LMFDB - Embedded newform 820.2.bm.a.687.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [820,2,Mod(23,820)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(820, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 15, 18]))

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

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

chi := DirichletCharacter("820.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 820 = 2^{2} \cdot 5 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	820.bm (of order $20$, degree $8$, 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:	$6.54773296574$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label			687.1
Root			$0.587785 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	820.687
Dual form			820.2.bm.a.783.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.221232 + 1.39680i) q^{2} +(-1.90211 + 0.618034i) q^{4} +(-2.20582 - 0.366554i) q^{5} +(-1.28408 - 2.52015i) q^{8} +3.00000i q^{9} +O(q^{10})$	\(q+(0.221232 + 1.39680i) q^{2} +(-1.90211 + 0.618034i) q^{4} +(-2.20582 - 0.366554i) q^{5} +(-1.28408 - 2.52015i) q^{8} +3.00000i q^{9} +(0.0240055 - 3.16219i) q^{10} +(-0.131418 - 0.829740i) q^{13} +(3.23607 - 2.35114i) q^{16} +(-3.72925 - 7.31906i) q^{17} +(-4.19041 + 0.663695i) q^{18} +(4.42226 - 0.666045i) q^{20} +(4.73128 + 1.61710i) q^{25} +(1.12991 - 0.367130i) q^{26} +(-3.09017 - 9.51057i) q^{29} +(4.00000 + 4.00000i) q^{32} +(9.39825 - 6.82823i) q^{34} +(-1.85410 - 5.70634i) q^{36} +(1.66427 - 3.26632i) q^{37} +(1.90868 + 6.02967i) q^{40} +(6.17499 - 1.69394i) q^{41} +(1.09966 - 6.61746i) q^{45} +(-6.65740 - 2.16312i) q^{49} +(-1.21206 + 6.96641i) q^{50} +(0.762779 + 1.49704i) q^{52} +(5.99354 - 11.7630i) q^{53} +(12.6007 - 6.42040i) q^{58} +(-10.6942 + 7.76980i) q^{61} +(-4.70228 + 6.47214i) q^{64} +(-0.0142600 + 1.87843i) q^{65} +(11.6169 + 11.6169i) q^{68} +(7.56044 - 3.85224i) q^{72} +(-1.35610 + 1.35610i) q^{73} +(4.93059 + 1.60205i) q^{74} +(-8.00000 + 4.00000i) q^{80} -9.00000 q^{81} +(3.73221 + 8.25049i) q^{82} +(5.54322 + 17.5115i) q^{85} +(-8.09017 + 5.87785i) q^{89} +(9.48656 + 0.0720166i) q^{90} +(-17.5261 - 8.93001i) q^{97} +(1.54862 - 9.77762i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 4 q^{5} - 4 q^{8}+O(q^{10}) $ 8 * q + 2 * q^2 - 4 * q^5 - 4 * q^8	$ 8 q + 2 q^{2} - 4 q^{5} - 4 q^{8} + 6 q^{10} - 10 q^{13} + 8 q^{16} + 10 q^{17} - 6 q^{18} - 4 q^{20} - 6 q^{25} - 10 q^{26} + 20 q^{29} + 32 q^{32} + 30 q^{34} + 12 q^{36} + 46 q^{37} - 4 q^{40} + 8 q^{41} - 6 q^{45} + 14 q^{50} - 40 q^{52} - 10 q^{53} - 20 q^{58} - 24 q^{61} + 50 q^{65} + 20 q^{68} - 12 q^{72} - 22 q^{73} + 50 q^{74} - 64 q^{80} - 72 q^{81} - 18 q^{82} + 30 q^{85} - 20 q^{89} - 6 q^{90} - 80 q^{97} + 14 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 - 4 * q^5 - 4 * q^8 + 6 * q^10 - 10 * q^13 + 8 * q^16 + 10 * q^17 - 6 * q^18 - 4 * q^20 - 6 * q^25 - 10 * q^26 + 20 * q^29 + 32 * q^32 + 30 * q^34 + 12 * q^36 + 46 * q^37 - 4 * q^40 + 8 * q^41 - 6 * q^45 + 14 * q^50 - 40 * q^52 - 10 * q^53 - 20 * q^58 - 24 * q^61 + 50 * q^65 + 20 * q^68 - 12 * q^72 - 22 * q^73 + 50 * q^74 - 64 * q^80 - 72 * q^81 - 18 * q^82 + 30 * q^85 - 20 * q^89 - 6 * q^90 - 80 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/820\mathbb{Z}\right)^\times$.

$n$	$411$	$621$	$657$
$\chi(n)$	$-1$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{1}{4}\right)$

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 $

$2$	0.221232	+	1.39680i	0.156434	+	0.987688i
$2$	0.221232	+	1.39680i	0.156434	+	0.987688i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$	−1.90211	+	0.618034i	−0.951057	+	0.309017i
$5$	−2.20582	−	0.366554i	−0.986472	−	0.163928i
$5$	−2.20582	−	0.366554i	−0.986472	−	0.163928i
$6$		0			0
$7$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$7$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$8$	−1.28408	−	2.52015i	−0.453990	−	0.891007i
$9$			3.00000i			1.00000i
$10$	0.0240055	−	3.16219i	0.00759122	−	0.999971i
$11$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$11$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$12$		0			0
$13$	−0.131418	−	0.829740i	−0.0364488	−	0.230129i	0.962739	−	0.270434i	$-0.0871670\pi$
$13$	−0.131418	−	0.829740i	−0.0364488	−	0.230129i	−0.999187	+	0.0403050i	$0.987167\pi$
$14$		0			0
$15$		0			0
$16$	3.23607	−	2.35114i	0.809017	−	0.587785i
$17$	−3.72925	−	7.31906i	−0.904476	−	1.77513i	−0.530456	−	0.847713i	$-0.677979\pi$
$17$	−3.72925	−	7.31906i	−0.904476	−	1.77513i	−0.374020	−	0.927421i	$-0.622021\pi$
$18$	−4.19041	+	0.663695i	−0.987688	+	0.156434i
$19$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$19$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$20$	4.42226	−	0.666045i	0.988847	−	0.148932i
$21$		0			0
$22$		0			0
$23$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$23$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$24$		0			0
$25$	4.73128	+	1.61710i	0.946255	+	0.323420i
$26$	1.12991	−	0.367130i	0.221593	−	0.0720001i
$27$		0			0
$28$		0			0
$29$	−3.09017	−	9.51057i	−0.573830	−	1.76607i	−0.640129	−	0.768268i	$-0.721119\pi$
$29$	−3.09017	−	9.51057i	−0.573830	−	1.76607i	0.0662984	−	0.997800i	$-0.478881\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$		0			0
$34$	9.39825	−	6.82823i	1.61179	−	1.17103i
$35$		0			0
$36$	−1.85410	−	5.70634i	−0.309017	−	0.951057i
$37$	1.66427	−	3.26632i	0.273605	−	0.536979i	−0.712789	−	0.701378i	$-0.752568\pi$
$37$	1.66427	−	3.26632i	0.273605	−	0.536979i	0.986394	+	0.164399i	$0.0525685\pi$
$38$		0			0
$39$		0			0
$40$	1.90868	+	6.02967i	0.301788	+	0.953375i
$41$	6.17499	−	1.69394i	0.964372	−	0.264550i
$41$	6.17499	−	1.69394i	0.964372	−	0.264550i
$42$		0			0
$43$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$43$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$44$		0			0
$45$	1.09966	−	6.61746i	0.163928	−	0.986472i
$46$		0			0
$47$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$47$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$48$		0			0
$49$	−6.65740	−	2.16312i	−0.951057	−	0.309017i
$50$	−1.21206	+	6.96641i	−0.171412	+	0.985200i
$51$		0			0
$52$	0.762779	+	1.49704i	0.105778	+	0.207602i
$53$	5.99354	−	11.7630i	0.823276	−	1.61577i	0.0358519	−	0.999357i	$-0.488586\pi$
$53$	5.99354	−	11.7630i	0.823276	−	1.61577i	0.787424	−	0.616412i	$-0.211414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	12.6007	−	6.42040i	1.65456	−	0.843039i
$59$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$59$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$60$		0			0
$61$	−10.6942	+	7.76980i	−1.36925	+	0.994821i	−0.371458	+	0.928450i	$0.621142\pi$
$61$	−10.6942	+	7.76980i	−1.36925	+	0.994821i	−0.997795	+	0.0663709i	$0.978858\pi$
$62$		0			0
$63$		0			0
$64$	−4.70228	+	6.47214i	−0.587785	+	0.809017i
$65$	−0.0142600	+	1.87843i	−0.00176873	+	0.232990i
$66$		0			0
$67$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$67$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$68$	11.6169	+	11.6169i	1.40875	+	1.40875i
$69$		0			0
$70$		0			0
$71$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$71$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$72$	7.56044	−	3.85224i	0.891007	−	0.453990i
$73$	−1.35610	+	1.35610i	−0.158719	+	0.158719i	−0.781999	−	0.623280i	$-0.785800\pi$
$73$	−1.35610	+	1.35610i	−0.158719	+	0.158719i	0.623280	+	0.781999i	$0.285800\pi$
$74$	4.93059	+	1.60205i	0.573169	+	0.186234i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−8.00000	+	4.00000i	−0.894427	+	0.447214i
$81$	−9.00000			−1.00000
$82$	3.73221	+	8.25049i	0.412154	+	0.911114i
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	5.54322	+	17.5115i	0.601247	+	1.89939i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.09017	+	5.87785i	−0.857556	+	0.623051i	−0.927219	−	0.374519i	$-0.877808\pi$
$89$	−8.09017	+	5.87785i	−0.857556	+	0.623051i	0.0696627	+	0.997571i	$0.477808\pi$
$90$	9.48656	+	0.0720166i	0.999971	+	0.00759122i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−17.5261	−	8.93001i	−1.77951	−	0.906705i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−17.5261	−	8.93001i	−1.77951	−	0.906705i	−0.865698	−	0.500567i	$-0.833125\pi$
$98$	1.54862	−	9.77762i	0.156434	−	0.987688i
$99$		0			0
$100$	−9.99885	−	0.151820i	−0.999885	−	0.0151820i
$101$	−8.81958	−	12.1391i	−0.877581	−	1.20789i	−0.977085	−	0.212850i	$-0.931726\pi$
$101$	−8.81958	−	12.1391i	−0.877581	−	1.20789i	0.0995037	−	0.995037i	$-0.468274\pi$
$102$		0			0
$103$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$103$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$104$	−1.92232	+	1.39664i	−0.188499	+	0.136952i
$105$		0			0
$106$	17.7565	+	5.76944i	1.72467	+	0.560378i
$107$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$107$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$108$		0			0
$109$	−12.6536			−1.21200			−0.605999	−	0.795465i	$-0.707226\pi$
$109$	−12.6536			−1.21200			−0.605999	+	0.795465i	$0.707226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.96084	+	17.5866i	0.842965	+	1.65441i	0.752577	+	0.658505i	$0.228811\pi$
$113$	8.96084	+	17.5866i	0.842965	+	1.65441i	0.0903879	+	0.995907i	$0.471189\pi$
$114$		0			0
$115$		0			0
$116$	11.7557	+	16.1803i	1.09149	+	1.50231i
$117$	2.48922	−	0.394254i	0.230129	−	0.0364488i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.89919	+	6.46564i	0.809017	+	0.587785i
$122$	−13.2188	−	13.2188i	−1.19677	−	1.19677i
$123$		0			0
$124$		0			0
$125$	−9.84359	−	5.30130i	−0.880437	−	0.474163i
$126$		0			0
$127$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$127$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$128$	−10.0806	−	5.13632i	−0.891007	−	0.453990i
$129$		0			0
$130$	−2.62695	+	0.395650i	−0.230399	+	0.0347008i
$131$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$131$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−13.6565	+	18.7965i	−1.17103	+	1.61179i
$137$	−2.02214	+	2.02214i	−0.172763	+	0.172763i	−0.788192	−	0.615429i	$-0.788983\pi$
$137$	−2.02214	+	2.02214i	−0.172763	+	0.172763i	0.615429	+	0.788192i	$0.288983\pi$
$138$		0			0
$139$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$139$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	7.05342	+	9.70820i	0.587785	+	0.809017i
$145$	3.33023	+	22.1113i	0.276560	+	1.83624i
$146$	−2.19421	−	1.59419i	−0.181594	−	0.131936i
$147$		0			0
$148$	−1.14694	+	7.24148i	−0.0942777	+	0.595246i
$149$	2.20467	+	6.78527i	0.180613	+	0.555871i	0.999845	−	0.0175917i	$-0.00559989\pi$
$149$	2.20467	+	6.78527i	0.180613	+	0.555871i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$151$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$152$		0			0
$153$	21.9572	−	11.1877i	1.77513	−	0.904476i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.91869	−	24.7416i	0.312745	−	1.97460i	0.128615	−	0.991695i	$-0.458947\pi$
$157$	3.91869	−	24.7416i	0.312745	−	1.97460i	0.184131	−	0.982902i	$-0.441053\pi$
$158$		0			0
$159$		0			0
$160$	−7.35706	−	10.2895i	−0.581627	−	0.813456i
$161$		0			0
$162$	−1.99109	−	12.5712i	−0.156434	−	0.987688i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$	−10.6986	+	7.03843i	−0.835422	+	0.549609i
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$	11.6925	−	3.79914i	0.899426	−	0.292241i
$170$	−23.2338	+	11.6169i	−1.78195	+	0.890974i
$171$		0			0
$172$		0			0
$173$	17.7160	−	17.7160i	1.34692	−	1.34692i	0.457933	−	0.888986i	$-0.348590\pi$
$173$	17.7160	−	17.7160i	1.34692	−	1.34692i	0.888986	−	0.457933i	$-0.151410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−10.0000	−	10.0000i	−0.749532	−	0.749532i
$179$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$179$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$180$	1.99814	+	13.2668i	0.148932	+	0.988847i
$181$	−22.1590	−	7.19990i	−1.64707	−	0.535164i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−22.1590	−	7.19990i	−1.64707	−	0.535164i	−0.978101	+	0.208130i	$0.933262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.86836	+	6.59486i	−0.357929	+	0.484864i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−9.76735	+	19.1695i	−0.703069	+	1.37985i	0.212287	+	0.977207i	$0.431909\pi$
$193$	−9.76735	+	19.1695i	−0.703069	+	1.37985i	−0.915357	+	0.402644i	$0.868091\pi$
$194$	8.59612	−	26.4561i	0.617166	−	1.89944i
$195$		0			0
$196$	14.0000			1.00000
$197$	−2.14268	+	1.09175i	−0.152660	+	0.0777840i	−0.528651	−	0.848839i	$-0.677302\pi$
$197$	−2.14268	+	1.09175i	−0.152660	+	0.0777840i	0.375992	+	0.926623i	$0.377302\pi$
$198$		0			0
$199$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$199$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$200$	−2.00000	−	14.0000i	−0.141421	−	0.989949i
$201$		0			0
$202$	15.0048	−	15.0048i	1.05573	−	1.05573i
$203$		0			0
$204$		0			0
$205$	−14.2418	+	1.47307i	−0.994693	+	0.102884i
$206$		0			0
$207$		0			0
$208$	−2.37611	−	2.37611i	−0.164754	−	0.164754i
$209$		0			0
$210$		0			0
$211$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$211$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$212$	−4.13046	+	26.0787i	−0.283681	+	1.79109i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−2.79938	−	17.6746i	−0.189598	−	1.19708i
$219$		0			0
$220$		0			0
$221$	−5.58283	+	4.05616i	−0.375542	+	0.272847i
$222$		0			0
$223$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$223$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$224$		0			0
$225$	−4.85130	+	14.1938i	−0.323420	+	0.946255i
$226$	−22.5826	+	16.4072i	−1.50217	+	1.09139i
$227$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$227$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$228$		0			0
$229$	−1.68917	+	5.19874i	−0.111624	+	0.343543i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−1.68917	+	5.19874i	−0.111624	+	0.343543i	0.879604	+	0.475706i	$0.157808\pi$
$230$		0			0
$231$		0			0
$232$	−20.0000	+	20.0000i	−1.31306	+	1.31306i
$233$	−1.83587	−	11.5912i	−0.120272	−	0.759367i	−0.971930	−	0.235269i	$-0.924403\pi$
$233$	−1.83587	−	11.5912i	−0.120272	−	0.759367i	0.851658	−	0.524097i	$-0.175597\pi$
$234$	1.10139	+	3.38973i	0.0720001	+	0.221593i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$239$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$240$		0			0
$241$	7.44907	+	22.9259i	0.479837	+	1.47679i	0.839322	+	0.543635i	$0.182952\pi$
$241$	7.44907	+	22.9259i	0.479837	+	1.47679i	−0.359485	+	0.933151i	$0.617048\pi$
$242$	−7.06243	+	13.8608i	−0.453990	+	0.891007i
$243$		0			0
$244$	15.5396	−	21.3884i	0.994821	−	1.36925i
$245$	13.8921	+	7.21174i	0.887535	+	0.460741i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	5.22715	−	14.9224i	0.330594	−	0.943773i
$251$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$251$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	−	15.2169i	0.309017	−	0.951057i
$257$	−9.63059	−	18.9011i	−0.600740	−	1.17902i	−0.968480	−	0.249090i	$-0.919869\pi$
$257$	−9.63059	−	18.9011i	−0.600740	−	1.17902i	0.367740	−	0.929928i	$-0.380131\pi$
$258$		0			0
$259$		0			0
$260$	−1.13381	−	3.58180i	−0.0703158	−	0.222134i
$261$	28.5317	−	9.27051i	1.76607	−	0.573830i
$262$		0			0
$263$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$263$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$264$		0			0
$265$	−17.5324	+	23.7501i	−1.07701	+	1.45895i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.9029	+	21.8884i	−0.969615	+	1.33456i	−0.0273737	+	0.999625i	$0.508714\pi$
$269$	−15.9029	+	21.8884i	−0.969615	+	1.33456i	−0.942241	+	0.334935i	$0.891286\pi$
$270$		0			0
$271$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$271$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$272$	−29.2762	−	14.9170i	−1.77513	−	0.904476i
$273$		0			0
$274$	−3.27189	−	2.37717i	−0.197662	−	0.143610i
$275$		0			0
$276$		0			0
$277$	27.1499	−	13.8336i	1.63128	−	0.831180i	0.632905	−	0.774229i	$-0.281862\pi$
$277$	27.1499	−	13.8336i	1.63128	−	0.831180i	0.998377	−	0.0569502i	$-0.0181376\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.0722	−	22.1215i	0.958786	−	1.31966i	0.0112742	−	0.999936i	$-0.496411\pi$
$281$	16.0722	−	22.1215i	0.958786	−	1.31966i	0.947512	−	0.319720i	$-0.103589\pi$
$282$		0			0
$283$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$283$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−12.0000	+	12.0000i	−0.707107	+	0.707107i
$289$	−29.6690	+	40.8359i	−1.74524	+	2.40211i
$290$	−30.1484	+	9.54339i	−1.77037	+	0.560407i
$291$		0			0
$292$	1.74133	−	3.41756i	0.101904	−	0.199998i
$293$	−5.15540	−	32.5499i	−0.301182	−	1.90159i	−0.418023	−	0.908436i	$-0.637277\pi$
$293$	−5.15540	−	32.5499i	−0.301182	−	1.90159i	0.116841	−	0.993151i	$-0.462723\pi$
$294$		0			0
$295$		0			0
$296$	−10.3687			−0.602666
$297$		0			0
$298$	−8.98993	+	4.58060i	−0.520773	+	0.265347i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	26.4375	−	13.2188i	1.51381	−	0.756905i
$306$	20.4847	+	28.1948i	1.17103	+	1.61179i
$307$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$307$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$311$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$312$		0			0
$313$	−16.0510	+	31.5018i	−0.907255	+	1.78059i	−0.418016	+	0.908440i	$0.637274\pi$
$313$	−16.0510	+	31.5018i	−0.907255	+	1.78059i	−0.489240	+	0.872149i	$0.662726\pi$
$314$	35.4261			1.99921
$315$		0			0
$316$		0			0
$317$	−6.79188	−	13.3298i	−0.381470	−	0.748677i	0.617822	−	0.786318i	$-0.288015\pi$
$317$	−6.79188	−	13.3298i	−0.381470	−	0.748677i	−0.999291	+	0.0376418i	$0.988015\pi$
$318$		0			0
$319$		0			0
$320$	12.7448	−	12.5527i	0.712454	−	0.701719i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	17.1190	−	5.56231i	0.951057	−	0.309017i
$325$	0.720000	−	4.13825i	0.0399384	−	0.229549i
$326$		0			0
$327$		0			0
$328$	−12.1982	−	13.3867i	−0.673531	−	0.739159i
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	9.79895	+	4.99281i	0.536979	+	0.273605i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.03151	+	9.03151i	0.491978	+	0.491978i	0.908929	−	0.416951i	$-0.136901\pi$
$337$	9.03151	+	9.03151i	0.491978	+	0.491978i	−0.416951	+	0.908929i	$0.636901\pi$
$338$	7.89340	+	15.4917i	0.429344	+	0.842636i
$339$		0			0
$340$	−21.3665	−	29.8829i	−1.15876	−	1.62063i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	28.6650	+	20.8264i	1.54104	+	1.11963i
$347$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$347$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$348$		0			0
$349$	13.6641	+	18.8070i	0.731423	+	1.00672i	0.999066	+	0.0431990i	$0.0137549\pi$
$349$	13.6641	+	18.8070i	0.731423	+	1.00672i	−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.64010	−	10.3552i	0.0872935	−	0.551150i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	1.64010	−	10.3552i	0.0872935	−	0.551150i	0.992112	−	0.125353i	$-0.0400062\pi$
$354$		0			0
$355$		0			0
$356$	11.7557	−	16.1803i	0.623051	−	0.857556i
$357$		0			0
$358$		0			0
$359$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$359$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$360$	−18.0890	+	5.72603i	−0.953375	+	0.301788i
$361$	5.87132	−	18.0701i	0.309017	−	0.951057i
$362$	5.15456	−	32.5446i	0.270918	−	1.71051i
$363$		0			0
$364$		0			0
$365$	3.48838	−	2.49422i	0.182590	−	0.130553i
$366$		0			0
$367$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$367$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$368$		0			0
$369$	5.08183	+	18.5250i	0.264550	+	0.964372i
$370$	−10.2888	−	5.34115i	−0.534887	−	0.277673i
$371$		0			0
$372$		0			0
$373$	25.6768	+	13.0830i	1.32949	+	0.677411i	0.967048	−	0.254593i	$-0.0819416\pi$
$373$	25.6768	+	13.0830i	1.32949	+	0.677411i	0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.48520	+	3.81390i	−0.385507	+	0.196426i
$378$		0			0
$379$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$379$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$	−28.9369	−	9.40215i	−1.47285	−	0.478557i
$387$		0			0
$388$	38.8557	+	6.15414i	1.97260	+	0.312429i
$389$	9.25813	+	12.7427i	0.469406	+	0.646082i	0.976426	−	0.215852i	$-0.0692530\pi$
$389$	9.25813	+	12.7427i	0.469406	+	0.646082i	−0.507020	+	0.861934i	$0.669253\pi$
$390$		0			0
$391$		0			0
$392$	3.09724	+	19.5552i	0.156434	+	0.987688i
$393$		0			0
$394$	−1.99899	−	2.75137i	−0.100708	−	0.138612i
$395$		0			0
$396$		0			0
$397$	38.9242	−	6.16498i	1.95355	−	0.309412i	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.9242	−	6.16498i	1.95355	−	0.309412i	0.999966	−	0.00828030i	$-0.00263573\pi$
$398$		0			0
$399$		0			0
$400$	19.1128	−	5.89085i	0.955638	−	0.294542i
$401$	25.1294			1.25490			0.627452	−	0.778655i	$-0.284098\pi$
$401$	25.1294			1.25490			0.627452	+	0.778655i	$0.284098\pi$
$402$		0			0
$403$		0			0
$404$	24.2782	+	17.6392i	1.20789	+	0.877581i
$405$	19.8524	+	3.29898i	0.986472	+	0.163928i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	36.1882i		−	1.78939i	−0.446678	−	0.894695i	$-0.647393\pi$
$409$		−	36.1882i		−	1.78939i	0.446678	−	0.894695i	$-0.352607\pi$
$410$	−5.20833	−	19.5671i	−0.257221	−	0.966353i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	2.79329	−	3.84463i	0.136952	−	0.188499i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	7.43015	+	2.41420i	0.362123	+	0.117661i	0.484427	−	0.874832i	$-0.339028\pi$
$421$	7.43015	+	2.41420i	0.362123	+	0.117661i	−0.122304	+	0.992493i	$0.539028\pi$
$422$		0			0
$423$		0			0
$424$	−37.3406			−1.81342
$425$	−5.80844	−	40.6591i	−0.281751	−	1.97226i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$		0			0
$433$	28.6660	−	4.54024i	1.37760	−	0.218190i	0.576683	−	0.816968i	$-0.304347\pi$
$433$	28.6660	−	4.54024i	1.37760	−	0.218190i	0.800915	+	0.598778i	$0.204347\pi$
$434$		0			0
$435$		0			0
$436$	24.0686	−	7.82037i	1.15268	−	0.374528i
$437$		0			0
$438$		0			0
$439$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$439$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$440$		0			0
$441$	6.48936	−	19.9722i	0.309017	−	0.951057i
$442$	−6.90076	−	6.90076i	−0.328236	−	0.328236i
$443$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$443$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$444$		0			0
$445$	20.0000	−	10.0000i	0.948091	−	0.474045i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−20.4771	+	28.1842i	−0.966372	+	1.33010i	−0.0225137	+	0.999747i	$0.507167\pi$
$449$	−20.4771	+	28.1842i	−0.966372	+	1.33010i	−0.943858	+	0.330350i	$0.892833\pi$
$450$	−20.8992	−	3.63619i	−0.985200	−	0.171412i
$451$		0			0
$452$	−27.9137	−	27.9137i	−1.31295	−	1.31295i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.7926	+	1.86776i	−0.551632	+	0.0873700i	−0.426027	−	0.904710i	$-0.640087\pi$
$457$	−11.7926	+	1.86776i	−0.551632	+	0.0873700i	−0.125605	+	0.992080i	$0.540087\pi$
$458$	−7.63532	−	1.20932i	−0.356775	−	0.0565076i
$459$		0			0
$460$		0			0
$461$	9.50653	−	29.2581i	0.442763	−	1.36269i	−0.442155	−	0.896939i	$-0.645786\pi$
$461$	9.50653	−	29.2581i	0.442763	−	1.36269i	0.884918	−	0.465746i	$-0.154214\pi$
$462$		0			0
$463$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$463$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$464$	−32.3607	−	23.5114i	−1.50231	−	1.09149i
$465$		0			0
$466$	15.7845	−	5.12870i	0.731203	−	0.237582i
$467$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$467$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$468$	−4.49112	+	2.28834i	−0.207602	+	0.105778i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	35.2889	+	17.9806i	1.61577	+	0.823276i
$478$		0			0
$479$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$479$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$480$		0			0
$481$	−2.92891	−	0.951660i	−0.133547	−	0.0433920i
$482$	−30.3749	+	15.4768i	−1.38354	+	0.704949i
$483$		0			0
$484$	−20.9232	−	6.79837i	−0.951057	−	0.309017i
$485$	35.3862	+	26.1223i	1.60680	+	1.18615i
$486$		0			0
$487$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$487$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$488$	33.3132	+	16.9739i	1.50802	+	0.768374i
$489$		0			0
$490$	−7.00000	+	21.0000i	−0.316228	+	0.948683i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−58.0844	+	58.0844i	−2.61599	+	2.61599i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$499$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$500$	22.0000	+	4.00000i	0.983870	+	0.178885i
$501$		0			0
$502$		0			0
$503$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$503$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$504$		0			0
$505$	15.0048	+	30.0095i	0.667703	+	1.33541i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.49197	+	16.9025i	−0.243427	+	0.749192i	0.752464	+	0.658633i	$0.228865\pi$
$509$	−5.49197	+	16.9025i	−0.243427	+	0.749192i	−0.995891	+	0.0905584i	$0.971135\pi$
$510$		0			0
$511$		0			0
$512$	22.3488	+	3.53971i	0.987688	+	0.156434i
$513$		0			0
$514$	24.2705	−	17.6336i	1.07053	−	0.777783i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	4.75223	−	2.37611i	0.208399	−	0.104199i
$521$	−31.6549	+	10.2853i	−1.38683	+	0.450607i	−0.904907	−	0.425609i	$-0.860060\pi$
$521$	−31.6549	+	10.2853i	−1.38683	+	0.450607i	−0.481919	+	0.876216i	$0.660060\pi$
$522$	19.2612	+	37.8022i	0.843039	+	1.65456i
$523$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$523$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.8743	−	7.10739i	0.951057	−	0.309017i
$530$	−37.0529	−	19.2351i	−1.60947	−	0.835518i
$531$		0			0
$532$		0			0
$533$	−2.21704	−	4.90103i	−0.0960306	−	0.212287i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−34.0920	−	17.3707i	−1.46981	−	0.748906i
$539$		0			0
$540$		0			0
$541$	14.1327	−	43.4960i	0.607613	−	1.87004i	0.129892	−	0.991528i	$-0.458537\pi$
$541$	14.1327	−	43.4960i	0.607613	−	1.87004i	0.477721	−	0.878512i	$-0.341463\pi$
$542$		0			0
$543$		0			0
$544$	14.3593	−	44.1932i	0.615648	−	1.89477i
$545$	27.9116	+	4.63823i	1.19560	+	0.198680i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	2.59659	−	5.09609i	0.110921	−	0.217694i
$549$	−23.3094	−	32.0826i	−0.994821	−	1.36925i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	25.3292	+	34.8627i	1.07614	+	1.48117i
$555$		0			0
$556$		0			0
$557$	−3.21020	−	6.30037i	−0.136020	−	0.266955i	0.812942	−	0.582345i	$-0.197865\pi$
$557$	−3.21020	−	6.30037i	−0.136020	−	0.266955i	−0.948962	+	0.315390i	$0.897865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	34.4550	+	17.5557i	1.45340	+	0.740542i
$563$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$563$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$564$		0			0
$565$	−13.3195	−	42.0776i	−0.560357	−	1.77022i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−43.8215	+	14.2385i	−1.83709	+	0.596908i	−0.838444	+	0.544988i	$0.816534\pi$
$569$	−43.8215	+	14.2385i	−1.83709	+	0.596908i	−0.998651	+	0.0519200i	$0.983466\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−19.4164	−	14.1068i	−0.809017	−	0.587785i
$577$	−6.70636	+	6.70636i	−0.279190	+	0.279190i	−0.832785	−	0.553596i	$-0.813255\pi$
$577$	−6.70636	+	6.70636i	−0.279190	+	0.279190i	0.553596	+	0.832785i	$0.313255\pi$
$578$	−63.6034	−	32.4076i	−2.64555	−	1.34798i
$579$		0			0
$580$	−20.0000	−	40.0000i	−0.830455	−	1.66091i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	5.15889	+	1.67623i	0.213477	+	0.0693627i
$585$	−5.63529	−	0.0427799i	−0.232990	−	0.00176873i
$586$	44.3253	−	14.4022i	1.83106	−	0.594947i
$587$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$587$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−2.29388	−	14.4830i	−0.0942777	−	0.595246i
$593$	34.7070	+	5.49706i	1.42525	+	0.225737i	0.820940	−	0.571014i	$-0.193450\pi$
$593$	34.7070	+	5.49706i	1.42525	+	0.225737i	0.604307	+	0.796751i	$0.293450\pi$
$594$		0			0
$595$		0			0
$596$	−8.38705	−	11.5438i	−0.343547	−	0.472852i
$597$		0			0
$598$		0			0
$599$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$599$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$600$		0			0
$601$			42.5605i			1.73608i	0.496494	+	0.868040i	$0.334621\pi$
$601$			42.5605i			1.73608i	−0.496494	+	0.868040i	$0.665379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−17.2600	−	17.5241i	−0.701719	−	0.712454i
$606$		0			0
$607$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$607$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$608$		0			0
$609$		0			0
$610$	24.3128	+	34.0036i	0.984398	+	1.37677i
$611$		0			0
$612$	−34.8506	+	34.8506i	−1.40875	+	1.40875i
$613$	−46.9270	+	7.43250i	−1.89536	+	0.300196i	−0.991753	−	0.128162i	$-0.959092\pi$
$613$	−46.9270	+	7.43250i	−1.89536	+	0.300196i	−0.903609	+	0.428358i	$0.859092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.01435	−	25.3456i	−0.161611	−	1.02037i	−0.926523	−	0.376239i	$-0.877217\pi$
$617$	−4.01435	−	25.3456i	−0.161611	−	1.02037i	0.764911	−	0.644136i	$-0.222783\pi$
$618$		0			0
$619$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$619$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	19.7700	+	15.3019i	0.790799	+	0.612076i
$626$	−47.5528	−	15.4508i	−1.90059	−	0.617540i
$627$		0			0
$628$	7.83738	+	49.4832i	0.312745	+	1.97460i
$629$	−30.1129			−1.20068
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$		0			0
$633$		0			0
$634$	17.1165	−	12.4359i	0.679784	−	0.493892i
$635$		0			0
$636$		0			0
$637$	−0.919926	+	5.80818i	−0.0364488	+	0.230129i
$638$		0			0
$639$		0			0
$640$	20.3532	+	15.0249i	0.804532	+	0.593910i
$641$	−42.9432	+	13.9531i	−1.69615	+	0.551114i	−0.987934	−	0.154878i	$-0.950502\pi$
$641$	−42.9432	+	13.9531i	−1.69615	+	0.551114i	−0.708220	+	0.705992i	$0.750502\pi$
$642$		0			0
$643$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$643$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	11.5567	+	22.6813i	0.453990	+	0.891007i
$649$		0			0
$650$	5.93960	+	0.0901854i	0.232970	+	0.00353736i
$651$		0			0
$652$		0			0
$653$	−19.3751	−	19.3751i	−0.758206	−	0.758206i	0.217789	−	0.975996i	$-0.430115\pi$
$653$	−19.3751	−	19.3751i	−0.758206	−	0.758206i	−0.975996	+	0.217789i	$0.930115\pi$
$654$		0			0
$655$		0			0
$656$	16.0000	−	20.0000i	0.624695	−	0.780869i
$657$	−4.06829	−	4.06829i	−0.158719	−	0.158719i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−15.8405	−	48.7521i	−0.616125	−	1.89624i	−0.382752	−	0.923851i	$-0.625024\pi$
$661$	−15.8405	−	48.7521i	−0.616125	−	1.89624i	−0.233373	−	0.972387i	$-0.574976\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−4.80614	+	14.7918i	−0.186234	+	0.573169i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	26.8109	−	13.6608i	1.03348	−	0.526586i	0.146898	−	0.989152i	$-0.453071\pi$
$673$	26.8109	−	13.6608i	1.03348	−	0.526586i	0.886585	+	0.462566i	$0.153071\pi$
$674$	−10.6172	+	14.6133i	−0.408958	+	0.562883i
$675$		0			0
$676$	−19.8925	+	14.4528i	−0.765097	+	0.555876i
$677$	−44.8651	−	7.10594i	−1.72431	−	0.273103i	−0.785829	−	0.618444i	$-0.787763\pi$
$677$	−44.8651	−	7.10594i	−1.72431	−	0.273103i	−0.938477	+	0.345341i	$0.887763\pi$
$678$		0			0
$679$		0			0
$680$	37.0136	−	36.4559i	1.41941	−	1.39802i
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	5.20170	−	3.71925i	0.198747	−	0.142105i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10.5479	−	3.42721i	−0.401842	−	0.130566i
$690$		0			0
$691$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$691$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$692$	−22.7487	+	44.6468i	−0.864776	+	1.69722i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−35.4262	−	38.8780i	−1.34186	−	1.47261i
$698$	−23.2468	+	23.2468i	−0.879903	+	0.879903i
$699$		0			0
$700$		0			0
$701$	11.1836	+	34.4197i	0.422400	+	1.30001i	0.905462	+	0.424428i	$0.139525\pi$
$701$	11.1836	+	34.4197i	0.422400	+	1.30001i	−0.483061	+	0.875587i	$0.660475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	14.8270			0.558020
$707$		0			0
$708$		0			0
$709$	−0.491968	−	1.51412i	−0.0184762	−	0.0568640i	0.941393	−	0.337311i	$-0.109517\pi$
$709$	−0.491968	−	1.51412i	−0.0184762	−	0.0568640i	−0.959870	+	0.280447i	$0.909517\pi$
$710$		0			0
$711$		0			0
$712$	25.2015	+	12.8408i	0.944465	+	0.481229i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$719$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$720$	−12.0000	−	24.0000i	−0.447214	−	0.894427i
$721$		0			0
$722$	26.5392	+	4.20340i	0.987688	+	0.156434i
$723$		0			0
$724$	46.5987			1.73183
$725$	0.759100	−	49.9942i	0.0281923	−	1.85674i
$726$		0			0
$727$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$727$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$	4.25567	+	4.32078i	0.157510	+	0.159919i
$731$		0			0
$732$		0			0
$733$	−18.6679	−	9.51176i	−0.689514	−	0.351325i	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	−18.6679	−	9.51176i	−0.689514	−	0.351325i	−0.763386	+	0.645943i	$0.776464\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−24.7515	+	11.1966i	−0.911114	+	0.412154i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	5.18433	−	15.5530i	0.190580	−	0.571739i
$741$		0			0
$742$		0			0
$743$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$743$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$744$		0			0
$745$	−2.37593	−	15.7752i	−0.0870475	−	0.577959i
$746$	−12.5938	+	38.7598i	−0.461092	+	1.41910i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$751$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$752$		0			0
$753$		0			0
$754$	−6.98322	−	9.61158i	−0.254314	−	0.350033i
$755$		0			0
$756$		0			0
$757$	−8.47493	+	53.5086i	−0.308027	+	1.94480i	0.0190839	+	0.999818i	$0.493925\pi$
$757$	−8.47493	+	53.5086i	−0.308027	+	1.94480i	−0.327111	+	0.944986i	$0.606075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	43.8925	+	31.8897i	1.59110	+	1.15600i	0.902351	+	0.431003i	$0.141840\pi$
$761$	43.8925	+	31.8897i	1.59110	+	1.15600i	0.688749	+	0.724999i	$0.258160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−52.5345	+	16.6297i	−1.89939	+	0.601247i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	52.2788	+	16.9864i	1.88522	+	0.612546i	0.983700	+	0.179815i	$0.0575500\pi$
$769$	52.2788	+	16.9864i	1.88522	+	0.612546i	0.901523	+	0.432731i	$0.142450\pi$
$770$		0			0
$771$		0			0
$772$	6.73120	−	42.4991i	0.242261	−	1.52958i
$773$	1.10616	+	6.98401i	0.0397858	+	0.251197i	0.999563	−	0.0295658i	$-0.00941245\pi$
$773$	1.10616	+	6.98401i	0.0397858	+	0.251197i	−0.959777	+	0.280763i	$0.909412\pi$
$774$		0			0
$775$		0			0
$776$			55.6353i			1.99719i
$777$		0			0
$778$	−15.7509	+	15.7509i	−0.564696	+	0.564696i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−26.6296	+	8.65248i	−0.951057	+	0.309017i
$785$	−17.7130	+	53.1391i	−0.632206	+	1.89662i
$786$		0			0
$787$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$787$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$788$	3.40088	−	3.40088i	0.121151	−	0.121151i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	7.85232	+	7.85232i	0.278844	+	0.278844i
$794$	17.2225	+	53.0055i	0.611205	+	1.88109i
$795$		0			0
$796$		0			0
$797$	−4.67172	−	29.4961i	−0.165481	−	1.04480i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−4.67172	−	29.4961i	−0.165481	−	1.04480i	0.755487	−	0.655164i	$-0.227401\pi$
$798$		0			0
$799$		0			0
$800$	12.4567	+	25.3935i	0.440411	+	0.897796i
$801$	−17.6336	−	24.2705i	−0.623051	−	0.857556i
$802$	5.55943	+	35.1009i	0.196310	+	1.23945i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−19.2673	+	37.8142i	−0.677822	+	1.33030i
$809$	15.5031	−	47.7135i	0.545059	−	1.67752i	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	15.5031	−	47.7135i	0.545059	−	1.67752i	0.720850	−	0.693091i	$-0.243752\pi$
$810$	−0.216050	+	28.4597i	−0.00759122	+	0.999971i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	50.5477	−	8.00597i	1.76736	−	0.279922i
$819$		0			0
$820$	26.1792	−	11.6039i	0.914217	−	0.405225i
$821$	−38.9003			−1.35763			−0.678816	−	0.734309i	$-0.737507\pi$
$821$	−38.9003			−1.35763			−0.678816	+	0.734309i	$0.737507\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$827$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$828$		0			0
$829$		−	35.7080i		−	1.24019i	−0.784526	−	0.620096i	$-0.787094\pi$
$829$		−	35.7080i		−	1.24019i	0.784526	−	0.620096i	$-0.212906\pi$
$830$		0			0
$831$		0			0
$832$	5.98816	+	3.05112i	0.207602	+	0.105778i
$833$	8.99508	+	56.7927i	0.311661	+	1.96775i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$839$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$840$		0			0
$841$	−57.4402	+	41.7328i	−1.98070	+	1.43906i
$842$	−1.72838	+	10.9125i	−0.0595638	+	0.376071i
$843$		0			0
$844$		0			0
$845$	−27.1842	+	4.09427i	−0.935165	+	0.140847i
$846$		0			0
$847$		0			0
$848$	−8.26093	−	52.1574i	−0.283681	−	1.79109i
$849$		0			0
$850$	55.5077	−	17.1083i	1.90390	−	0.586811i
$851$		0			0
$852$		0			0
$853$	26.3236	+	51.6630i	0.901304	+	1.76891i	0.558512	+	0.829496i	$0.311372\pi$
$853$	26.3236	+	51.6630i	0.901304	+	1.76891i	0.342792	+	0.939411i	$0.388628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−26.5754	+	52.1572i	−0.907799	+	1.78166i	−0.436069	+	0.899913i	$0.643630\pi$
$857$	−26.5754	+	52.1572i	−0.907799	+	1.78166i	−0.471731	+	0.881743i	$0.656370\pi$
$858$		0			0
$859$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$859$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$863$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$864$		0			0
$865$	−45.5721	+	32.5844i	−1.54950	+	1.10790i
$866$	12.6836	+	39.0362i	0.431008	+	1.32651i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	16.2483	+	31.8890i	0.550235	+	1.07990i
$873$	26.7900	−	52.5784i	0.906705	−	1.77951i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	54.7265	+	8.66783i	1.84798	+	0.292692i	0.979260	−	0.202606i	$-0.0649409\pi$
$877$	54.7265	+	8.66783i	1.84798	+	0.292692i	0.868723	+	0.495297i	$0.164941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−25.8885	+	18.8091i	−0.872207	+	0.633696i	−0.931178	−	0.364564i	$-0.881218\pi$
$881$	−25.8885	+	18.8091i	−0.872207	+	0.633696i	0.0589711	+	0.998260i	$0.481218\pi$
$882$	29.3328	+	4.64587i	0.987688	+	0.156434i
$883$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$883$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$884$	8.11233	−	11.1657i	0.272847	−	0.375542i
$885$		0			0
$886$		0			0
$887$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$887$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$888$		0			0
$889$		0			0
$890$	18.3927	+	25.7237i	0.616523	+	0.862261i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−43.8980	−	22.3671i	−1.46490	−	0.746401i
$899$		0			0
$900$	0.455460	−	29.9965i	0.0151820	−	0.999885i
$901$	−108.445			−3.61284
$902$		0			0
$903$		0			0
$904$	32.8145	−	45.1653i	1.09139	−	1.50217i
$905$	46.2396	+	24.0041i	1.53706	+	0.797924i
$906$		0			0
$907$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$907$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$908$		0			0
$909$	36.4173	−	26.4587i	1.20789	−	0.877581i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	−5.21777	−	16.0587i	−0.172589	−	0.531173i
$915$		0			0
$916$		−	10.9326i		−	0.361222i
$917$		0			0
$918$		0			0
$919$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$919$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$920$		0			0
$921$		0			0
$922$	42.9709	+	6.80592i	1.41517	+	0.224141i
$923$		0			0
$924$		0			0
$925$	13.1561	−	12.7626i	0.432570	−	0.419630i
$926$		0			0
$927$		0			0
$928$	25.6816	−	50.4029i	0.843039	−	1.65456i
$929$	31.3879			1.02980			0.514902	−	0.857249i	$-0.327828\pi$
$929$	31.3879			1.02980			0.514902	+	0.857249i	$0.327828\pi$
$930$		0			0
$931$		0			0
$932$	10.6558	+	20.9132i	0.349043	+	0.685035i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−4.18993	−	5.76695i	−0.136952	−	0.188499i
$937$	6.98401	−	1.10616i	0.228158	−	0.0361366i	−0.0413087	−	0.999146i	$-0.513153\pi$
$937$	6.98401	−	1.10616i	0.228158	−	0.0361366i	0.269466	+	0.963010i	$0.413153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.888418	−	0.645473i	−0.0289616	−	0.0210418i	0.573210	−	0.819408i	$-0.305698\pi$
$941$	−0.888418	−	0.645473i	−0.0289616	−	0.0210418i	−0.602172	+	0.798366i	$0.705698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$947$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$948$		0			0
$949$	1.30342	+	0.946992i	0.0423109	+	0.0307407i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.3236	−	51.6630i	−0.852706	−	1.67353i	−0.732486	−	0.680782i	$-0.761640\pi$
$953$	−26.3236	−	51.6630i	−0.852706	−	1.67353i	−0.120219	−	0.992747i	$-0.538360\pi$
$954$	−17.3083	+	53.2695i	−0.560378	+	1.72467i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−25.0795	−	18.2213i	−0.809017	−	0.587785i
$962$	0.681314	−	4.30165i	0.0219664	−	0.138691i
$963$		0			0
$964$	−28.3379	−	39.0038i	−0.912704	−	1.25623i
$965$	28.5717	−	38.7042i	0.919754	−	1.24593i
$966$		0			0
$967$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$967$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$968$	4.86710	−	30.7296i	0.156434	−	0.987688i
$969$		0			0
$970$	−28.6591	+	55.2065i	−0.920188	+	1.77258i
$971$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$971$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−16.3393	+	50.2872i	−0.523008	+	1.60965i
$977$	−7.74311	+	48.8881i	−0.247724	+	1.56407i	0.479421	+	0.877585i	$0.340847\pi$
$977$	−7.74311	+	48.8881i	−0.247724	+	1.56407i	−0.727145	+	0.686483i	$0.759153\pi$
$978$		0			0
$979$		0			0
$980$	−30.8815	−	5.13175i	−0.986472	−	0.163928i
$981$		−	37.9609i		−	1.21200i
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	5.12655	−	1.62280i	0.163346	−	0.0517066i
$986$	−93.9825	−	68.2823i	−2.99301	−	2.17455i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$991$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	2.12681	−	0.336854i	0.0673568	−	0.0106683i	−0.122665	−	0.992448i	$-0.539144\pi$
$997$	2.12681	−	0.336854i	0.0673568	−	0.0106683i	0.190022	+	0.981780i	$0.439144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	820.2.bm.a.687.1	✓	8
4.3	odd	2	CM	820.2.bm.a.687.1	✓	8
5.3	odd	4		820.2.bm.b.523.1	yes	8
20.3	even	4		820.2.bm.b.523.1	yes	8
41.4	even	10		820.2.bm.b.127.1	yes	8
164.127	odd	10		820.2.bm.b.127.1	yes	8
205.168	odd	20	inner	820.2.bm.a.783.1	yes	8
820.783	even	20	inner	820.2.bm.a.783.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
820.2.bm.a.687.1	✓	8	1.1	even	1	trivial
820.2.bm.a.687.1	✓	8	4.3	odd	2	CM
820.2.bm.a.783.1	yes	8	205.168	odd	20	inner
820.2.bm.a.783.1	yes	8	820.783	even	20	inner
820.2.bm.b.127.1	yes	8	41.4	even	10
820.2.bm.b.127.1	yes	8	164.127	odd	10
820.2.bm.b.523.1	yes	8	5.3	odd	4
820.2.bm.b.523.1	yes	8	20.3	even	4

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:	\( N \)	\(=\)	\( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	820.bm (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.221232 + 1.39680i) q^{2} +(-1.90211 + 0.618034i) q^{4} +(-2.20582 - 0.366554i) q^{5} +(-1.28408 - 2.52015i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(0.221232 + 1.39680i) q^{2} +(-1.90211 + 0.618034i) q^{4} +(-2.20582 - 0.366554i) q^{5} +(-1.28408 - 2.52015i) q^{8} +3.00000i q^{9} +(0.0240055 - 3.16219i) q^{10} +(-0.131418 - 0.829740i) q^{13} +(3.23607 - 2.35114i) q^{16} +(-3.72925 - 7.31906i) q^{17} +(-4.19041 + 0.663695i) q^{18} +(4.42226 - 0.666045i) q^{20} +(4.73128 + 1.61710i) q^{25} +(1.12991 - 0.367130i) q^{26} +(-3.09017 - 9.51057i) q^{29} +(4.00000 + 4.00000i) q^{32} +(9.39825 - 6.82823i) q^{34} +(-1.85410 - 5.70634i) q^{36} +(1.66427 - 3.26632i) q^{37} +(1.90868 + 6.02967i) q^{40} +(6.17499 - 1.69394i) q^{41} +(1.09966 - 6.61746i) q^{45} +(-6.65740 - 2.16312i) q^{49} +(-1.21206 + 6.96641i) q^{50} +(0.762779 + 1.49704i) q^{52} +(5.99354 - 11.7630i) q^{53} +(12.6007 - 6.42040i) q^{58} +(-10.6942 + 7.76980i) q^{61} +(-4.70228 + 6.47214i) q^{64} +(-0.0142600 + 1.87843i) q^{65} +(11.6169 + 11.6169i) q^{68} +(7.56044 - 3.85224i) q^{72} +(-1.35610 + 1.35610i) q^{73} +(4.93059 + 1.60205i) q^{74} +(-8.00000 + 4.00000i) q^{80} -9.00000 q^{81} +(3.73221 + 8.25049i) q^{82} +(5.54322 + 17.5115i) q^{85} +(-8.09017 + 5.87785i) q^{89} +(9.48656 + 0.0720166i) q^{90} +(-17.5261 - 8.93001i) q^{97} +(1.54862 - 9.77762i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 4 q^{5} - 4 q^{8}+O(q^{10}) \) 8 * q + 2 * q^2 - 4 * q^5 - 4 * q^8	\( 8 q + 2 q^{2} - 4 q^{5} - 4 q^{8} + 6 q^{10} - 10 q^{13} + 8 q^{16} + 10 q^{17} - 6 q^{18} - 4 q^{20} - 6 q^{25} - 10 q^{26} + 20 q^{29} + 32 q^{32} + 30 q^{34} + 12 q^{36} + 46 q^{37} - 4 q^{40} + 8 q^{41} - 6 q^{45} + 14 q^{50} - 40 q^{52} - 10 q^{53} - 20 q^{58} - 24 q^{61} + 50 q^{65} + 20 q^{68} - 12 q^{72} - 22 q^{73} + 50 q^{74} - 64 q^{80} - 72 q^{81} - 18 q^{82} + 30 q^{85} - 20 q^{89} - 6 q^{90} - 80 q^{97} + 14 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 - 4 * q^5 - 4 * q^8 + 6 * q^10 - 10 * q^13 + 8 * q^16 + 10 * q^17 - 6 * q^18 - 4 * q^20 - 6 * q^25 - 10 * q^26 + 20 * q^29 + 32 * q^32 + 30 * q^34 + 12 * q^36 + 46 * q^37 - 4 * q^40 + 8 * q^41 - 6 * q^45 + 14 * q^50 - 40 * q^52 - 10 * q^53 - 20 * q^58 - 24 * q^61 + 50 * q^65 + 20 * q^68 - 12 * q^72 - 22 * q^73 + 50 * q^74 - 64 * q^80 - 72 * q^81 - 18 * q^82 + 30 * q^85 - 20 * q^89 - 6 * q^90 - 80 * q^97 + 14 * q^98

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