	−	0.438446i	$-0.144471\pi$
$103$	0.707107	+	1.22474i	0.0696733	+	0.120678i	−0.829084	+	0.559124i	$0.811138\pi$
$104$	4.24264			0.416025
$105$		0			0
$106$	2.00000			0.194257
$107$	7.00000	+	12.1244i	0.676716	+	1.17211i	0.975964	+	0.217931i	$0.0699306\pi$
$107$	7.00000	+	12.1244i	0.676716	+	1.17211i	−0.299249	+	0.954175i	$0.596736\pi$
$108$	−2.82843	+	4.89898i	−0.272166	+	0.471405i
$109$	7.00000	−	12.1244i	0.670478	−	1.16130i	−0.307290	−	0.951616i	$-0.599422\pi$
$109$	7.00000	−	12.1244i	0.670478	−	1.16130i	0.977769	−	0.209687i	$-0.0672444\pi$
$110$		0			0
$111$	5.65685			0.536925
$112$		0			0
$113$	16.0000			1.50515			0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000			1.50515			0.752577	+	0.658505i	$0.228811\pi$
$114$	6.00000	+	10.3923i	0.561951	+	0.973329i
$115$		0			0
$116$	2.00000	−	3.46410i	0.185695	−	0.321634i
$117$	−10.6066	−	18.3712i	−0.980581	−	1.69842i
$118$	11.3137			1.04151
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$	4.94975	−	8.57321i	0.448129	−	0.776182i
$123$	−4.00000	+	6.92820i	−0.360668	+	0.624695i
$124$	3.53553	+	6.12372i	0.317500	+	0.549927i
$125$		0			0
$126$		0			0
$127$	−20.0000			−1.77471			−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000			−1.77471			−0.887357	+	0.461084i	$0.847461\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$	−14.1421	+	24.4949i	−1.24515	+	2.15666i
$130$		0			0
$131$	−3.53553	−	6.12372i	−0.308901	−	0.535032i	0.669221	−	0.743063i	$-0.266628\pi$
$131$	−3.53553	−	6.12372i	−0.308901	−	0.535032i	−0.978122	+	0.208031i	$0.933295\pi$
$132$	−2.82843			−0.246183
$133$		0			0
$134$	8.00000			0.691095
$135$		0			0
$136$	1.41421	−	2.44949i	0.121268	−	0.210042i
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	0.965250	+	0.261329i	$0.0841608\pi$
$138$	−8.48528	−	14.6969i	−0.722315	−	1.25109i
$139$	18.3848			1.55938			0.779688	−	0.626168i	$-0.215378\pi$
$139$	18.3848			1.55938			0.779688	+	0.626168i	$0.215378\pi$
$140$		0			0
$141$	−36.0000			−3.03175
$142$	−8.00000	−	13.8564i	−0.671345	−	1.16280i
$143$	2.12132	−	3.67423i	0.177394	−	0.307255i
$144$	−2.50000	+	4.33013i	−0.208333	+	0.360844i
$145$		0			0
$146$	−8.48528			−0.702247
$147$		0			0
$148$	2.00000			0.164399
$149$	−4.00000	−	6.92820i	−0.327693	−	0.567581i	0.654361	−	0.756182i	$-0.272938\pi$
$149$	−4.00000	−	6.92820i	−0.327693	−	0.567581i	−0.982054	+	0.188602i	$0.939604\pi$
$150$	7.07107	−	12.2474i	0.577350	−	1.00000i
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$	2.12132	+	3.67423i	0.172062	+	0.298020i
$153$	−14.1421			−1.14332
$154$		0			0
$155$		0			0
$156$	−6.00000	−	10.3923i	−0.480384	−	0.832050i
$157$	−4.24264	+	7.34847i	−0.338600	+	0.586472i	−0.984170	−	0.177229i	$-0.943287\pi$
$157$	−4.24264	+	7.34847i	−0.338600	+	0.586472i	0.645570	+	0.763701i	$0.276620\pi$
$158$	4.00000	−	6.92820i	0.318223	−	0.551178i
$159$	−2.82843	−	4.89898i	−0.224309	−	0.388514i
$160$		0			0
$161$		0			0
$162$	1.00000			0.0785674
$163$	2.00000	+	3.46410i	0.156652	+	0.271329i	0.933659	−	0.358162i	$-0.116597\pi$
$163$	2.00000	+	3.46410i	0.156652	+	0.271329i	−0.777007	+	0.629492i	$0.783263\pi$
$164$	−1.41421	+	2.44949i	−0.110432	+	0.191273i
$165$		0			0
$166$	6.36396	+	11.0227i	0.493939	+	0.855528i
$167$	−14.1421			−1.09435			−0.547176	−	0.837018i	$-0.684297\pi$
$167$	−14.1421			−1.09435			−0.547176	+	0.837018i	$0.684297\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$	10.6066	−	18.3712i	0.811107	−	1.40488i
$172$	−5.00000	+	8.66025i	−0.381246	+	0.660338i
$173$	−3.53553	−	6.12372i	−0.268802	−	0.465578i	0.699751	−	0.714387i	$-0.253294\pi$
$173$	−3.53553	−	6.12372i	−0.268802	−	0.465578i	−0.968553	+	0.248809i	$0.919961\pi$
$174$	−11.3137			−0.857690
$175$		0			0
$176$	−1.00000			−0.0753778
$177$	−16.0000	−	27.7128i	−1.20263	−	2.08302i
$178$	−3.53553	+	6.12372i	−0.264999	+	0.458993i
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	−0.549825	−	0.835280i	$-0.685306\pi$
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	0.998286	+	0.0585225i	$0.0186389\pi$
$180$		0			0
$181$	11.3137			0.840941			0.420471	−	0.907306i	$-0.361865\pi$
$181$	11.3137			0.840941			0.420471	+	0.907306i	$0.361865\pi$
$182$		0			0
$183$	−28.0000			−2.06982
$184$	−3.00000	−	5.19615i	−0.221163	−	0.383065i
$185$		0			0
$186$	10.0000	−	17.3205i	0.733236	−	1.27000i
$187$	−1.41421	−	2.44949i	−0.103418	−	0.179124i
$188$	−12.7279			−0.928279
$189$		0			0
$190$		0			0
$191$	−5.00000	−	8.66025i	−0.361787	−	0.626634i	0.626468	−	0.779447i	$-0.284500\pi$
$191$	−5.00000	−	8.66025i	−0.361787	−	0.626634i	−0.988255	+	0.152813i	$0.951167\pi$
$192$	−1.41421	+	2.44949i	−0.102062	+	0.176777i
$193$	−3.00000	+	5.19615i	−0.215945	+	0.374027i	−0.953564	−	0.301189i	$-0.902616\pi$
$193$	−3.00000	+	5.19615i	−0.215945	+	0.374027i	0.737620	+	0.675216i	$0.235950\pi$
$194$	3.53553	+	6.12372i	0.253837	+	0.439658i
$195$		0			0
$196$		0			0
$197$	−2.00000			−0.142494			−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$	2.50000	+	4.33013i	0.177667	+	0.307729i
$199$	−4.94975	+	8.57321i	−0.350878	+	0.607739i	−0.986404	−	0.164341i	$-0.947450\pi$
$199$	−4.94975	+	8.57321i	−0.350878	+	0.607739i	0.635525	+	0.772080i	$0.280784\pi$
$200$	2.50000	−	4.33013i	0.176777	−	0.306186i
$201$	−11.3137	−	19.5959i	−0.798007	−	1.38219i
$202$	−1.41421			−0.0995037
$203$		0			0
$204$	−8.00000			−0.560112
$205$		0			0
$206$	0.707107	−	1.22474i	0.0492665	−	0.0853320i
$207$	−15.0000	+	25.9808i	−1.04257	+	1.80579i
$208$	−2.12132	−	3.67423i	−0.147087	−	0.254762i
$209$	4.24264			0.293470
$210$		0			0
$211$	26.0000			1.78991			0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000			1.78991			0.894957	+	0.446153i	$0.147206\pi$
$212$	−1.00000	−	1.73205i	−0.0686803	−	0.118958i
$213$	−22.6274	+	39.1918i	−1.55041	+	2.68538i
$214$	7.00000	−	12.1244i	0.478510	−	0.828804i
$215$		0			0
$216$	5.65685			0.384900
$217$		0			0
$218$	−14.0000			−0.948200
$219$	12.0000	+	20.7846i	0.810885	+	1.40449i
$220$		0			0
$221$	6.00000	−	10.3923i	0.403604	−	0.699062i
$222$	−2.82843	−	4.89898i	−0.189832	−	0.328798i
$223$	21.2132			1.42054			0.710271	−	0.703929i	$-0.248573\pi$
$223$	21.2132			1.42054			0.710271	+	0.703929i	$0.248573\pi$
$224$		0			0
$225$	−25.0000			−1.66667
$226$	−8.00000	−	13.8564i	−0.532152	−	0.921714i
$227$	3.53553	−	6.12372i	0.234662	−	0.406446i	−0.724513	−	0.689261i	$-0.757935\pi$
$227$	3.53553	−	6.12372i	0.234662	−	0.406446i	0.959174	+	0.282816i	$0.0912685\pi$
$228$	6.00000	−	10.3923i	0.397360	−	0.688247i
$229$	11.3137	+	19.5959i	0.747631	+	1.29493i	0.948955	+	0.315410i	$0.102142\pi$
$229$	11.3137	+	19.5959i	0.747631	+	1.29493i	−0.201325	+	0.979525i	$0.564525\pi$
$230$		0			0
$231$		0			0
$232$	−4.00000			−0.262613
$233$	−13.0000	−	22.5167i	−0.851658	−	1.47512i	−0.879711	−	0.475509i	$-0.842264\pi$
$233$	−13.0000	−	22.5167i	−0.851658	−	1.47512i	0.0280525	−	0.999606i	$-0.491069\pi$
$234$	−10.6066	+	18.3712i	−0.693375	+	1.20096i
$235$		0			0
$236$	−5.65685	−	9.79796i	−0.368230	−	0.637793i
$237$	−22.6274			−1.46981
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	4.24264	−	7.34847i	0.273293	−	0.473357i	−0.696410	−	0.717644i	$-0.745221\pi$
$241$	4.24264	−	7.34847i	0.273293	−	0.473357i	0.969703	+	0.244287i	$0.0785540\pi$
$242$	−0.500000	+	0.866025i	−0.0321412	+	0.0556702i
$243$	7.07107	+	12.2474i	0.453609	+	0.785674i
$244$	−9.89949			−0.633750
$245$		0			0
$246$	8.00000			0.510061
$247$	9.00000	+	15.5885i	0.572656	+	0.991870i
$248$	3.53553	−	6.12372i	0.224507	−	0.388857i
$249$	18.0000	−	31.1769i	1.14070	−	1.97576i
$250$		0			0
$251$	−5.65685			−0.357057			−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685			−0.357057			−0.178529	+	0.983935i	$0.557134\pi$
$252$		0			0
$253$	−6.00000			−0.377217
$254$	10.0000	+	17.3205i	0.627456	+	1.08679i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−3.53553	−	6.12372i	−0.220541	−	0.381987i	0.734432	−	0.678683i	$-0.237449\pi$
$257$	−3.53553	−	6.12372i	−0.220541	−	0.381987i	−0.954972	+	0.296695i	$0.904115\pi$
$258$	28.2843			1.76090
$259$		0			0
$260$		0			0
$261$	10.0000	+	17.3205i	0.618984	+	1.07211i
$262$	−3.53553	+	6.12372i	−0.218426	+	0.378325i
$263$	−2.00000	+	3.46410i	−0.123325	+	0.213606i	−0.921077	−	0.389380i	$-0.872689\pi$
$263$	−2.00000	+	3.46410i	−0.123325	+	0.213606i	0.797752	+	0.602986i	$0.206023\pi$
$264$	1.41421	+	2.44949i	0.0870388	+	0.150756i
$265$		0			0
$266$		0			0
$267$	20.0000			1.22398
$268$	−4.00000	−	6.92820i	−0.244339	−	0.423207i
$269$	−5.65685	+	9.79796i	−0.344904	+	0.597392i	−0.985336	−	0.170623i	$-0.945422\pi$
$269$	−5.65685	+	9.79796i	−0.344904	+	0.597392i	0.640432	+	0.768015i	$0.278755\pi$
$270$		0			0
$271$	−4.24264	−	7.34847i	−0.257722	−	0.446388i	0.707909	−	0.706303i	$-0.249639\pi$
$271$	−4.24264	−	7.34847i	−0.257722	−	0.446388i	−0.965631	+	0.259916i	$0.916305\pi$
$272$	−2.82843			−0.171499
$273$		0			0
$274$	−6.00000			−0.362473
$275$	−2.50000	−	4.33013i	−0.150756	−	0.261116i
$276$	−8.48528	+	14.6969i	−0.510754	+	0.884652i
$277$	4.00000	−	6.92820i	0.240337	−	0.416275i	−0.720473	−	0.693482i	$-0.756075\pi$
$277$	4.00000	−	6.92820i	0.240337	−	0.416275i	0.960810	+	0.277207i	$0.0894088\pi$
$278$	−9.19239	−	15.9217i	−0.551323	−	0.954919i
$279$	−35.3553			−2.11667
$280$		0			0
$281$	22.0000			1.31241			0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000			1.31241			0.656205	+	0.754583i	$0.272161\pi$
$282$	18.0000	+	31.1769i	1.07188	+	1.85656i
$283$	−3.53553	+	6.12372i	−0.210166	+	0.364018i	−0.951766	−	0.306824i	$-0.900734\pi$
$283$	−3.53553	+	6.12372i	−0.210166	+	0.364018i	0.741601	+	0.670842i	$0.234067\pi$
$284$	−8.00000	+	13.8564i	−0.474713	+	0.822226i
$285$		0			0
$286$	−4.24264			−0.250873
$287$		0			0
$288$	5.00000			0.294628
$289$	4.50000	+	7.79423i	0.264706	+	0.458484i
$290$		0			0
$291$	10.0000	−	17.3205i	0.586210	−	1.01535i
$292$	4.24264	+	7.34847i	0.248282	+	0.430037i
$293$	−21.2132			−1.23929			−0.619644	−	0.784883i	$-0.712723\pi$
$293$	−21.2132			−1.23929			−0.619644	+	0.784883i	$0.712723\pi$
$294$		0			0
$295$		0			0
$296$	−1.00000	−	1.73205i	−0.0581238	−	0.100673i
$297$	2.82843	−	4.89898i	0.164122	−	0.284268i
$298$	−4.00000	+	6.92820i	−0.231714	+	0.401340i
$299$	−12.7279	−	22.0454i	−0.736075	−	1.27492i
$300$	−14.1421			−0.816497
$301$		0			0
$302$	−8.00000			−0.460348
$303$	2.00000	+	3.46410i	0.114897	+	0.199007i
$304$	2.12132	−	3.67423i	0.121666	−	0.210732i
$305$		0			0
$306$	7.07107	+	12.2474i	0.404226	+	0.700140i
$307$	21.2132			1.21070			0.605351	−	0.795959i	$-0.293033\pi$
$307$	21.2132			1.21070			0.605351	+	0.795959i	$0.293033\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		0			0
$311$	0.707107	−	1.22474i	0.0400963	−	0.0694489i	−0.845281	−	0.534322i	$-0.820567\pi$
$311$	0.707107	−	1.22474i	0.0400963	−	0.0694489i	0.885377	+	0.464873i	$0.153900\pi$
$312$	−6.00000	+	10.3923i	−0.339683	+	0.588348i
$313$	14.8492	+	25.7196i	0.839329	+	1.45376i	0.890457	+	0.455068i	$0.150385\pi$
$313$	14.8492	+	25.7196i	0.839329	+	1.45376i	−0.0511279	+	0.998692i	$0.516282\pi$
$314$	8.48528			0.478852
$315$		0			0
$316$	−8.00000			−0.450035
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	−0.999980	−	0.00635137i	$-0.997978\pi$
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	0.494489	−	0.869184i	$-0.335355\pi$
$318$	−2.82843	+	4.89898i	−0.158610	+	0.274721i
$319$	−2.00000	+	3.46410i	−0.111979	+	0.193952i
$320$		0			0
$321$	−39.5980			−2.21014
$322$		0			0
$323$	12.0000			0.667698
$324$	−0.500000	−	0.866025i	−0.0277778	−	0.0481125i
$325$	10.6066	−	18.3712i	0.588348	−	1.01905i
$326$	2.00000	−	3.46410i	0.110770	−	0.191859i
$327$	19.7990	+	34.2929i	1.09489	+	1.89640i
$328$	2.82843			0.156174
$329$		0			0
$330$		0			0
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	0.734905	−	0.678170i	$-0.237227\pi$
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	−0.954765	+	0.297361i	$0.903893\pi$
$332$	6.36396	−	11.0227i	0.349268	−	0.604949i
$333$	−5.00000	+	8.66025i	−0.273998	+	0.474579i
$334$	7.07107	+	12.2474i	0.386912	+	0.670151i
$335$		0			0
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$	−2.50000	−	4.33013i	−0.135982	−	0.235528i
$339$	−22.6274	+	39.1918i	−1.22895	+	2.12861i
$340$		0			0
$341$	−3.53553	−	6.12372i	−0.191460	−	0.331618i
$342$	−21.2132			−1.14708
$343$		0			0
$344$	10.0000			0.539164
$345$		0			0
$346$	−3.53553	+	6.12372i	−0.190071	+	0.329213i
$347$	−13.0000	+	22.5167i	−0.697877	+	1.20876i	0.271325	+	0.962488i	$0.412538\pi$
$347$	−13.0000	+	22.5167i	−0.697877	+	1.20876i	−0.969201	+	0.246270i	$0.920795\pi$
$348$	5.65685	+	9.79796i	0.303239	+	0.525226i
$349$	−26.8701			−1.43832			−0.719161	−	0.694844i	$-0.755473\pi$
$349$	−26.8701			−1.43832			−0.719161	+	0.694844i	$0.755473\pi$
$350$		0			0
$351$	24.0000			1.28103
$352$	0.500000	+	0.866025i	0.0266501	+	0.0461593i
$353$	−7.77817	+	13.4722i	−0.413990	+	0.717053i	−0.995322	−	0.0966144i	$-0.969199\pi$
$353$	−7.77817	+	13.4722i	−0.413990	+	0.717053i	0.581331	+	0.813667i	$0.302532\pi$
$354$	−16.0000	+	27.7128i	−0.850390	+	1.47292i
$355$		0			0
$356$	7.07107			0.374766
$357$		0			0
$358$	−12.0000			−0.634220
$359$	−6.00000	−	10.3923i	−0.316668	−	0.548485i	0.663123	−	0.748511i	$-0.269231\pi$
$359$	−6.00000	−	10.3923i	−0.316668	−	0.548485i	−0.979791	+	0.200026i	$0.935897\pi$
$360$		0			0
$361$	0.500000	−	0.866025i	0.0263158	−	0.0455803i
$362$	−5.65685	−	9.79796i	−0.297318	−	0.514969i
$363$	2.82843			0.148454
$364$		0			0
$365$		0			0
$366$	14.0000	+	24.2487i	0.731792	+	1.26750i
$367$	10.6066	−	18.3712i	0.553660	−	0.958967i	−0.444346	−	0.895855i	$-0.646564\pi$
$367$	10.6066	−	18.3712i	0.553660	−	0.958967i	0.998006	−	0.0631123i	$-0.0201026\pi$
$368$	−3.00000	+	5.19615i	−0.156386	+	0.270868i
$369$	−7.07107	−	12.2474i	−0.368105	−	0.637577i
$370$		0			0
$371$		0			0
$372$	−20.0000			−1.03695
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$	−1.41421	+	2.44949i	−0.0731272	+	0.126660i
$375$		0			0
$376$	6.36396	+	11.0227i	0.328196	+	0.568453i
$377$	−16.9706			−0.874028
$378$		0			0
$379$	24.0000			1.23280			0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000			1.23280			0.616399	+	0.787434i	$0.288591\pi$
$380$		0			0
$381$	28.2843	−	48.9898i	1.44905	−	2.50982i
$382$	−5.00000	+	8.66025i	−0.255822	+	0.443097i
$383$	7.77817	+	13.4722i	0.397446	+	0.688397i	0.993410	−	0.114614i	$-0.0365632\pi$
$383$	7.77817	+	13.4722i	0.397446	+	0.688397i	−0.595964	+	0.803011i	$0.703230\pi$
$384$	2.82843			0.144338
$385$		0			0
$386$	6.00000			0.305392
$387$	−25.0000	−	43.3013i	−1.27082	−	2.20113i
$388$	3.53553	−	6.12372i	0.179490	−	0.310885i
$389$	15.0000	−	25.9808i	0.760530	−	1.31728i	−0.182047	−	0.983290i	$-0.558272\pi$
$389$	15.0000	−	25.9808i	0.760530	−	1.31728i	0.942578	−	0.333987i	$-0.108394\pi$
$390$		0			0
$391$	−16.9706			−0.858238
$392$		0			0
$393$	20.0000			1.00887
$394$	1.00000	+	1.73205i	0.0503793	+	0.0872595i
$395$		0			0
$396$	2.50000	−	4.33013i	0.125630	−	0.217597i
$397$	−12.7279	−	22.0454i	−0.638796	−	1.10643i	−0.985697	−	0.168526i	$-0.946099\pi$
$397$	−12.7279	−	22.0454i	−0.638796	−	1.10643i	0.346901	−	0.937902i	$-0.387234\pi$
$398$	9.89949			0.496217
$399$		0			0
$400$	−5.00000			−0.250000
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	0.948272	+	0.317460i	$0.102830\pi$
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	−0.199207	+	0.979957i	$0.563837\pi$
$402$	−11.3137	+	19.5959i	−0.564276	+	0.977356i
$403$	15.0000	−	25.9808i	0.747203	−	1.29419i
$404$	0.707107	+	1.22474i	0.0351799	+	0.0609333i
$405$		0			0
$406$		0			0
$407$	−2.00000			−0.0991363
$408$	4.00000	+	6.92820i	0.198030	+	0.342997i
$409$	−14.1421	+	24.4949i	−0.699284	+	1.21119i	0.269432	+	0.963020i	$0.413164\pi$
$409$	−14.1421	+	24.4949i	−0.699284	+	1.21119i	−0.968715	+	0.248175i	$0.920169\pi$
$410$		0			0
$411$	8.48528	+	14.6969i	0.418548	+	0.724947i
$412$	−1.41421			−0.0696733
$413$		0			0
$414$	30.0000			1.47442
$415$		0			0
$416$	−2.12132	+	3.67423i	−0.104006	+	0.180144i
$417$	−26.0000	+	45.0333i	−1.27323	+	2.20529i
$418$	−2.12132	−	3.67423i	−0.103757	−	0.179713i
$419$	14.1421			0.690889			0.345444	−	0.938439i	$-0.387728\pi$
$419$	14.1421			0.690889			0.345444	+	0.938439i	$0.387728\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$	−13.0000	−	22.5167i	−0.632830	−	1.09609i
$423$	31.8198	−	55.1135i	1.54713	−	2.67971i
$424$	−1.00000	+	1.73205i	−0.0485643	+	0.0841158i
$425$	−7.07107	−	12.2474i	−0.342997	−	0.594089i
$426$	45.2548			2.19260
$427$		0			0
$428$	−14.0000			−0.676716
$429$	6.00000	+	10.3923i	0.289683	+	0.501745i
$430$		0			0
$431$	−4.00000	+	6.92820i	−0.192673	+	0.333720i	−0.946135	−	0.323772i	$-0.895049\pi$
$431$	−4.00000	+	6.92820i	−0.192673	+	0.333720i	0.753462	+	0.657491i	$0.228382\pi$
$432$	−2.82843	−	4.89898i	−0.136083	−	0.235702i
$433$	29.6985			1.42722			0.713609	−	0.700544i	$-0.247059\pi$
$433$	29.6985			1.42722			0.713609	+	0.700544i	$0.247059\pi$
$434$		0			0
$435$		0			0
$436$	7.00000	+	12.1244i	0.335239	+	0.580651i
$437$	12.7279	−	22.0454i	0.608859	−	1.05457i
$438$	12.0000	−	20.7846i	0.573382	−	0.993127i
$439$	−12.7279	−	22.0454i	−0.607471	−	1.05217i	−0.991656	−	0.128914i	$-0.958851\pi$
$439$	−12.7279	−	22.0454i	−0.607471	−	1.05217i	0.384185	−	0.923256i	$-0.374482\pi$
$440$		0			0
$441$		0			0
$442$	−12.0000			−0.570782
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	0.687557	−	0.726130i	$-0.258683\pi$
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	−0.972626	+	0.232377i	$0.925350\pi$
$444$	−2.82843	+	4.89898i	−0.134231	+	0.232495i
$445$		0			0
$446$	−10.6066	−	18.3712i	−0.502237	−	0.869900i
$447$	22.6274			1.07024
$448$		0			0
$449$	16.0000			0.755087			0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000			0.755087			0.377543	+	0.925992i	$0.376769\pi$
$450$	12.5000	+	21.6506i	0.589256	+	1.02062i
$451$	1.41421	−	2.44949i	0.0665927	−	0.115342i
$452$	−8.00000	+	13.8564i	−0.376288	+	0.651751i
$453$	11.3137	+	19.5959i	0.531564	+	0.920697i
$454$	−7.07107			−0.331862
$455$		0			0
$456$	−12.0000			−0.561951
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	0.999857	+	0.0168929i	$0.00537742\pi$
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	−0.485299	+	0.874348i	$0.661289\pi$
$458$	11.3137	−	19.5959i	0.528655	−	0.915657i
$459$	8.00000	−	13.8564i	0.373408	−	0.646762i
$460$		0			0
$461$	1.41421			0.0658665			0.0329332	−	0.999458i	$-0.489515\pi$
$461$	1.41421			0.0658665			0.0329332	+	0.999458i	$0.489515\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$	2.00000	+	3.46410i	0.0928477	+	0.160817i
$465$		0			0
$466$	−13.0000	+	22.5167i	−0.602213	+	1.04306i
$467$	−12.7279	−	22.0454i	−0.588978	−	1.02014i	−0.994367	−	0.105995i	$-0.966197\pi$
$467$	−12.7279	−	22.0454i	−0.588978	−	1.02014i	0.405389	−	0.914144i	$-0.367136\pi$
$468$	21.2132			0.980581
$469$		0			0
$470$		0			0
$471$	−12.0000	−	20.7846i	−0.552931	−	0.957704i
$472$	−5.65685	+	9.79796i	−0.260378	+	0.450988i
$473$	5.00000	−	8.66025i	0.229900	−	0.398199i
$474$	11.3137	+	19.5959i	0.519656	+	0.900070i
$475$	21.2132			0.973329
$476$		0			0
$477$	10.0000			0.457869
$478$		0			0
$479$	14.1421	−	24.4949i	0.646171	−	1.11920i	−0.337859	−	0.941197i	$-0.609703\pi$
$479$	14.1421	−	24.4949i	0.646171	−	1.11920i	0.984030	−	0.178004i	$-0.0569639\pi$
$480$		0			0
$481$	−4.24264	−	7.34847i	−0.193448	−	0.335061i
$482$	−8.48528			−0.386494
$483$		0			0
$484$	1.00000			0.0454545
$485$		0			0
$486$	7.07107	−	12.2474i	0.320750	−	0.555556i
$487$	−19.0000	+	32.9090i	−0.860972	+	1.49125i	0.0100195	+	0.999950i	$0.496811\pi$
$487$	−19.0000	+	32.9090i	−0.860972	+	1.49125i	−0.870992	+	0.491298i	$0.836523\pi$
$488$	4.94975	+	8.57321i	0.224065	+	0.388091i
$489$	−11.3137			−0.511624
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$	−4.00000	−	6.92820i	−0.180334	−	0.312348i
$493$	−5.65685	+	9.79796i	−0.254772	+	0.441278i
$494$	9.00000	−	15.5885i	0.404929	−	0.701358i
$495$		0			0
$496$	−7.07107			−0.317500
$497$		0			0
$498$	−36.0000			−1.61320
$499$	−18.0000	−	31.1769i	−0.805791	−	1.39567i	−0.915756	−	0.401735i	$-0.868407\pi$
$499$	−18.0000	−	31.1769i	−0.805791	−	1.39567i	0.109965	−	0.993935i	$-0.464926\pi$
$500$		0			0
$501$	20.0000	−	34.6410i	0.893534	−	1.54765i
$502$	2.82843	+	4.89898i	0.126239	+	0.218652i
$503$	19.7990			0.882793			0.441397	−	0.897312i	$-0.354483\pi$
$503$	19.7990			0.882793			0.441397	+	0.897312i	$0.354483\pi$
$504$		0			0
$505$		0			0
$506$	3.00000	+	5.19615i	0.133366	+	0.230997i
$507$	−7.07107	+	12.2474i	−0.314037	+	0.543928i
$508$	10.0000	−	17.3205i	0.443678	−	0.768473i
$509$	−8.48528	−	14.6969i	−0.376103	−	0.651430i	0.614388	−	0.789004i	$-0.289403\pi$
$509$	−8.48528	−	14.6969i	−0.376103	−	0.651430i	−0.990492	+	0.137574i	$0.956070\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			0.0441942
$513$	12.0000	+	20.7846i	0.529813	+	0.917663i
$514$	−3.53553	+	6.12372i	−0.155946	+	0.270106i
$515$		0			0
$516$	−14.1421	−	24.4949i	−0.622573	−	1.07833i
$517$	12.7279			0.559773
$518$		0			0
$519$	20.0000			0.877903
$520$		0			0
$521$	0.707107	−	1.22474i	0.0309789	−	0.0536570i	−0.850120	−	0.526589i	$-0.823471\pi$
$521$	0.707107	−	1.22474i	0.0309789	−	0.0536570i	0.881099	+	0.472931i	$0.156804\pi$
$522$	10.0000	−	17.3205i	0.437688	−	0.758098i
$523$	−2.12132	−	3.67423i	−0.0927589	−	0.160663i	0.815912	−	0.578176i	$-0.196235\pi$
$523$	−2.12132	−	3.67423i	−0.0927589	−	0.160663i	−0.908671	+	0.417513i	$0.862902\pi$
$524$	7.07107			0.308901
$525$		0			0
$526$	4.00000			0.174408
$527$	−10.0000	−	17.3205i	−0.435607	−	0.754493i
$528$	1.41421	−	2.44949i	0.0615457	−	0.106600i
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$	56.5685			2.45487
$532$		0			0
$533$	12.0000			0.519778
$534$	−10.0000	−	17.3205i	−0.432742	−	0.749532i
$535$		0			0
$536$	−4.00000	+	6.92820i	−0.172774	+	0.299253i
$537$	16.9706	+	29.3939i	0.732334	+	1.26844i
$538$	11.3137			0.487769
$539$		0			0
$540$		0			0
$541$	15.0000	+	25.9808i	0.644900	+	1.11700i	0.984325	+	0.176367i	$0.0564345\pi$
$541$	15.0000	+	25.9808i	0.644900	+	1.11700i	−0.339424	+	0.940633i	$0.610232\pi$
$542$	−4.24264	+	7.34847i	−0.182237	+	0.315644i
$543$	−16.0000	+	27.7128i	−0.686626	+	1.18927i
$544$	1.41421	+	2.44949i	0.0606339	+	0.105021i
$545$		0			0
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$	3.00000	+	5.19615i	0.128154	+	0.221969i
$549$	24.7487	−	42.8661i	1.05625	−	1.82948i
$550$	−2.50000	+	4.33013i	−0.106600	+	0.184637i
$551$	−8.48528	−	14.6969i	−0.361485	−	0.626111i
$552$	16.9706			0.722315
$553$		0			0
$554$	−8.00000			−0.339887
$555$		0			0
$556$	−9.19239	+	15.9217i	−0.389844	+	0.675230i
$557$	3.00000	−	5.19615i	0.127114	−	0.220168i	−0.795443	−	0.606028i	$-0.792762\pi$
$557$	3.00000	−	5.19615i	0.127114	−	0.220168i	0.922557	+	0.385860i	$0.126095\pi$
$558$	17.6777	+	30.6186i	0.748355	+	1.29619i
$559$	42.4264			1.79445
$560$		0			0
$561$	8.00000			0.337760
$562$	−11.0000	−	19.0526i	−0.464007	−	0.803684i
$563$	−20.5061	+	35.5176i	−0.864229	+	1.49689i	0.00358183	+	0.999994i	$0.498860\pi$
$563$	−20.5061	+	35.5176i	−0.864229	+	1.49689i	−0.867811	+	0.496895i	$0.834473\pi$
$564$	18.0000	−	31.1769i	0.757937	−	1.31278i
$565$		0			0
$566$	7.07107			0.297219
$567$		0			0
$568$	16.0000			0.671345
$569$	−21.0000	−	36.3731i	−0.880366	−	1.52484i	−0.850935	−	0.525271i	$-0.823964\pi$
$569$	−21.0000	−	36.3731i	−0.880366	−	1.52484i	−0.0294311	−	0.999567i	$-0.509370\pi$
$570$		0			0
$571$	−10.0000	+	17.3205i	−0.418487	+	0.724841i	−0.995788	−	0.0916910i	$-0.970773\pi$
$571$	−10.0000	+	17.3205i	−0.418487	+	0.724841i	0.577301	+	0.816532i	$0.304106\pi$
$572$	2.12132	+	3.67423i	0.0886969	+	0.153627i
$573$	28.2843			1.18159
$574$		0			0
$575$	−30.0000			−1.25109
$576$	−2.50000	−	4.33013i	−0.104167	−	0.180422i
$577$	4.94975	−	8.57321i	0.206061	−	0.356908i	−0.744409	−	0.667723i	$-0.767269\pi$
$577$	4.94975	−	8.57321i	0.206061	−	0.356908i	0.950470	+	0.310816i	$0.100602\pi$
$578$	4.50000	−	7.79423i	0.187175	−	0.324197i
$579$	−8.48528	−	14.6969i	−0.352636	−	0.610784i
$580$		0			0
$581$		0			0
$582$	−20.0000			−0.829027
$583$	1.00000	+	1.73205i	0.0414158	+	0.0717342i
$584$	4.24264	−	7.34847i	0.175562	−	0.304082i
$585$		0			0
$586$	10.6066	+	18.3712i	0.438155	+	0.758906i
$587$	5.65685			0.233483			0.116742	−	0.993162i	$-0.462755\pi$
$587$	5.65685			0.233483			0.116742	+	0.993162i	$0.462755\pi$
$588$		0			0
$589$	30.0000			1.23613
$590$		0			0
$591$	2.82843	−	4.89898i	0.116346	−	0.201517i
$592$	−1.00000	+	1.73205i	−0.0410997	+	0.0711868i
$593$	−4.24264	−	7.34847i	−0.174224	−	0.301765i	0.765668	−	0.643236i	$-0.222408\pi$
$593$	−4.24264	−	7.34847i	−0.174224	−	0.301765i	−0.939893	+	0.341470i	$0.889075\pi$
$594$	−5.65685			−0.232104
$595$		0			0
$596$	8.00000			0.327693
$597$	−14.0000	−	24.2487i	−0.572982	−	0.992434i
$598$	−12.7279	+	22.0454i	−0.520483	+	0.901504i
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$	7.07107	+	12.2474i	0.288675	+	0.500000i
$601$	11.3137			0.461496			0.230748	−	0.973014i	$-0.425883\pi$
$601$	11.3137			0.461496			0.230748	+	0.973014i	$0.425883\pi$
$602$		0			0
$603$	40.0000			1.62893
$604$	4.00000	+	6.92820i	0.162758	+	0.281905i
$605$		0			0
$606$	2.00000	−	3.46410i	0.0812444	−	0.140720i
$607$	8.48528	+	14.6969i	0.344407	+	0.596530i	0.985246	−	0.171145i	$-0.0547467\pi$
$607$	8.48528	+	14.6969i	0.344407	+	0.596530i	−0.640839	+	0.767675i	$0.721413\pi$
$608$	−4.24264			−0.172062
$609$		0			0
$610$		0			0
$611$	27.0000	+	46.7654i	1.09230	+	1.89192i
$612$	7.07107	−	12.2474i	0.285831	−	0.495074i
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	−0.658012	−	0.753007i	$-0.728603\pi$
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	0.981129	+	0.193352i	$0.0619359\pi$
$614$	−10.6066	−	18.3712i	−0.428048	−	0.741400i
$615$		0			0
$616$		0			0
$617$	28.0000			1.12724			0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000			1.12724			0.563619	+	0.826035i	$0.309409\pi$
$618$	2.00000	+	3.46410i	0.0804518	+	0.139347i
$619$	7.07107	−	12.2474i	0.284210	−	0.492267i	−0.688207	−	0.725514i	$-0.741602\pi$
$619$	7.07107	−	12.2474i	0.284210	−	0.492267i	0.972417	+	0.233248i	$0.0749353\pi$
$620$		0			0
$621$	−16.9706	−	29.3939i	−0.681005	−	1.17954i
$622$	−1.41421			−0.0567048
$623$		0			0
$624$	12.0000			0.480384
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$	14.8492	−	25.7196i	0.593495	−	1.02796i
$627$	−6.00000	+	10.3923i	−0.239617	+	0.415029i
$628$	−4.24264	−	7.34847i	−0.169300	−	0.293236i
$629$	−5.65685			−0.225554
$630$		0			0
$631$	−30.0000			−1.19428			−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000			−1.19428			−0.597141	+	0.802137i	$0.703697\pi$
$632$	4.00000	+	6.92820i	0.159111	+	0.275589i
$633$	−36.7696	+	63.6867i	−1.46146	+	2.53132i
$634$	−9.00000	+	15.5885i	−0.357436	+	0.619097i
$635$		0			0
$636$	5.65685			0.224309
$637$		0			0
$638$	4.00000			0.158362
$639$	−40.0000	−	69.2820i	−1.58238	−	2.74075i
$640$		0			0
$641$	−22.0000	+	38.1051i	−0.868948	+	1.50506i	−0.00587459	+	0.999983i	$0.501870\pi$
$641$	−22.0000	+	38.1051i	−0.868948	+	1.50506i	−0.863073	+	0.505079i	$0.831463\pi$
$642$	19.7990	+	34.2929i	0.781404	+	1.35343i
$643$	−33.9411			−1.33851			−0.669254	−	0.743034i	$-0.733386\pi$
$643$	−33.9411			−1.33851			−0.669254	+	0.743034i	$0.733386\pi$
$644$		0			0
$645$		0			0
$646$	−6.00000	−	10.3923i	−0.236067	−	0.408880i
$647$	−16.2635	+	28.1691i	−0.639382	+	1.10744i	0.346186	+	0.938166i	$0.387477\pi$
$647$	−16.2635	+	28.1691i	−0.639382	+	1.10744i	−0.985569	+	0.169277i	$0.945857\pi$
$648$	−0.500000	+	0.866025i	−0.0196419	+	0.0340207i
$649$	5.65685	+	9.79796i	0.222051	+	0.384604i
$650$	−21.2132			−0.832050
$651$		0			0
$652$	−4.00000			−0.156652
$653$	15.0000	+	25.9808i	0.586995	+	1.01671i	0.994623	+	0.103558i	$0.0330227\pi$
$653$	15.0000	+	25.9808i	0.586995	+	1.01671i	−0.407628	+	0.913148i	$0.633644\pi$
$654$	19.7990	−	34.2929i	0.774202	−	1.34096i
$655$		0			0
$656$	−1.41421	−	2.44949i	−0.0552158	−	0.0956365i
$657$	−42.4264			−1.65521
$658$		0			0
$659$	30.0000			1.16863			0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000			1.16863			0.584317	+	0.811525i	$0.301362\pi$
$660$		0			0
$661$	8.48528	−	14.6969i	0.330039	−	0.571645i	−0.652480	−	0.757806i	$-0.726271\pi$
$661$	8.48528	−	14.6969i	0.330039	−	0.571645i	0.982519	+	0.186161i	$0.0596047\pi$
$662$	−4.00000	+	6.92820i	−0.155464	+	0.269272i
$663$	16.9706	+	29.3939i	0.659082	+	1.14156i
$664$	−12.7279			−0.493939
$665$		0			0
$666$	10.0000			0.387492
$667$	12.0000	+	20.7846i	0.464642	+	0.804783i
$668$	7.07107	−	12.2474i	0.273588	−	0.473868i
$669$	−30.0000	+	51.9615i	−1.15987	+	2.00895i
$670$		0			0
$671$	9.89949			0.382166
$672$		0			0
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$	11.0000	+	19.0526i	0.423704	+	0.733877i
$675$	14.1421	−	24.4949i	0.544331	−	0.942809i
$676$	−2.50000	+	4.33013i	−0.0961538	+	0.166543i
$677$	−4.94975	−	8.57321i	−0.190234	−	0.329495i	0.755094	−	0.655617i	$-0.227591\pi$
$677$	−4.94975	−	8.57321i	−0.190234	−	0.329495i	−0.945328	+	0.326122i	$0.894258\pi$
$678$	45.2548			1.73800
$679$		0			0
$680$		0			0
$681$	10.0000	+	17.3205i	0.383201	+	0.663723i
$682$	−3.53553	+	6.12372i	−0.135383	+	0.234490i
$683$	−16.0000	+	27.7128i	−0.612223	+	1.06040i	0.378642	+	0.925543i	$0.376391\pi$
$683$	−16.0000	+	27.7128i	−0.612223	+	1.06040i	−0.990865	+	0.134858i	$0.956942\pi$
$684$	10.6066	+	18.3712i	0.405554	+	0.702439i
$685$		0			0
$686$		0			0
$687$	−64.0000			−2.44175
$688$	−5.00000	−	8.66025i	−0.190623	−	0.330169i
$689$	−4.24264	+	7.34847i	−0.161632	+	0.279954i
$690$		0			0
$691$	−19.7990	−	34.2929i	−0.753189	−	1.30456i	−0.946270	−	0.323379i	$-0.895181\pi$
$691$	−19.7990	−	34.2929i	−0.753189	−	1.30456i	0.193081	−	0.981183i	$-0.438152\pi$
$692$	7.07107			0.268802
$693$		0			0
$694$	26.0000			0.986947
$695$		0			0
$696$	5.65685	−	9.79796i	0.214423	−	0.371391i
$697$	4.00000	−	6.92820i	0.151511	−	0.262424i
$698$	13.4350	+	23.2702i	0.508523	+	0.880788i
$699$	73.5391			2.78150
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$	−12.0000	−	20.7846i	−0.452911	−	0.784465i
$703$	4.24264	−	7.34847i	0.160014	−	0.277153i
$704$	0.500000	−	0.866025i	0.0188445	−	0.0326396i
$705$		0			0
$706$	15.5563			0.585471
$707$		0			0
$708$	32.0000			1.20263
$709$	19.0000	+	32.9090i	0.713560	+	1.23592i	0.963512	+	0.267664i	$0.0862517\pi$
$709$	19.0000	+	32.9090i	0.713560	+	1.23592i	−0.249952	+	0.968258i	$0.580415\pi$
$710$		0			0
$711$	20.0000	−	34.6410i	0.750059	−	1.29914i
$712$	−3.53553	−	6.12372i	−0.132500	−	0.229496i
$713$	−42.4264			−1.58888
$714$		0			0
$715$		0			0
$716$	6.00000	+	10.3923i	0.224231	+	0.388379i
$717$		0			0
$718$	−6.00000	+	10.3923i	−0.223918	+	0.387837i
$719$	−16.2635	−	28.1691i	−0.606525	−	1.05053i	−0.991809	−	0.127733i	$-0.959230\pi$
$719$	−16.2635	−	28.1691i	−0.606525	−	1.05053i	0.385284	−	0.922798i	$-0.374103\pi$
$720$		0			0
$721$		0			0
$722$	−1.00000			−0.0372161
$723$	12.0000	+	20.7846i	0.446285	+	0.772988i
$724$	−5.65685	+	9.79796i	−0.210235	+	0.364138i
$725$	−10.0000	+	17.3205i	−0.371391	+	0.643268i
$726$	−1.41421	−	2.44949i	−0.0524864	−	0.0909091i
$727$	−46.6690			−1.73086			−0.865430	−	0.501031i	$-0.832954\pi$
$727$	−46.6690			−1.73086			−0.865430	+	0.501031i	$0.832954\pi$
$728$		0			0
$729$	−43.0000			−1.59259
$730$		0			0
$731$	14.1421	−	24.4949i	0.523066	−	0.905977i
$732$	14.0000	−	24.2487i	0.517455	−	0.896258i
$733$	−3.53553	−	6.12372i	−0.130588	−	0.226185i	0.793315	−	0.608811i	$-0.208353\pi$
$733$	−3.53553	−	6.12372i	−0.130588	−	0.226185i	−0.923903	+	0.382626i	$0.875020\pi$
$734$	−21.2132			−0.782994
$735$		0			0
$736$	6.00000			0.221163
$737$	4.00000	+	6.92820i	0.147342	+	0.255204i
$738$	−7.07107	+	12.2474i	−0.260290	+	0.450835i
$739$	6.00000	−	10.3923i	0.220714	−	0.382287i	−0.734311	−	0.678813i	$-0.762495\pi$
$739$	6.00000	−	10.3923i	0.220714	−	0.382287i	0.955025	+	0.296526i	$0.0958281\pi$
$740$		0			0
$741$	−50.9117			−1.87029
$742$		0			0
$743$	−8.00000			−0.293492			−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000			−0.293492			−0.146746	+	0.989174i	$0.546880\pi$
$744$	10.0000	+	17.3205i	0.366618	+	0.635001i
$745$		0			0
$746$	5.00000	−	8.66025i	0.183063	−	0.317074i
$747$	31.8198	+	55.1135i	1.16423	+	2.01650i
$748$	2.82843			0.103418
$749$		0			0
$750$		0			0
$751$	1.00000	+	1.73205i	0.0364905	+	0.0632034i	0.883694	−	0.468065i	$-0.155049\pi$
$751$	1.00000	+	1.73205i	0.0364905	+	0.0632034i	−0.847203	+	0.531269i	$0.821715\pi$
$752$	6.36396	−	11.0227i	0.232070	−	0.401957i
$753$	8.00000	−	13.8564i	0.291536	−	0.504956i
$754$	8.48528	+	14.6969i	0.309016	+	0.535231i
$755$		0			0
$756$		0			0
$757$	46.0000			1.67190			0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000			1.67190			0.835949	+	0.548807i	$0.184918\pi$
$758$	−12.0000	−	20.7846i	−0.435860	−	0.754931i
$759$	8.48528	−	14.6969i	0.307996	−	0.533465i
$760$		0			0
$761$	16.9706	+	29.3939i	0.615182	+	1.06553i	0.990352	+	0.138571i	$0.0442510\pi$
$761$	16.9706	+	29.3939i	0.615182	+	1.06553i	−0.375170	+	0.926956i	$0.622416\pi$
$762$	−56.5685			−2.04926
$763$		0			0
$764$	10.0000			0.361787
$765$		0			0
$766$	7.77817	−	13.4722i	0.281037	−	0.486770i
$767$	−24.0000	+	41.5692i	−0.866590	+	1.50098i
$768$	−1.41421	−	2.44949i	−0.0510310	−	0.0883883i
$769$	5.65685			0.203991			0.101996	−	0.994785i	$-0.467477\pi$
$769$	5.65685			0.203991			0.101996	+	0.994785i	$0.467477\pi$
$770$		0			0
$771$	20.0000			0.720282
$772$	−3.00000	−	5.19615i	−0.107972	−	0.187014i
$773$	24.0416	−	41.6413i	0.864717	−	1.49773i	−0.00261021	−	0.999997i	$-0.500831\pi$
$773$	24.0416	−	41.6413i	0.864717	−	1.49773i	0.867328	−	0.497738i	$-0.165836\pi$
$774$	−25.0000	+	43.3013i	−0.898606	+	1.55643i
$775$	−17.6777	−	30.6186i	−0.635001	−	1.09985i
$776$	−7.07107			−0.253837
$777$		0			0
$778$	−30.0000			−1.07555
$779$	6.00000	+	10.3923i	0.214972	+	0.372343i
$780$		0			0
$781$	8.00000	−	13.8564i	0.286263	−	0.495821i
$782$	8.48528	+	14.6969i	0.303433	+	0.525561i
$783$	−22.6274

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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-1.41421 + 2.44949i) q^{3} +(-0.500000 + 0.866025i) q^{4} +2.82843 q^{6} +1.00000 q^{8} +(-2.50000 - 4.33013i) q^{9} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-1.41421 + 2.44949i) q^{3} +(-0.500000 + 0.866025i) q^{4} +2.82843 q^{6} +1.00000 q^{8} +(-2.50000 - 4.33013i) q^{9} +(0.500000 - 0.866025i) q^{11} +(-1.41421 - 2.44949i) q^{12} +4.24264 q^{13} +(-0.500000 - 0.866025i) q^{16} +(1.41421 - 2.44949i) q^{17} +(-2.50000 + 4.33013i) q^{18} +(2.12132 + 3.67423i) q^{19} -1.00000 q^{22} +(-3.00000 - 5.19615i) q^{23} +(-1.41421 + 2.44949i) q^{24} +(2.50000 - 4.33013i) q^{25} +(-2.12132 - 3.67423i) q^{26} +5.65685 q^{27} -4.00000 q^{29} +(3.53553 - 6.12372i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(1.41421 + 2.44949i) q^{33} -2.82843 q^{34} +5.00000 q^{36} +(-1.00000 - 1.73205i) q^{37} +(2.12132 - 3.67423i) q^{38} +(-6.00000 + 10.3923i) q^{39} +2.82843 q^{41} +10.0000 q^{43} +(0.500000 + 0.866025i) q^{44} +(-3.00000 + 5.19615i) q^{46} +(6.36396 + 11.0227i) q^{47} +2.82843 q^{48} -5.00000 q^{50} +(4.00000 + 6.92820i) q^{51} +(-2.12132 + 3.67423i) q^{52} +(-1.00000 + 1.73205i) q^{53} +(-2.82843 - 4.89898i) q^{54} -12.0000 q^{57} +(2.00000 + 3.46410i) q^{58} +(-5.65685 + 9.79796i) q^{59} +(4.94975 + 8.57321i) q^{61} -7.07107 q^{62} +1.00000 q^{64} +(1.41421 - 2.44949i) q^{66} +(-4.00000 + 6.92820i) q^{67} +(1.41421 + 2.44949i) q^{68} +16.9706 q^{69} +16.0000 q^{71} +(-2.50000 - 4.33013i) q^{72} +(4.24264 - 7.34847i) q^{73} +(-1.00000 + 1.73205i) q^{74} +(7.07107 + 12.2474i) q^{75} -4.24264 q^{76} +12.0000 q^{78} +(4.00000 + 6.92820i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-1.41421 - 2.44949i) q^{82} -12.7279 q^{83} +(-5.00000 - 8.66025i) q^{86} +(5.65685 - 9.79796i) q^{87} +(0.500000 - 0.866025i) q^{88} +(-3.53553 - 6.12372i) q^{89} +6.00000 q^{92} +(10.0000 + 17.3205i) q^{93} +(6.36396 - 11.0227i) q^{94} +(-1.41421 - 2.44949i) q^{96} -7.07107 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 10 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 - 10 * q^9	\( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 10 q^{9} + 2 q^{11} - 2 q^{16} - 10 q^{18} - 4 q^{22} - 12 q^{23} + 10 q^{25} - 16 q^{29} - 2 q^{32} + 20 q^{36} - 4 q^{37} - 24 q^{39} + 40 q^{43} + 2 q^{44} - 12 q^{46} - 20 q^{50} + 16 q^{51} - 4 q^{53} - 48 q^{57} + 8 q^{58} + 4 q^{64} - 16 q^{67} + 64 q^{71} - 10 q^{72} - 4 q^{74} + 48 q^{78} + 16 q^{79} - 2 q^{81} - 20 q^{86} + 2 q^{88} + 24 q^{92} + 40 q^{93} - 20 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 - 10 * q^9 + 2 * q^11 - 2 * q^16 - 10 * q^18 - 4 * q^22 - 12 * q^23 + 10 * q^25 - 16 * q^29 - 2 * q^32 + 20 * q^36 - 4 * q^37 - 24 * q^39 + 40 * q^43 + 2 * q^44 - 12 * q^46 - 20 * q^50 + 16 * q^51 - 4 * q^53 - 48 * q^57 + 8 * q^58 + 4 * q^64 - 16 * q^67 + 64 * q^71 - 10 * q^72 - 4 * q^74 + 48 * q^78 + 16 * q^79 - 2 * q^81 - 20 * q^86 + 2 * q^88 + 24 * q^92 + 40 * q^93 - 20 * q^99

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