LMFDB - Embedded newform 810.3.j.a.269.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(269,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.j (of order $6$, 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:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.81622204416.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 165x^{4} - 434x^{2} + 961 $ x^8 - 14x^6 + 165x^4 - 434*x^2 + 961
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.4
Root			$2.90379 - 1.67650i$ of defining polynomial
Character	$\chi$	$=$	810.269
Dual form			810.3.j.a.539.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(3.31368 - 3.74426i) q^{5} +(0.704577 + 0.406788i) q^{7} -2.82843 q^{8} +O(q^{10})$	\(q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(3.31368 - 3.74426i) q^{5} +(0.704577 + 0.406788i) q^{7} -2.82843 q^{8} +(-2.24264 - 6.70601i) q^{10} +(-13.3161 - 7.68808i) q^{11} +(-5.10300 + 2.94622i) q^{13} +(0.996422 - 0.575285i) q^{14} +(-2.00000 + 3.46410i) q^{16} -12.8995 q^{17} +1.24264 q^{19} +(-9.79894 - 1.99520i) q^{20} +(-18.8319 + 10.8726i) q^{22} +(2.39949 + 4.15605i) q^{23} +(-3.03902 - 24.8146i) q^{25} +8.33316i q^{26} -1.62715i q^{28} +(-36.9592 - 21.3384i) q^{29} +(-2.10660 - 3.64874i) q^{31} +(2.82843 + 4.89898i) q^{32} +(-9.12132 + 15.7986i) q^{34} +(3.85786 - 1.29016i) q^{35} +70.3144i q^{37} +(0.878680 - 1.52192i) q^{38} +(-9.37251 + 10.5904i) q^{40} +(6.09942 - 3.52150i) q^{41} +(-35.5500 - 20.5248i) q^{43} +30.7523i q^{44} +6.78680 q^{46} +(39.8995 - 69.1080i) q^{47} +(-24.1690 - 41.8620i) q^{49} +(-32.5405 - 13.8245i) q^{50} +(10.2060 + 5.89243i) q^{52} +63.7279 q^{53} +(-72.9117 + 24.3833i) q^{55} +(-1.99284 - 1.15057i) q^{56} +(-52.2682 + 30.1770i) q^{58} +(-31.7353 + 18.3224i) q^{59} +(41.4706 - 71.8291i) q^{61} -5.95837 q^{62} +8.00000 q^{64} +(-5.87830 + 28.8698i) q^{65} +(-77.0786 + 44.5013i) q^{67} +(12.8995 + 22.3426i) q^{68} +(1.14781 - 5.63718i) q^{70} -69.6982i q^{71} +89.6188i q^{73} +(86.1172 + 49.7198i) q^{74} +(-1.24264 - 2.15232i) q^{76} +(-6.25483 - 10.8337i) q^{77} +(-67.0477 + 116.130i) q^{79} +(6.34315 + 18.9675i) q^{80} -9.96031i q^{82} +(-54.5122 + 94.4179i) q^{83} +(-42.7448 + 48.2991i) q^{85} +(-50.2753 + 29.0265i) q^{86} +(37.6638 + 21.7452i) q^{88} -137.514i q^{89} -4.79394 q^{91} +(4.79899 - 8.31209i) q^{92} +(-56.4264 - 97.7334i) q^{94} +(4.11772 - 4.65277i) q^{95} +(-78.4877 - 45.3149i) q^{97} -68.3604 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 12 q^{5}+O(q^{10}) $ 8 * q - 8 * q^4 - 12 * q^5	$ 8 q - 8 q^{4} - 12 q^{5} + 16 q^{10} - 16 q^{16} - 24 q^{17} - 24 q^{19} - 24 q^{20} - 60 q^{23} + 12 q^{25} + 68 q^{31} - 56 q^{34} + 144 q^{35} + 24 q^{38} - 16 q^{40} + 224 q^{46} + 240 q^{47} + 180 q^{49} - 96 q^{50} + 408 q^{53} - 176 q^{55} + 196 q^{61} - 240 q^{62} + 64 q^{64} + 24 q^{65} + 24 q^{68} - 80 q^{70} + 24 q^{76} + 312 q^{77} - 180 q^{79} + 96 q^{80} - 108 q^{83} - 20 q^{85} + 912 q^{91} - 120 q^{92} - 112 q^{94} + 60 q^{95} - 1056 q^{98}+O(q^{100}) $ 8 * q - 8 * q^4 - 12 * q^5 + 16 * q^10 - 16 * q^16 - 24 * q^17 - 24 * q^19 - 24 * q^20 - 60 * q^23 + 12 * q^25 + 68 * q^31 - 56 * q^34 + 144 * q^35 + 24 * q^38 - 16 * q^40 + 224 * q^46 + 240 * q^47 + 180 * q^49 - 96 * q^50 + 408 * q^53 - 176 * q^55 + 196 * q^61 - 240 * q^62 + 64 * q^64 + 24 * q^65 + 24 * q^68 - 80 * q^70 + 24 * q^76 + 312 * q^77 - 180 * q^79 + 96 * q^80 - 108 * q^83 - 20 * q^85 + 912 * q^91 - 120 * q^92 - 112 * q^94 + 60 * q^95 - 1056 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$	0.707107	−	1.22474i	0.353553	−	0.612372i
$2$	0.707107	−	1.22474i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−1.00000	−	1.73205i	−0.250000	−	0.433013i
$5$	3.31368	−	3.74426i	0.662736	−	0.748853i
$5$	3.31368	−	3.74426i	0.662736	−	0.748853i
$6$		0			0
$7$	0.704577	+	0.406788i	0.100654	+	0.0581125i	0.549482	−	0.835506i	$-0.314825\pi$
$7$	0.704577	+	0.406788i	0.100654	+	0.0581125i	−0.448828	+	0.893618i	$0.648158\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$	−2.24264	−	6.70601i	−0.224264	−	0.670601i
$11$	−13.3161	−	7.68808i	−1.21056	−	0.698917i	−0.247678	−	0.968842i	$-0.579667\pi$
$11$	−13.3161	−	7.68808i	−1.21056	−	0.698917i	−0.962881	+	0.269926i	$0.913001\pi$
$12$		0			0
$13$	−5.10300	+	2.94622i	−0.392538	+	0.226632i	−0.683259	−	0.730176i	$-0.739438\pi$
$13$	−5.10300	+	2.94622i	−0.392538	+	0.226632i	0.290721	+	0.956808i	$0.406105\pi$
$14$	0.996422	−	0.575285i	0.0711730	−	0.0410918i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.125000	+	0.216506i
$17$	−12.8995			−0.758794			−0.379397	−	0.925234i	$-0.623869\pi$
$17$	−12.8995			−0.758794			−0.379397	+	0.925234i	$0.623869\pi$
$18$		0			0
$19$	1.24264			0.0654021			0.0327011	−	0.999465i	$-0.489589\pi$
$19$	1.24264			0.0654021			0.0327011	+	0.999465i	$0.489589\pi$
$20$	−9.79894	−	1.99520i	−0.489947	−	0.0997601i
$21$		0			0
$22$	−18.8319	+	10.8726i	−0.855994	+	0.494209i
$23$	2.39949	+	4.15605i	0.104326	+	0.180698i	0.913463	−	0.406923i	$-0.133398\pi$
$23$	2.39949	+	4.15605i	0.104326	+	0.180698i	−0.809137	+	0.587620i	$0.800065\pi$
$24$		0			0
$25$	−3.03902	−	24.8146i	−0.121561	−	0.992584i
$26$			8.33316i			0.320506i
$27$		0			0
$28$		−	1.62715i		−	0.0581125i
$29$	−36.9592	−	21.3384i	−1.27445	−	0.735807i	−0.298631	−	0.954369i	$-0.596530\pi$
$29$	−36.9592	−	21.3384i	−1.27445	−	0.735807i	−0.975823	+	0.218562i	$0.929863\pi$
$30$		0			0
$31$	−2.10660	−	3.64874i	−0.0679549	−	0.117701i	0.830046	−	0.557695i	$-0.188314\pi$
$31$	−2.10660	−	3.64874i	−0.0679549	−	0.117701i	−0.898001	+	0.439994i	$0.854981\pi$
$32$	2.82843	+	4.89898i	0.0883883	+	0.153093i
$33$		0			0
$34$	−9.12132	+	15.7986i	−0.268274	+	0.464664i
$35$	3.85786	−	1.29016i	0.110225	−	0.0368616i
$36$		0			0
$37$			70.3144i			1.90039i	0.311657	+	0.950195i	$0.399116\pi$
$37$			70.3144i			1.90039i	−0.311657	+	0.950195i	$0.600884\pi$
$38$	0.878680	−	1.52192i	0.0231231	−	0.0400505i
$39$		0			0
$40$	−9.37251	+	10.5904i	−0.234313	+	0.264759i
$41$	6.09942	−	3.52150i	0.148766	−	0.0858903i	−0.423769	−	0.905770i	$-0.639293\pi$
$41$	6.09942	−	3.52150i	0.148766	−	0.0858903i	0.572535	+	0.819880i	$0.305960\pi$
$42$		0			0
$43$	−35.5500	−	20.5248i	−0.826745	−	0.477321i	0.0259920	−	0.999662i	$-0.491726\pi$
$43$	−35.5500	−	20.5248i	−0.826745	−	0.477321i	−0.852737	+	0.522341i	$0.825059\pi$
$44$			30.7523i			0.698917i
$45$		0			0
$46$	6.78680			0.147539
$47$	39.8995	−	69.1080i	0.848925	−	1.47038i	−0.0332427	−	0.999447i	$-0.510583\pi$
$47$	39.8995	−	69.1080i	0.848925	−	1.47038i	0.882168	−	0.470935i	$-0.156083\pi$
$48$		0			0
$49$	−24.1690	−	41.8620i	−0.493246	−	0.854327i
$50$	−32.5405	−	13.8245i	−0.650809	−	0.276491i
$51$		0			0
$52$	10.2060	+	5.89243i	0.196269	+	0.113316i
$53$	63.7279			1.20241			0.601207	−	0.799093i	$-0.294687\pi$
$53$	63.7279			1.20241			0.601207	+	0.799093i	$0.294687\pi$
$54$		0			0
$55$	−72.9117	+	24.3833i	−1.32567	+	0.443333i
$56$	−1.99284	−	1.15057i	−0.0355865	−	0.0205459i
$57$		0			0
$58$	−52.2682	+	30.1770i	−0.901175	+	0.520294i
$59$	−31.7353	+	18.3224i	−0.537886	+	0.310549i	−0.744222	−	0.667932i	$-0.767179\pi$
$59$	−31.7353	+	18.3224i	−0.537886	+	0.310549i	0.206336	+	0.978481i	$0.433846\pi$
$60$		0			0
$61$	41.4706	−	71.8291i	0.679845	−	1.17753i	−0.295182	−	0.955441i	$-0.595380\pi$
$61$	41.4706	−	71.8291i	0.679845	−	1.17753i	0.975027	−	0.222086i	$-0.0712864\pi$
$62$	−5.95837			−0.0961027
$63$		0			0
$64$	8.00000			0.125000
$65$	−5.87830	+	28.8698i	−0.0904354	+	0.444151i
$66$		0			0
$67$	−77.0786	+	44.5013i	−1.15043	+	0.664199i	−0.948991	−	0.315303i	$-0.897894\pi$
$67$	−77.0786	+	44.5013i	−1.15043	+	0.664199i	−0.201435	+	0.979502i	$0.564561\pi$
$68$	12.8995	+	22.3426i	0.189698	+	0.328567i
$69$		0			0
$70$	1.14781	−	5.63718i	0.0163973	−	0.0805311i
$71$		−	69.6982i		−	0.981665i	−0.871254	−	0.490833i	$-0.836693\pi$
$71$		−	69.6982i		−	0.981665i	0.871254	−	0.490833i	$-0.163307\pi$
$72$		0			0
$73$			89.6188i			1.22766i	0.789440	+	0.613828i	$0.210371\pi$
$73$			89.6188i			1.22766i	−0.789440	+	0.613828i	$0.789629\pi$
$74$	86.1172	+	49.7198i	1.16375	+	0.671889i
$75$		0			0
$76$	−1.24264	−	2.15232i	−0.0163505	−	0.0283200i
$77$	−6.25483	−	10.8337i	−0.0812316	−	0.140697i
$78$		0			0
$79$	−67.0477	+	116.130i	−0.848705	+	1.47000i	0.0336586	+	0.999433i	$0.489284\pi$
$79$	−67.0477	+	116.130i	−0.848705	+	1.47000i	−0.882364	+	0.470567i	$0.844049\pi$
$80$	6.34315	+	18.9675i	0.0792893	+	0.237093i
$81$		0			0
$82$		−	9.96031i		−	0.121467i
$83$	−54.5122	+	94.4179i	−0.656773	+	1.13756i	0.324673	+	0.945826i	$0.394746\pi$
$83$	−54.5122	+	94.4179i	−0.656773	+	1.13756i	−0.981446	+	0.191738i	$0.938587\pi$
$84$		0			0
$85$	−42.7448	+	48.2991i	−0.502880	+	0.568225i
$86$	−50.2753	+	29.0265i	−0.584597	+	0.337517i
$87$		0			0
$88$	37.6638	+	21.7452i	0.427997	+	0.247104i
$89$		−	137.514i		−	1.54510i	−0.634953	−	0.772551i	$-0.718980\pi$
$89$		−	137.514i		−	1.54510i	0.634953	−	0.772551i	$-0.281020\pi$
$90$		0			0
$91$	−4.79394			−0.0526807
$92$	4.79899	−	8.31209i	0.0521629	−	0.0903489i
$93$		0			0
$94$	−56.4264	−	97.7334i	−0.600281	−	1.03972i
$95$	4.11772	−	4.65277i	0.0433444	−	0.0489766i
$96$		0			0
$97$	−78.4877	−	45.3149i	−0.809152	−	0.467164i	0.0375094	−	0.999296i	$-0.488058\pi$
$97$	−78.4877	−	45.3149i	−0.809152	−	0.467164i	−0.846661	+	0.532132i	$0.821391\pi$
$98$	−68.3604			−0.697555
$99$		0			0
$100$	−39.9411	+	30.0783i	−0.399411	+	0.300783i
$101$	43.0586	+	24.8599i	0.426323	+	0.246138i	0.697779	−	0.716313i	$-0.254172\pi$
$101$	43.0586	+	24.8599i	0.426323	+	0.246138i	−0.271456	+	0.962451i	$0.587505\pi$
$102$		0			0
$103$	−111.582	+	64.4220i	−1.08332	+	0.625456i	−0.931790	−	0.362997i	$-0.881754\pi$
$103$	−111.582	+	64.4220i	−1.08332	+	0.625456i	−0.151531	+	0.988453i	$0.548420\pi$
$104$	14.4335	−	8.33316i	0.138783	−	0.0801265i
$105$		0			0
$106$	45.0624	−	78.0504i	0.425117	−	0.736325i
$107$	6.42641			0.0600599			0.0300299	−	0.999549i	$-0.490440\pi$
$107$	6.42641			0.0600599			0.0300299	+	0.999549i	$0.490440\pi$
$108$		0			0
$109$	108.279			0.993387			0.496694	−	0.867926i	$-0.334547\pi$
$109$	108.279			0.993387			0.496694	+	0.867926i	$0.334547\pi$
$110$	−21.6930	+	106.540i	−0.197209	+	0.968544i
$111$		0			0
$112$	−2.81831	+	1.62715i	−0.0251635	+	0.0145281i
$113$	15.2132	+	26.3500i	0.134630	+	0.233186i	0.925456	−	0.378855i	$-0.123682\pi$
$113$	15.2132	+	26.3500i	0.134630	+	0.233186i	−0.790826	+	0.612041i	$0.790349\pi$
$114$		0			0
$115$	23.5125	+	4.78748i	0.204457	+	0.0416302i
$116$			85.3536i			0.735807i
$117$		0			0
$118$			51.8235i			0.439182i
$119$	−9.08869	−	5.24736i	−0.0763755	−	0.0440954i
$120$		0			0
$121$	57.7132	+	99.9622i	0.476969	+	0.826134i
$122$	−58.6482	−	101.582i	−0.480723	−	0.832637i
$123$		0			0
$124$	−4.21320	+	7.29748i	−0.0339774	+	0.0588507i
$125$	−102.983	−	70.8488i	−0.823862	−	0.566790i
$126$		0			0
$127$		−	133.725i		−	1.05296i	−0.850189	−	0.526478i	$-0.823512\pi$
$127$		−	133.725i		−	1.05296i	0.850189	−	0.526478i	$-0.176488\pi$
$128$	5.65685	−	9.79796i	0.0441942	−	0.0765466i
$129$		0			0
$130$	31.2015	+	27.6134i	0.240012	+	0.212411i
$131$	53.5064	−	30.8919i	0.408446	−	0.235816i	−0.281676	−	0.959510i	$-0.590890\pi$
$131$	53.5064	−	30.8919i	0.408446	−	0.235816i	0.690122	+	0.723693i	$0.257557\pi$
$132$		0			0
$133$	0.875536	+	0.505491i	0.00658298	+	0.00380068i
$134$			125.869i			0.939319i
$135$		0			0
$136$	36.4853			0.268274
$137$	81.9594	−	141.958i	0.598244	−	1.03619i	−0.394836	−	0.918751i	$-0.629199\pi$
$137$	81.9594	−	141.958i	0.598244	−	1.03619i	0.993080	−	0.117437i	$-0.0374679\pi$
$138$		0			0
$139$	−91.1249	−	157.833i	−0.655575	−	1.13549i	−0.981749	−	0.190179i	$-0.939093\pi$
$139$	−91.1249	−	157.833i	−0.655575	−	1.13549i	0.326175	−	0.945310i	$-0.394240\pi$
$140$	−6.09248	−	5.39186i	−0.0435177	−	0.0385133i
$141$		0			0
$142$	−85.3625	−	49.2841i	−0.601145	−	0.347071i
$143$	90.6030			0.633588
$144$		0			0
$145$	−202.368	+	67.6763i	−1.39564	+	0.466733i
$146$	109.760	+	63.3701i	0.751782	+	0.434042i
$147$		0			0
$148$	121.788	−	70.3144i	0.822893	−	0.475097i
$149$	164.142	−	94.7675i	1.10163	−	0.636023i	0.164978	−	0.986297i	$-0.447245\pi$
$149$	164.142	−	94.7675i	1.10163	−	0.636023i	0.936647	+	0.350274i	$0.113911\pi$
$150$		0			0
$151$	56.6690	−	98.1537i	0.375292	−	0.650024i	−0.615079	−	0.788466i	$-0.710876\pi$
$151$	56.6690	−	98.1537i	0.375292	−	0.650024i	0.990371	+	0.138441i	$0.0442092\pi$
$152$	−3.51472			−0.0231231
$153$		0			0
$154$	−17.6913			−0.114879
$155$	−20.6425	−	4.20310i	−0.133177	−	0.0271167i
$156$		0			0
$157$	98.0242	−	56.5943i	0.624358	−	0.360473i	−0.154206	−	0.988039i	$-0.549282\pi$
$157$	98.0242	−	56.5943i	0.624358	−	0.360473i	0.778564	+	0.627566i	$0.215949\pi$
$158$	94.8198	+	164.233i	0.600125	+	1.03945i
$159$		0			0
$160$	27.7156	+	5.64328i	0.173222	+	0.0352705i
$161$			3.90434i			0.0242506i
$162$		0			0
$163$		−	152.414i		−	0.935053i	−0.883979	−	0.467527i	$-0.845145\pi$
$163$		−	152.414i		−	0.935053i	0.883979	−	0.467527i	$-0.154855\pi$
$164$	−12.1988	−	7.04300i	−0.0743832	−	0.0429451i
$165$		0			0
$166$	77.0919	+	133.527i	0.464409	+	0.804380i
$167$	−52.9264	−	91.6712i	−0.316925	−	0.548929i	0.662920	−	0.748690i	$-0.269317\pi$
$167$	−52.9264	−	91.6712i	−0.316925	−	0.548929i	−0.979845	+	0.199761i	$0.935984\pi$
$168$		0			0
$169$	−67.1396	+	116.289i	−0.397276	+	0.688102i
$170$	28.9289	+	86.5041i	0.170170	+	0.508848i
$171$		0			0
$172$			82.0993i			0.477321i
$173$	−55.0061	+	95.2734i	−0.317954	+	0.550713i	−0.980061	−	0.198697i	$-0.936329\pi$
$173$	−55.0061	+	95.2734i	−0.317954	+	0.550713i	0.662107	+	0.749410i	$0.269663\pi$
$174$		0			0
$175$	7.95305	−	18.7200i	0.0454460	−	0.106972i
$176$	53.2646	−	30.7523i	0.302640	−	0.174729i
$177$		0			0
$178$	−168.420	−	97.2371i	−0.946178	−	0.546276i
$179$			118.407i			0.661492i	0.943720	+	0.330746i	$0.107300\pi$
$179$			118.407i			0.661492i	−0.943720	+	0.330746i	$0.892700\pi$
$180$		0			0
$181$	172.397			0.952469			0.476235	−	0.879318i	$-0.342001\pi$
$181$	172.397			0.952469			0.476235	+	0.879318i	$0.342001\pi$
$182$	−3.38983	+	5.87135i	−0.0186254	+	0.0322602i
$183$		0			0
$184$	−6.78680	−	11.7551i	−0.0368848	−	0.0638863i
$185$	263.276	+	233.000i	1.42311	+	1.25946i
$186$		0			0
$187$	171.772	+	99.1724i	0.918565	+	0.530334i
$188$	−159.598			−0.848925
$189$		0			0
$190$	−2.78680	−	8.33316i	−0.0146674	−	0.0438587i
$191$	−250.622	−	144.697i	−1.31216	−	0.757574i	−0.329704	−	0.944084i	$-0.606949\pi$
$191$	−250.622	−	144.697i	−1.31216	−	0.757574i	−0.982453	+	0.186510i	$0.940282\pi$
$192$		0			0
$193$	110.707	−	63.9165i	0.573609	−	0.331173i	−0.184980	−	0.982742i	$-0.559222\pi$
$193$	110.707	−	63.9165i	0.573609	−	0.331173i	0.758590	+	0.651569i	$0.225889\pi$
$194$	−110.998	+	64.0850i	−0.572157	+	0.330335i
$195$		0			0
$196$	−48.3381	+	83.7240i	−0.246623	+	0.427163i
$197$	280.414			1.42342			0.711711	−	0.702472i	$-0.247920\pi$
$197$	280.414			1.42342			0.711711	+	0.702472i	$0.247920\pi$
$198$		0			0
$199$	156.426			0.786062			0.393031	−	0.919525i	$-0.371426\pi$
$199$	156.426			0.786062			0.393031	+	0.919525i	$0.371426\pi$
$200$	8.59565	+	70.1863i	0.0429783	+	0.350931i
$201$		0			0
$202$	60.8941	−	35.1572i	0.301456	−	0.174046i
$203$	−17.3604	−	30.0691i	−0.0855192	−	0.148124i
$204$		0			0
$205$	7.02611	−	34.5070i	0.0342737	−	0.168327i
$206$			182.213i			0.884528i
$207$		0			0
$208$		−	23.5697i		−	0.113316i
$209$	−16.5472	−	9.55352i	−0.0791731	−	0.0457106i
$210$		0			0
$211$	−11.2538	−	19.4921i	−0.0533355	−	0.0923798i	0.838125	−	0.545478i	$-0.183652\pi$
$211$	−11.2538	−	19.4921i	−0.0533355	−	0.0923798i	−0.891460	+	0.453098i	$0.850319\pi$
$212$	−63.7279	−	110.380i	−0.300603	−	0.520660i
$213$		0			0
$214$	4.54416	−	7.87071i	0.0212344	−	0.0367790i
$215$	−194.652	+	65.0959i	−0.905357	+	0.302772i
$216$		0			0
$217$		−	3.42776i		−	0.0157961i
$218$	76.5650	−	132.614i	0.351215	−	0.608323i
$219$		0			0
$220$	115.145	+	101.903i	0.523386	+	0.463197i
$221$	65.8261	−	38.0047i	0.297856	−	0.171967i
$222$		0			0
$223$	307.289	+	177.413i	1.37798	+	0.795575i	0.991916	−	0.126899i	$-0.0405024\pi$
$223$	307.289	+	177.413i	1.37798	+	0.795575i	0.386060	+	0.922474i	$0.373836\pi$
$224$			4.60228i			0.0205459i
$225$		0			0
$226$	43.0294			0.190396
$227$	207.549	−	359.485i	0.914312	−	1.58363i	0.106406	−	0.994323i	$-0.466066\pi$
$227$	207.549	−	359.485i	0.914312	−	1.58363i	0.807906	−	0.589312i	$-0.200601\pi$
$228$		0			0
$229$	110.110	+	190.716i	0.480830	+	0.832823i	0.999758	−	0.0219954i	$-0.00700192\pi$
$229$	110.110	+	190.716i	0.480830	+	0.832823i	−0.518928	+	0.854818i	$0.673669\pi$
$230$	22.4893	−	25.4116i	0.0977795	−	0.110485i
$231$		0			0
$232$	104.536	+	60.3541i	0.450588	+	0.260147i
$233$	10.0345			0.0430665			0.0215332	−	0.999768i	$-0.493145\pi$
$233$	10.0345			0.0430665			0.0215332	+	0.999768i	$0.493145\pi$
$234$		0			0
$235$	−126.544	−	378.396i	−0.538486	−	1.61020i
$236$	63.4706	+	36.6448i	0.268943	+	0.155274i
$237$		0			0
$238$	−12.8533	+	7.42088i	−0.0540056	+	0.0311802i
$239$	−99.0414	+	57.1816i	−0.414399	+	0.239253i	−0.692678	−	0.721247i	$-0.743569\pi$
$239$	−99.0414	+	57.1816i	−0.414399	+	0.239253i	0.278279	+	0.960500i	$0.410236\pi$
$240$		0			0
$241$	31.8604	−	55.1838i	0.132201	−	0.228978i	−0.792324	−	0.610101i	$-0.791129\pi$
$241$	31.8604	−	55.1838i	0.132201	−	0.228978i	0.924525	+	0.381122i	$0.124462\pi$
$242$	163.238			0.674535
$243$		0			0
$244$	−165.882			−0.679845
$245$	−236.831	−	48.2221i	−0.966657	−	0.196825i
$246$		0			0
$247$	−6.34119	+	3.66109i	−0.0256728	+	0.0148222i
$248$	5.95837	+	10.3202i	0.0240257	+	0.0416137i
$249$		0			0
$250$	−159.592	+	76.0299i	−0.638366	+	0.304120i
$251$		−	229.631i		−	0.914866i	−0.889244	−	0.457433i	$-0.848769\pi$
$251$		−	229.631i		−	0.914866i	0.889244	−	0.457433i	$-0.151231\pi$
$252$		0			0
$253$		−	73.7901i		−	0.291660i
$254$	−163.779	−	94.5581i	−0.644801	−	0.372276i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.0312500	−	0.0541266i
$257$	−194.114	−	336.215i	−0.755306	−	1.30823i	−0.945222	−	0.326429i	$-0.894155\pi$
$257$	−194.114	−	336.215i	−0.755306	−	1.30823i	0.189915	−	0.981800i	$-0.439179\pi$
$258$		0			0
$259$	−28.6030	+	49.5419i	−0.110436	+	0.191281i
$260$	55.8823	−	18.6883i	0.214932	−	0.0718780i
$261$		0			0
$262$		−	87.3755i		−	0.333494i
$263$	−113.439	+	196.481i	−0.431325	+	0.747078i	−0.996988	−	0.0775594i	$-0.975287\pi$
$263$	−113.439	+	196.481i	−0.431325	+	0.747078i	0.565662	+	0.824637i	$0.308621\pi$
$264$		0			0
$265$	211.174	−	238.614i	0.796883	−	0.900431i
$266$	1.23819	−	0.714872i	0.00465487	−	0.00268749i
$267$		0			0
$268$	154.157	+	89.0027i	0.575213	+	0.332100i
$269$		−	226.772i		−	0.843019i	−0.906824	−	0.421509i	$-0.861500\pi$
$269$		−	226.772i		−	0.843019i	0.906824	−	0.421509i	$-0.138500\pi$
$270$		0			0
$271$	322.889			1.19147			0.595737	−	0.803180i	$-0.296860\pi$
$271$	322.889			1.19147			0.595737	+	0.803180i	$0.296860\pi$
$272$	25.7990	−	44.6852i	0.0948492	−	0.164284i
$273$		0			0
$274$	−115.908	−	200.759i	−0.423022	−	0.732696i
$275$	−150.309	+	353.799i	−0.546577	+	1.28654i
$276$		0			0
$277$	173.694	+	100.282i	0.627053	+	0.362029i	0.779610	−	0.626266i	$-0.215417\pi$
$277$	173.694	+	100.282i	0.627053	+	0.362029i	−0.152557	+	0.988295i	$0.548751\pi$
$278$	−257.740			−0.927123
$279$		0			0
$280$	−10.9117	+	3.64911i	−0.0389703	+	0.0130326i
$281$	110.878	+	64.0152i	0.394582	+	0.227812i	0.684144	−	0.729347i	$-0.260176\pi$
$281$	110.878	+	64.0152i	0.394582	+	0.227812i	−0.289562	+	0.957159i	$0.593510\pi$
$282$		0			0
$283$	363.934	−	210.118i	1.28599	−	0.742465i	0.308051	−	0.951370i	$-0.400323\pi$
$283$	363.934	−	210.118i	1.28599	−	0.742465i	0.977936	+	0.208905i	$0.0669898\pi$
$284$	−120.721	+	69.6982i	−0.425073	+	0.245416i
$285$		0			0
$286$	64.0660	−	110.966i	0.224007	−	0.387992i
$287$	5.73001			0.0199652
$288$		0			0
$289$	−122.603			−0.424232
$290$	−60.2093	+	295.703i	−0.207618	+	1.01967i
$291$		0			0
$292$	155.224	−	89.6188i	0.531590	−	0.306914i
$293$	−68.7168	−	119.021i	−0.234528	−	0.406215i	0.724607	−	0.689162i	$-0.242021\pi$
$293$	−68.7168	−	119.021i	−0.234528	−	0.406215i	−0.959135	+	0.282947i	$0.908688\pi$
$294$		0			0
$295$	−36.5569	+	179.540i	−0.123922	+	0.608610i
$296$		−	198.879i		−	0.671889i
$297$		0			0
$298$		−	268.043i		−	0.899473i
$299$	−24.4892	−	14.1389i	−0.0819038	−	0.0472872i
$300$		0			0
$301$	−16.6985	−	28.9226i	−0.0554767	−	0.0960885i
$302$	−80.1421	−	138.810i	−0.265371	−	0.459637i
$303$		0			0
$304$	−2.48528	+	4.30463i	−0.00817527	+	0.0141600i
$305$	−131.527	−	393.296i	−0.431236	−	1.28949i
$306$		0			0
$307$			117.849i			0.383872i	0.981407	+	0.191936i	$0.0614766\pi$
$307$			117.849i			0.383872i	−0.981407	+	0.191936i	$0.938523\pi$
$308$	−12.5097	+	21.6674i	−0.0406158	+	0.0703486i
$309$		0			0
$310$	−19.7441	+	22.3097i	−0.0636908	+	0.0719668i
$311$	−432.429	+	249.663i	−1.39045	+	0.802774i	−0.993364	−	0.115009i	$-0.963310\pi$
$311$	−432.429	+	249.663i	−1.39045	+	0.802774i	−0.397081	+	0.917783i	$0.629977\pi$
$312$		0			0
$313$	−425.916	−	245.903i	−1.36076	−	0.785633i	−0.371031	−	0.928620i	$-0.620996\pi$
$313$	−425.916	−	245.903i	−1.36076	−	0.785633i	−0.989724	+	0.142988i	$0.954329\pi$
$314$		−	160.073i		−	0.509786i
$315$		0			0
$316$	268.191			0.848705
$317$	120.700	−	209.058i	0.380756	−	0.659488i	−0.610415	−	0.792082i	$-0.708997\pi$
$317$	120.700	−	209.058i	0.380756	−	0.659488i	0.991170	+	0.132594i	$0.0423306\pi$
$318$		0			0
$319$	328.103	+	568.290i	1.02853	+	1.78147i
$320$	26.5095	−	29.9541i	0.0828421	−	0.0936066i
$321$		0			0
$322$	4.78182	+	2.76079i	0.0148504	+	0.00857387i
$323$	−16.0294			−0.0496267
$324$		0			0
$325$	88.6173	+	117.675i	0.272669	+	0.362078i
$326$	−186.668	−	107.773i	−0.572601	−	0.330591i
$327$		0			0
$328$	−17.2518	+	9.96031i	−0.0525968	+	0.0303668i
$329$	56.2245	−	32.4612i	0.170895	−	0.0986664i
$330$		0			0
$331$	43.9045	−	76.0449i	0.132642	−	0.229743i	−0.792052	−	0.610453i	$-0.790987\pi$
$331$	43.9045	−	76.0449i	0.132642	−	0.229743i	0.924694	+	0.380711i	$0.124321\pi$
$332$	218.049			0.656773
$333$		0			0
$334$	−149.698			−0.448199
$335$	−88.7892	+	436.066i	−0.265042	+	1.30169i
$336$		0			0
$337$	391.563	−	226.069i	1.16191	−	0.670828i	0.210148	−	0.977670i	$-0.432605\pi$
$337$	391.563	−	226.069i	1.16191	−	0.670828i	0.951761	+	0.306841i	$0.0992721\pi$
$338$	94.9497	+	164.458i	0.280916	+	0.486561i
$339$		0			0
$340$	126.401	+	25.7371i	0.371769	+	0.0756974i
$341$			64.7829i			0.189979i
$342$		0			0
$343$		−	79.1919i		−	0.230880i
$344$	100.551	+	58.0529i	0.292298	+	0.168759i
$345$		0			0
$346$	77.7904	+	134.737i	0.224828	+	0.389413i
$347$	−94.7056	−	164.035i	−0.272927	−	0.472723i	0.696683	−	0.717379i	$-0.254658\pi$
$347$	−94.7056	−	164.035i	−0.272927	−	0.472723i	−0.969610	+	0.244656i	$0.921325\pi$
$348$		0			0
$349$	77.8970	−	134.922i	0.223200	−	0.386595i	−0.732578	−	0.680684i	$-0.761683\pi$
$349$	77.8970	−	134.922i	0.223200	−	0.386595i	0.955778	+	0.294089i	$0.0950162\pi$
$350$	−17.3036	−	22.9775i	−0.0494389	−	0.0656500i
$351$		0			0
$352$		−	86.9807i		−	0.247104i
$353$	−131.693	+	228.100i	−0.373069	+	0.646175i	−0.990036	−	0.140814i	$-0.955028\pi$
$353$	−131.693	+	228.100i	−0.373069	+	0.646175i	0.616967	+	0.786989i	$0.288361\pi$
$354$		0			0
$355$	−260.969	−	230.958i	−0.735123	−	0.650585i
$356$	−238.181	+	137.514i	−0.669049	+	0.386275i
$357$		0			0
$358$	145.018	+	83.7264i	0.405079	+	0.233873i
$359$			79.6346i			0.221823i	0.993830	+	0.110912i	$0.0353771\pi$
$359$			79.6346i			0.221823i	−0.993830	+	0.110912i	$0.964623\pi$
$360$		0			0
$361$	−359.456			−0.995723
$362$	121.903	−	211.142i	0.336749	−	0.583266i
$363$		0			0
$364$	4.79394	+	8.30335i	0.0131702	+	0.0228114i
$365$	335.557	+	296.968i	0.919333	+	0.813612i
$366$		0			0
$367$	−412.188	−	237.977i	−1.12313	−	0.648438i	−0.180930	−	0.983496i	$-0.557911\pi$
$367$	−412.188	−	237.977i	−1.12313	−	0.648438i	−0.942197	+	0.335058i	$0.891244\pi$
$368$	−19.1960			−0.0521629
$369$		0			0
$370$	471.529	−	157.690i	1.27440	−	0.426189i
$371$	44.9012	+	25.9237i	0.121028	+	0.0698753i
$372$		0			0
$373$	53.5064	−	30.8919i	0.143449	−	0.0828201i	−0.426558	−	0.904460i	$-0.640274\pi$
$373$	53.5064	−	30.8919i	0.143449	−	0.0828201i	0.570007	+	0.821640i	$0.306941\pi$
$374$	242.922	−	140.251i	0.649523	−	0.375002i
$375$		0			0
$376$	−112.853	+	195.467i	−0.300140	+	0.519859i
$377$	251.470			0.667030
$378$		0			0
$379$	−142.345			−0.375581			−0.187791	−	0.982209i	$-0.560133\pi$
$379$	−142.345			−0.375581			−0.187791	+	0.982209i	$0.560133\pi$
$380$	−12.1766	−	2.47932i	−0.0320436	−	0.00652452i
$381$		0			0
$382$	−354.433	+	204.632i	−0.927835	+	0.535686i
$383$	−178.760	−	309.621i	−0.466736	−	0.808410i	0.532542	−	0.846404i	$-0.321237\pi$
$383$	−178.760	−	309.621i	−0.466736	−	0.808410i	−0.999278	+	0.0379931i	$0.987904\pi$
$384$		0			0
$385$	−61.2907	−	12.4797i	−0.159197	−	0.0324147i
$386$		−	180.783i		−	0.468350i
$387$		0			0
$388$			181.260i			0.467164i
$389$	356.005	+	205.539i	0.915179	+	0.528379i	0.882094	−	0.471074i	$-0.156133\pi$
$389$	356.005	+	205.539i	0.915179	+	0.528379i	0.0330850	+	0.999453i	$0.489467\pi$
$390$		0			0
$391$	−30.9523	−	53.6109i	−0.0791618	−	0.137112i
$392$	68.3604	+	118.404i	0.174389	+	0.302050i
$393$		0			0
$394$	198.283	−	343.436i	0.503256	−	0.871665i
$395$	212.647	+	635.863i	0.538346	+	1.60978i
$396$		0			0
$397$		−	457.462i		−	1.15230i	−0.817345	−	0.576149i	$-0.804555\pi$
$397$		−	457.462i		−	1.15230i	0.817345	−	0.576149i	$-0.195445\pi$
$398$	110.610	−	191.582i	0.277915	−	0.481363i
$399$		0			0
$400$	92.0383	+	39.1017i	0.230096	+	0.0977543i
$401$	404.679	−	233.642i	1.00917	−	0.582647i	0.0982247	−	0.995164i	$-0.468684\pi$
$401$	404.679	−	233.642i	1.00917	−	0.582647i	0.910950	+	0.412517i	$0.135350\pi$
$402$		0			0
$403$	21.5000	+	12.4130i	0.0533498	+	0.0308015i
$404$		−	99.4396i		−	0.246138i
$405$		0			0
$406$	−49.1026			−0.120942
$407$	540.583	−	936.317i	1.32821	−	2.30053i
$408$		0			0
$409$	224.287	+	388.476i	0.548378	+	0.949819i	0.998386	+	0.0567946i	$0.0180880\pi$
$409$	224.287	+	388.476i	0.548378	+	0.949819i	−0.450007	+	0.893025i	$0.648579\pi$
$410$	−37.2940	−	33.0053i	−0.0909611	−	0.0805007i
$411$		0			0
$412$	223.164	+	128.844i	0.541661	+	0.312728i
$413$	−29.8133			−0.0721871
$414$		0			0
$415$	172.889	+	516.979i	0.416601	+	1.24573i
$416$	−28.8669	−	16.6663i	−0.0693916	−	0.0400633i
$417$		0			0
$418$	−23.4013	+	13.5107i	−0.0559839	+	0.0323223i
$419$	−312.492	+	180.417i	−0.745804	+	0.430590i	−0.824176	−	0.566334i	$-0.808361\pi$
$419$	−312.492	+	180.417i	−0.745804	+	0.430590i	0.0783720	+	0.996924i	$0.475028\pi$
$420$		0			0
$421$	−178.529	+	309.222i	−0.424060	+	0.734494i	−0.996332	−	0.0855699i	$-0.972729\pi$
$421$	−178.529	+	309.222i	−0.424060	+	0.734494i	0.572272	+	0.820064i	$0.306062\pi$
$422$	−31.8305			−0.0754278
$423$		0			0
$424$	−180.250			−0.425117
$425$	39.2018	+	320.096i	0.0922396	+	0.753167i
$426$		0			0
$427$	58.4384	−	33.7394i	0.136858	−	0.0790151i
$428$	−6.42641	−	11.1309i	−0.0150150	−	0.0260067i
$429$		0			0
$430$	−57.9137	+	284.429i	−0.134683	+	0.661462i
$431$			655.947i			1.52192i	0.648800	+	0.760959i	$0.275271\pi$
$431$			655.947i			1.52192i	−0.648800	+	0.760959i	$0.724729\pi$
$432$		0			0
$433$			164.372i			0.379612i	0.981822	+	0.189806i	$0.0607859\pi$
$433$			164.372i			0.379612i	−0.981822	+	0.189806i	$0.939214\pi$
$434$	−4.19813	−	2.42379i	−0.00967311	−	0.00558477i
$435$		0			0
$436$	−108.279	−	187.545i	−0.248347	−	0.430149i
$437$	2.98171	+	5.16447i	0.00682314	+	0.0118180i
$438$		0			0
$439$	66.1432	−	114.563i	0.150668	−	0.260964i	−0.780805	−	0.624774i	$-0.785191\pi$
$439$	66.1432	−	114.563i	0.150668	−	0.260964i	0.931473	+	0.363810i	$0.118524\pi$
$440$	206.225	−	68.9664i	0.468694	−	0.156742i
$441$		0			0
$442$		−	107.494i		−	0.243198i
$443$	−227.743	+	394.462i	−0.514092	+	0.890433i	0.485775	+	0.874084i	$0.338538\pi$
$443$	−227.743	+	394.462i	−0.514092	+	0.890433i	−0.999866	+	0.0163489i	$0.994796\pi$
$444$		0			0
$445$	−514.889	−	455.678i	−1.15705	−	1.02400i
$446$	434.572	−	250.900i	0.974376	−	0.562556i
$447$		0			0
$448$	5.63662	+	3.25430i	0.0125817	+	0.00726407i
$449$			415.749i			0.925944i	0.886373	+	0.462972i	$0.153217\pi$
$449$			415.749i			0.925944i	−0.886373	+	0.462972i	$0.846783\pi$
$450$		0			0
$451$	−108.294			−0.240121
$452$	30.4264	−	52.7001i	0.0673151	−	0.116593i
$453$		0			0
$454$	−293.518	−	508.389i	−0.646516	−	1.11980i
$455$	−15.8856	+	17.9498i	−0.0349134	+	0.0394501i
$456$		0			0
$457$	−262.962	−	151.821i	−0.575410	−	0.332213i	0.183897	−	0.982946i	$-0.441129\pi$
$457$	−262.962	−	151.821i	−0.575410	−	0.332213i	−0.759307	+	0.650732i	$0.774462\pi$
$458$	311.439			0.679997
$459$		0			0
$460$	−15.2203	−	45.5123i	−0.0330877	−	0.0989398i
$461$	371.343	+	214.395i	0.805516	+	0.465065i	0.845396	−	0.534140i	$-0.179364\pi$
$461$	371.343	+	214.395i	0.805516	+	0.465065i	−0.0398803	+	0.999204i	$0.512698\pi$
$462$		0			0
$463$	−705.364	+	407.242i	−1.52346	+	0.879572i	−0.523850	+	0.851811i	$0.675505\pi$
$463$	−705.364	+	407.242i	−1.52346	+	0.879572i	−0.999615	+	0.0277616i	$0.991162\pi$
$464$	147.837	−	85.3536i	0.318614	−	0.183952i
$465$		0			0
$466$	7.09545	−	12.2897i	0.0152263	−	0.0263727i
$467$	−607.118			−1.30004			−0.650019	−	0.759918i	$-0.725239\pi$
$467$	−607.118			−1.30004			−0.650019	+	0.759918i	$0.725239\pi$
$468$		0			0
$469$	−72.4104			−0.154393
$470$	−552.919	−	112.582i	−1.17642	−	0.239536i
$471$		0			0
$472$	89.7610	−	51.8235i	0.190172	−	0.109796i
$473$	315.593	+	546.623i	0.667215	+	1.15565i
$474$		0			0
$475$	−3.77641	−	30.8356i	−0.00795034	−	0.0649171i
$476$			20.9894i			0.0440954i
$477$		0			0
$478$			161.734i			0.338355i
$479$	−297.816	−	171.944i	−0.621746	−	0.358965i	0.155802	−	0.987788i	$-0.450204\pi$
$479$	−297.816	−	171.944i	−0.621746	−	0.358965i	−0.777549	+	0.628823i	$0.783537\pi$
$480$		0			0
$481$	−207.161	−	358.814i	−0.430689	−	0.745975i
$482$	−45.0574	−	78.0417i	−0.0934801	−	0.161912i
$483$		0			0
$484$	115.426	−	199.924i	0.238484	−	0.413067i
$485$	−429.754	+	143.720i	−0.886092	+	0.296329i
$486$		0			0
$487$			300.759i			0.617576i	0.951131	+	0.308788i	$0.0999233\pi$
$487$			300.759i			0.617576i	−0.951131	+	0.308788i	$0.900077\pi$
$488$	−117.296	+	203.163i	−0.240362	+	0.416319i
$489$		0			0
$490$	−226.525	+	255.959i	−0.462295	+	0.522366i
$491$	150.614	−	86.9568i	0.306749	−	0.177101i	−0.338722	−	0.940886i	$-0.609995\pi$
$491$	150.614	−	86.9568i	0.306749	−	0.177101i	0.645471	+	0.763785i	$0.276661\pi$
$492$		0			0
$493$	476.755	+	275.254i	0.967048	+	0.558326i
$494$			10.3551i			0.0209618i
$495$		0			0
$496$	16.8528			0.0339774
$497$	28.3524	−	49.1078i	0.0570470	−	0.0988084i
$498$		0			0
$499$	380.805	+	659.574i	0.763136	+	1.32179i	0.941226	+	0.337777i	$0.109675\pi$
$499$	380.805	+	659.574i	0.763136	+	1.32179i	−0.178090	+	0.984014i	$0.556992\pi$
$500$	−19.7310	+	249.220i	−0.0394619	+	0.498440i
$501$		0			0
$502$	−281.240	−	162.374i	−0.560239	−	0.323454i
$503$	−280.632			−0.557917			−0.278959	−	0.960303i	$-0.589989\pi$
$503$	−280.632			−0.557917			−0.278959	+	0.960303i	$0.589989\pi$
$504$		0			0
$505$	235.765	−	78.8450i	0.466860	−	0.156129i
$506$	−90.3740	−	52.1774i	−0.178605	−	0.103117i
$507$		0			0
$508$	−231.619	+	133.725i	−0.455943	+	0.263239i
$509$	548.347	−	316.588i	1.07730	−	0.621981i	0.147135	−	0.989116i	$-0.452995\pi$
$509$	548.347	−	316.588i	1.07730	−	0.621981i	0.930167	+	0.367136i	$0.119662\pi$
$510$		0			0
$511$	−36.4558	+	63.1434i	−0.0713422	+	0.123568i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	−549.037			−1.06816
$515$	−128.535	+	631.267i	−0.249582	+	1.22576i
$516$		0			0
$517$	−1062.62	+	613.501i	−2.05535	+	1.18666i
$518$	40.4508	+	70.0628i	0.0780903	+	0.135256i
$519$		0			0
$520$	16.6263	−	81.6561i	0.0319737	−	0.157031i
$521$		−	92.3966i		−	0.177345i	−0.996061	−	0.0886723i	$-0.971738\pi$
$521$		−	92.3966i		−	0.177345i	0.996061	−	0.0886723i	$-0.0282624\pi$
$522$		0			0
$523$			818.552i			1.56511i	0.622582	+	0.782554i	$0.286084\pi$
$523$			818.552i			1.56511i	−0.622582	+	0.782554i	$0.713916\pi$
$524$	−107.013	−	61.7838i	−0.204223	−	0.117908i
$525$		0			0
$526$	160.426	+	277.867i	0.304993	+	0.528264i
$527$	27.1741	+	47.0669i	0.0515638	+	0.0893110i
$528$		0			0
$529$	252.985	−	438.183i	0.478232	−	0.828323i
$530$	−142.919	−	427.360i	−0.269658	−	0.806340i
$531$		0			0
$532$		−	2.02196i		−	0.00380068i
$533$	−20.7502	+	35.9404i	−0.0389310	+	0.0674304i
$534$		0			0
$535$	21.2951	−	24.0622i	0.0398039	−	0.0449760i
$536$	218.011	−	125.869i	0.406737	−	0.234830i
$537$		0			0
$538$	−277.738	−	160.352i	−0.516241	−	0.298052i
$539$			743.254i			1.37895i
$540$		0			0
$541$	−376.985			−0.696830			−0.348415	−	0.937340i	$-0.613280\pi$
$541$	−376.985			−0.696830			−0.348415	+	0.937340i	$0.613280\pi$
$542$	228.317	−	395.457i	0.421250	−	0.729626i
$543$		0			0
$544$	−36.4853	−	63.1944i	−0.0670685	−	0.116166i
$545$	358.803	−	405.426i	0.658354	−	0.743901i
$546$		0			0
$547$	292.151	+	168.673i	0.534096	+	0.308360i	0.742683	−	0.669643i	$-0.233553\pi$
$547$	292.151	+	168.673i	0.534096	+	0.308360i	−0.208587	+	0.978004i	$0.566886\pi$
$548$	−327.838			−0.598244
$549$		0			0
$550$	327.029	+	434.263i	0.594599	+	0.789570i
$551$	−45.9270	−	26.5160i	−0.0833520	−	0.0481233i
$552$		0			0
$553$	−94.4806	+	54.5484i	−0.170851	+	0.0986408i
$554$	245.640	−	141.820i	0.443393	−	0.255993i
$555$		0			0
$556$	−182.250	+	315.666i	−0.327787	+	0.567744i
$557$	303.338			0.544593			0.272296	−	0.962213i	$-0.412217\pi$
$557$	303.338			0.544593			0.272296	+	0.962213i	$0.412217\pi$
$558$		0			0
$559$	241.882			0.432705
$560$	−3.24649	+	15.9443i	−0.00579731	+	0.0284720i
$561$		0			0
$562$	156.805	−	90.5311i	0.279012	−	0.161087i
$563$	−117.047	−	202.731i	−0.207898	−	0.360090i	0.743154	−	0.669120i	$-0.233329\pi$
$563$	−117.047	−	202.731i	−0.207898	−	0.360090i	−0.951052	+	0.309030i	$0.899996\pi$
$564$		0			0
$565$	149.073	+	30.3534i	0.263846	+	0.0537229i
$566$		−	594.302i		−	1.05000i
$567$		0			0
$568$			197.136i			0.347071i
$569$	−57.6129	−	33.2628i	−0.101253	−	0.0584584i	0.448518	−	0.893774i	$-0.351952\pi$
$569$	−57.6129	−	33.2628i	−0.101253	−	0.0584584i	−0.549771	+	0.835315i	$0.685285\pi$
$570$		0			0
$571$	−557.768	−	966.082i	−0.976826	−	1.69191i	−0.673773	−	0.738939i	$-0.735327\pi$
$571$	−557.768	−	966.082i	−0.976826	−	1.69191i	−0.303053	−	0.952974i	$-0.598006\pi$
$572$	−90.6030	−	156.929i	−0.158397	−	0.274351i
$573$		0			0
$574$	4.05173	−	7.01781i	0.00705877	−	0.0122261i
$575$	95.8385	−	72.1728i	0.166676	−	0.125518i
$576$		0			0
$577$			957.356i			1.65920i	0.558361	+	0.829598i	$0.311430\pi$
$577$			957.356i			1.65920i	−0.558361	+	0.829598i	$0.688570\pi$
$578$	−86.6934	+	150.157i	−0.149989	+	0.259788i
$579$		0			0
$580$	319.586	+	282.835i	0.551011	+	0.487646i
$581$	−76.8161	+	44.3498i	−0.132214	+	0.0763335i
$582$		0			0
$583$	−848.610	−	489.945i	−1.45559	−	0.840387i
$584$		−	253.480i		−	0.434042i
$585$		0			0
$586$	−194.360			−0.331673
$587$	39.9487	−	69.1932i	0.0680557	−	0.117876i	−0.829990	−	0.557779i	$-0.811654\pi$
$587$	39.9487	−	69.1932i	0.0680557	−	0.117876i	0.898045	+	0.439903i	$0.144987\pi$
$588$		0			0
$589$	−2.61775	−	4.53407i	−0.00444440	−	0.00769792i
$590$	194.041	+	171.727i	0.328883	+	0.291062i
$591$		0			0
$592$	−243.576	−	140.629i	−0.411446	−	0.237549i
$593$	−213.831			−0.360593			−0.180296	−	0.983612i	$-0.557706\pi$
$593$	−213.831			−0.360593			−0.180296	+	0.983612i	$0.557706\pi$
$594$		0			0
$595$	−49.7645	+	16.6424i	−0.0836378	+	0.0279704i
$596$	−328.284	−	189.535i	−0.550813	−	0.318012i
$597$		0			0
$598$	−34.6330	+	19.9954i	−0.0579147	+	0.0334371i
$599$	−792.840	+	457.746i	−1.32361	+	0.764184i	−0.984302	−	0.176492i	$-0.943525\pi$
$599$	−792.840	+	457.746i	−1.32361	+	0.764184i	−0.339304	+	0.940677i	$0.610192\pi$
$600$		0			0
$601$	−239.713	+	415.195i	−0.398857	+	0.690841i	−0.993585	−	0.113086i	$-0.963926\pi$
$601$	−239.713	+	415.195i	−0.398857	+	0.690841i	0.594728	+	0.803927i	$0.297260\pi$
$602$	−47.2304			−0.0784559
$603$		0			0
$604$	−226.676			−0.375292
$605$	565.528	+	115.150i	0.934757	+	0.190330i
$606$		0			0
$607$	−190.433	+	109.946i	−0.313727	+	0.181131i	−0.648593	−	0.761135i	$-0.724642\pi$
$607$	−190.433	+	109.946i	−0.313727	+	0.181131i	0.334866	+	0.942266i	$0.391309\pi$
$608$	3.51472	+	6.08767i	0.00578079	+	0.0100126i
$609$		0			0
$610$	−574.690	−	117.015i	−0.942115	−	0.191828i
$611$			470.210i			0.769575i
$612$		0			0
$613$		−	862.066i		−	1.40631i	−0.711038	−	0.703154i	$-0.751775\pi$
$613$		−	862.066i		−	1.40631i	0.711038	−	0.703154i	$-0.248225\pi$
$614$	144.335	+	83.3316i	0.235073	+	0.135719i
$615$		0			0
$616$	17.6913	+	30.6423i	0.0287197	+	0.0497440i
$617$	−347.712	−	602.254i	−0.563552	−	0.976101i	−0.997183	−	0.0750103i	$-0.976101\pi$
$617$	−347.712	−	602.254i	−0.563552	−	0.976101i	0.433631	−	0.901091i	$-0.357232\pi$
$618$		0			0
$619$	256.037	−	443.468i	0.413629	−	0.716427i	−0.581654	−	0.813436i	$-0.697594\pi$
$619$	256.037	−	443.468i	0.413629	−	0.716427i	0.995283	+	0.0970091i	$0.0309276\pi$
$620$	13.3625	+	39.9569i	0.0215524	+	0.0644466i
$621$		0			0
$622$			706.153i			1.13529i
$623$	55.9390	−	96.8892i	0.0897898	−	0.155520i
$624$		0			0
$625$	−606.529	+	150.824i	−0.970446	+	0.241319i
$626$	−602.337	+	347.759i	−0.962199	+	0.555526i
$627$		0			0
$628$	−196.048	−	113.189i	−0.312179	−	0.180237i
$629$		−	907.020i		−	1.44200i
$630$		0			0
$631$	−643.375			−1.01961			−0.509806	−	0.860290i	$-0.670283\pi$
$631$	−643.375			−1.01961			−0.509806	+	0.860290i	$0.670283\pi$
$632$	189.640	−	328.465i	0.300063	−	0.519724i
$633$		0			0
$634$	−170.695	−	295.652i	−0.269235	−	0.466328i
$635$	−500.703	−	443.123i	−0.788509	−	0.697832i
$636$		0			0
$637$	246.669	+	142.415i	0.387236	+	0.223571i
$638$	928.014			1.45457
$639$		0			0
$640$	−17.9411	−	53.6481i	−0.0280330	−	0.0838251i
$641$	170.542	+	98.4625i	0.266056	+	0.153608i	0.627094	−	0.778944i	$-0.284244\pi$
$641$	170.542	+	98.4625i	0.266056	+	0.153608i	−0.361038	+	0.932551i	$0.617578\pi$
$642$		0			0
$643$	205.208	−	118.477i	0.319141	−	0.184256i	−0.331868	−	0.943326i	$-0.607679\pi$
$643$	205.208	−	118.477i	0.319141	−	0.184256i	0.651010	+	0.759069i	$0.274346\pi$
$644$	6.76252	−	3.90434i	0.0105008	−	0.00606264i
$645$		0			0
$646$	−11.3345	+	19.6320i	−0.0175457	+	0.0303900i
$647$	17.3818			0.0268653			0.0134326	−	0.999910i	$-0.495724\pi$
$647$	17.3818			0.0268653			0.0134326	+	0.999910i	$0.495724\pi$
$648$		0			0
$649$	563.456			0.868191
$650$	206.784	−	25.3247i	0.318129	−	0.0389610i
$651$		0			0
$652$	−263.988	+	152.414i	−0.404890	+	0.233763i
$653$	392.852	+	680.439i	0.601611	+	1.04202i	0.992577	+	0.121615i	$0.0388074\pi$
$653$	392.852	+	680.439i	0.601611	+	1.04202i	−0.390967	+	0.920405i	$0.627859\pi$
$654$		0			0
$655$	61.6356	−	302.708i	0.0941002	−	0.462150i
$656$			28.1720i			0.0429451i
$657$		0			0
$658$		−	91.8143i		−	0.139535i
$659$	431.945	+	249.384i	0.655455	+	0.378427i	0.790543	−	0.612406i	$-0.209798\pi$
$659$	431.945	+	249.384i	0.655455	+	0.378427i	−0.135088	+	0.990834i	$0.543132\pi$
$660$		0			0
$661$	−151.566	−	262.521i	−0.229299	−	0.397157i	0.728302	−	0.685257i	$-0.240310\pi$
$661$	−151.566	−	262.521i	−0.229299	−	0.397157i	−0.957600	+	0.288100i	$0.906977\pi$
$662$	−62.0904	−	107.544i	−0.0937922	−	0.162453i
$663$		0			0
$664$	154.184	−	267.054i	0.232204	−	0.402190i
$665$	4.79394	−	1.60320i	0.00720893	−	0.00241083i
$666$		0			0
$667$		−	204.805i		−	0.307055i
$668$	−105.853	+	183.342i	−0.158462	+	0.274465i
$669$		0			0
$670$	471.286	+	417.089i	0.703412	+	0.622521i
$671$	−1104.46	+	637.658i	−1.64599	+	0.950310i
$672$		0			0
$673$	401.085	+	231.567i	0.595966	+	0.344081i	0.767453	−	0.641105i	$-0.221524\pi$
$673$	401.085	+	231.567i	0.595966	+	0.344081i	−0.171487	+	0.985186i	$0.554857\pi$
$674$		−	639.420i		−	0.948694i
$675$		0			0
$676$	268.558			0.397276
$677$	−155.532	+	269.389i	−0.229737	+	0.397916i	−0.957730	−	0.287668i	$-0.907120\pi$
$677$	−155.532	+	269.389i	−0.229737	+	0.397916i	0.727993	+	0.685584i	$0.240453\pi$
$678$		0			0
$679$	−36.8671	−	63.8557i	−0.0542962	−	0.0940437i
$680$	120.901	−	136.611i	0.177795	−	0.200898i
$681$		0			0
$682$	79.3425	+	45.8084i	0.116338	+	0.0671678i
$683$	232.009			0.339691			0.169846	−	0.985471i	$-0.445673\pi$
$683$	232.009			0.339691			0.169846	+	0.985471i	$0.445673\pi$
$684$		0			0
$685$	−259.940	−	777.281i	−0.379475	−	1.13472i
$686$	−96.9898	−	55.9971i	−0.141385	−	0.0816284i
$687$		0			0
$688$	142.200	−	82.0993i	0.206686	−	0.119330i
$689$	−325.203	+	187.756i	−0.471993	+	0.272506i
$690$		0			0
$691$	276.452	−	478.829i	0.400075	−	0.692950i	−0.593660	−	0.804716i	$-0.702317\pi$
$691$	276.452	−	478.829i	0.400075	−	0.692950i	0.993735	+	0.111766i	$0.0356507\pi$
$692$	220.024			0.317954
$693$		0			0
$694$	−267.868			−0.385977
$695$	−892.927	−	181.813i	−1.28479	−	0.261601i
$696$		0			0
$697$	−78.6794	+	45.4256i	−0.112883	+	0.0651730i
$698$	−110.163	−	190.808i	−0.157827	−	0.273364i
$699$		0			0
$700$	−40.3771	+	4.94495i	−0.0576816	+	0.00706421i
$701$		−	250.274i		−	0.357024i	−0.983938	−	0.178512i	$-0.942872\pi$
$701$		−	250.274i		−	0.357024i	0.983938	−	0.178512i	$-0.0571284\pi$
$702$		0			0
$703$			87.3755i			0.124290i
$704$	−106.529	−	61.5047i	−0.151320	−	0.0873646i
$705$		0			0
$706$	186.243	+	322.582i	0.263800	+	0.456915i
$707$	20.2254	+	35.0314i	0.0286074	+	0.0495494i
$708$		0			0
$709$	−585.286	+	1013.75i	−0.825510	+	1.42982i	0.0760195	+	0.997106i	$0.475779\pi$
$709$	−585.286	+	1013.75i	−0.825510	+	1.42982i	−0.901529	+	0.432718i	$0.857554\pi$
$710$	−467.397	+	156.308i	−0.658306	+	0.220152i
$711$		0			0
$712$			388.949i			0.546276i
$713$	10.1096	−	17.5103i	0.0141789	−	0.0245586i
$714$		0			0
$715$	300.230	−	339.242i	0.419902	−	0.474464i
$716$	205.087	−	118.407i	0.286434	−	0.165373i
$717$		0			0
$718$	97.5321	+	56.3102i	0.135839	+	0.0784264i
$719$			168.127i			0.233834i	0.993142	+	0.116917i	$0.0373012\pi$
$719$			168.127i			0.233834i	−0.993142	+	0.116917i	$0.962699\pi$
$720$		0			0
$721$	−104.824			−0.145387
$722$	−254.174	+	440.242i	−0.352041	+	0.609753i
$723$		0			0
$724$	−172.397	−	298.600i	−0.238117	−	0.412431i
$725$	−417.184	+	981.975i	−0.575426	+	1.35445i
$726$		0			0
$727$	331.545	+	191.417i	0.456045	+	0.263298i	0.710380	−	0.703819i	$-0.248523\pi$
$727$	331.545	+	191.417i	0.456045	+	0.263298i	−0.254335	+	0.967116i	$0.581857\pi$
$728$	13.5593			0.0186254
$729$		0			0
$730$	600.985	−	200.983i	0.823267	−	0.275319i
$731$	458.577	+	264.760i	0.627329	+	0.362188i
$732$		0			0
$733$	−1215.10	+	701.541i	−1.65771	+	0.957082i	−0.683948	+	0.729531i	$0.739738\pi$
$733$	−1215.10	+	701.541i	−1.65771	+	0.957082i	−0.973766	+	0.227550i	$0.926928\pi$
$734$	−582.921	+	336.550i	−0.794171	+	0.458515i
$735$		0			0
$736$	−13.5736	+	23.5102i	−0.0184424	+	0.0319431i
$737$	1368.52			1.85688
$738$		0			0
$739$	899.125			1.21668			0.608339	−	0.793677i	$-0.291836\pi$
$739$	899.125			1.21668			0.608339	+	0.793677i	$0.291836\pi$
$740$	140.291	−	689.006i	0.189583	−	0.931090i
$741$		0			0
$742$	63.4999	−	36.6617i	0.0855794	−	0.0494093i
$743$	158.294	+	274.172i	0.213046	+	0.369007i	0.952666	−	0.304017i	$-0.0983281\pi$
$743$	158.294	+	274.172i	0.213046	+	0.369007i	−0.739620	+	0.673025i	$0.764995\pi$
$744$		0			0
$745$	189.080	−	928.621i	0.253799	−	1.24647i
$746$		−	87.3755i		−	0.117125i
$747$		0			0
$748$		−	396.689i		−	0.530334i
$749$	4.52790	+	2.61418i	0.00604526	+	0.00349023i
$750$		0			0
$751$	143.216	+	248.058i	0.190701	+	0.330304i	0.945483	−	0.325672i	$-0.105591\pi$
$751$	143.216	+	248.058i	0.190701	+	0.330304i	−0.754782	+	0.655976i	$0.772257\pi$
$752$	159.598	+	276.432i	0.212231	+	0.367595i
$753$		0			0
$754$	177.816	−	307.987i	0.235831	−	0.408470i
$755$	−179.730	−	537.434i	−0.238053	−	0.711833i
$756$		0			0
$757$		−	783.221i		−	1.03464i	−0.855793	−	0.517319i	$-0.826930\pi$
$757$		−	783.221i		−	1.03464i	0.855793	−	0.517319i	$-0.173070\pi$
$758$	−100.653	+	174.337i	−0.132788	+	0.229996i
$759$		0			0
$760$	−11.6467	+	13.1600i	−0.0153246	+	0.0173158i
$761$	893.662	−	515.956i	1.17433	−	0.677997i	0.219631	−	0.975583i	$-0.429515\pi$
$761$	893.662	−	515.956i	1.17433	−	0.677997i	0.954695	+	0.297586i	$0.0961814\pi$
$762$		0			0
$763$	76.2910	+	44.0467i	0.0999883	+	0.0577282i
$764$			578.787i			0.757574i
$765$		0			0
$766$	−505.609			−0.660064
$767$	107.963	−	186.998i	0.140761	−	0.243805i
$768$		0			0
$769$	415.625	+	719.883i	0.540475	+	0.936129i	0.998877	+	0.0473845i	$0.0150886\pi$
$769$	415.625	+	719.883i	0.540475	+	0.936129i	−0.458402	+	0.888745i	$0.651578\pi$
$770$	−58.6235	+	66.2411i	−0.0761344	+	0.0860273i
$771$		0			0
$772$	−221.413	−	127.833i	−0.286805	−	0.165587i
$773$	−395.360			−0.511462			−0.255731	−	0.966748i	$-0.582316\pi$
$773$	−395.360			−0.511462			−0.255731	+	0.966748i	$0.582316\pi$
$774$		0			0
$775$	−84.1400	+	63.3631i	−0.108568	+	0.0817588i
$776$	221.997	+	128.170i	0.286078	+	0.165167i
$777$		0			0
$778$	503.466	−	290.676i	0.647129	−	0.373620i
$779$	7.57939	−	4.37596i	0.00972964	−	0.00561741i
$780$		0			0
$781$	−535.846

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.j (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(3.31368 - 3.74426i) q^{5} +(0.704577 + 0.406788i) q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(3.31368 - 3.74426i) q^{5} +(0.704577 + 0.406788i) q^{7} -2.82843 q^{8} +(-2.24264 - 6.70601i) q^{10} +(-13.3161 - 7.68808i) q^{11} +(-5.10300 + 2.94622i) q^{13} +(0.996422 - 0.575285i) q^{14} +(-2.00000 + 3.46410i) q^{16} -12.8995 q^{17} +1.24264 q^{19} +(-9.79894 - 1.99520i) q^{20} +(-18.8319 + 10.8726i) q^{22} +(2.39949 + 4.15605i) q^{23} +(-3.03902 - 24.8146i) q^{25} +8.33316i q^{26} -1.62715i q^{28} +(-36.9592 - 21.3384i) q^{29} +(-2.10660 - 3.64874i) q^{31} +(2.82843 + 4.89898i) q^{32} +(-9.12132 + 15.7986i) q^{34} +(3.85786 - 1.29016i) q^{35} +70.3144i q^{37} +(0.878680 - 1.52192i) q^{38} +(-9.37251 + 10.5904i) q^{40} +(6.09942 - 3.52150i) q^{41} +(-35.5500 - 20.5248i) q^{43} +30.7523i q^{44} +6.78680 q^{46} +(39.8995 - 69.1080i) q^{47} +(-24.1690 - 41.8620i) q^{49} +(-32.5405 - 13.8245i) q^{50} +(10.2060 + 5.89243i) q^{52} +63.7279 q^{53} +(-72.9117 + 24.3833i) q^{55} +(-1.99284 - 1.15057i) q^{56} +(-52.2682 + 30.1770i) q^{58} +(-31.7353 + 18.3224i) q^{59} +(41.4706 - 71.8291i) q^{61} -5.95837 q^{62} +8.00000 q^{64} +(-5.87830 + 28.8698i) q^{65} +(-77.0786 + 44.5013i) q^{67} +(12.8995 + 22.3426i) q^{68} +(1.14781 - 5.63718i) q^{70} -69.6982i q^{71} +89.6188i q^{73} +(86.1172 + 49.7198i) q^{74} +(-1.24264 - 2.15232i) q^{76} +(-6.25483 - 10.8337i) q^{77} +(-67.0477 + 116.130i) q^{79} +(6.34315 + 18.9675i) q^{80} -9.96031i q^{82} +(-54.5122 + 94.4179i) q^{83} +(-42.7448 + 48.2991i) q^{85} +(-50.2753 + 29.0265i) q^{86} +(37.6638 + 21.7452i) q^{88} -137.514i q^{89} -4.79394 q^{91} +(4.79899 - 8.31209i) q^{92} +(-56.4264 - 97.7334i) q^{94} +(4.11772 - 4.65277i) q^{95} +(-78.4877 - 45.3149i) q^{97} -68.3604 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 12 q^{5}+O(q^{10}) \) 8 * q - 8 * q^4 - 12 * q^5	\( 8 q - 8 q^{4} - 12 q^{5} + 16 q^{10} - 16 q^{16} - 24 q^{17} - 24 q^{19} - 24 q^{20} - 60 q^{23} + 12 q^{25} + 68 q^{31} - 56 q^{34} + 144 q^{35} + 24 q^{38} - 16 q^{40} + 224 q^{46} + 240 q^{47} + 180 q^{49} - 96 q^{50} + 408 q^{53} - 176 q^{55} + 196 q^{61} - 240 q^{62} + 64 q^{64} + 24 q^{65} + 24 q^{68} - 80 q^{70} + 24 q^{76} + 312 q^{77} - 180 q^{79} + 96 q^{80} - 108 q^{83} - 20 q^{85} + 912 q^{91} - 120 q^{92} - 112 q^{94} + 60 q^{95} - 1056 q^{98}+O(q^{100}) \) 8 * q - 8 * q^4 - 12 * q^5 + 16 * q^10 - 16 * q^16 - 24 * q^17 - 24 * q^19 - 24 * q^20 - 60 * q^23 + 12 * q^25 + 68 * q^31 - 56 * q^34 + 144 * q^35 + 24 * q^38 - 16 * q^40 + 224 * q^46 + 240 * q^47 + 180 * q^49 - 96 * q^50 + 408 * q^53 - 176 * q^55 + 196 * q^61 - 240 * q^62 + 64 * q^64 + 24 * q^65 + 24 * q^68 - 80 * q^70 + 24 * q^76 + 312 * q^77 - 180 * q^79 + 96 * q^80 - 108 * q^83 - 20 * q^85 + 912 * q^91 - 120 * q^92 - 112 * q^94 + 60 * q^95 - 1056 * q^98

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