LMFDB - Embedded newform 820.2.bm.a.127.1

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			127.1
Root			$-0.587785 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	820.127
Dual form			820.2.bm.a.523.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.39680 + 0.221232i) q^{2} +(1.90211 + 0.618034i) q^{4} +(-1.03025 + 1.98459i) q^{5} +(2.52015 + 1.28408i) q^{8} +3.00000i q^{9} +O(q^{10})$	\(q+(1.39680 + 0.221232i) q^{2} +(1.90211 + 0.618034i) q^{4} +(-1.03025 + 1.98459i) q^{5} +(2.52015 + 1.28408i) q^{8} +3.00000i q^{9} +(-1.87811 + 2.54415i) q^{10} +(-3.48662 - 0.552226i) q^{13} +(3.23607 + 2.35114i) q^{16} +(2.87515 + 1.46496i) q^{17} +(-0.663695 + 4.19041i) q^{18} +(-3.18619 + 3.13818i) q^{20} +(-2.87718 - 4.08924i) q^{25} +(-4.74794 - 1.54270i) q^{26} +(-3.09017 + 9.51057i) q^{29} +(4.00000 + 4.00000i) q^{32} +(3.69192 + 2.68233i) q^{34} +(-1.85410 + 5.70634i) q^{36} +(8.71769 - 4.44189i) q^{37} +(-5.14475 + 3.67853i) q^{40} +(0.297142 - 6.39623i) q^{41} +(-5.95376 - 3.09075i) q^{45} +(6.65740 - 2.16312i) q^{49} +(-3.11418 - 6.34838i) q^{50} +(-6.29064 - 3.20524i) q^{52} +(1.56877 - 0.799328i) q^{53} +(-6.42040 + 12.6007i) q^{58} +(4.69421 + 3.41054i) q^{61} +(4.70228 + 6.47214i) q^{64} +(4.68802 - 6.35056i) q^{65} +(4.56346 + 4.56346i) q^{68} +(-3.85224 + 7.56044i) q^{72} +(8.15447 - 8.15447i) q^{73} +(13.1596 - 4.27581i) q^{74} +(-8.00000 + 4.00000i) q^{80} -9.00000 q^{81} +(1.83010 - 8.86853i) q^{82} +(-5.86946 + 4.19671i) q^{85} +(-8.09017 - 5.87785i) q^{89} +(-7.63246 - 5.63432i) q^{90} +(-6.94600 - 13.6323i) q^{97} +(9.77762 - 1.54862i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/820\mathbb{Z}\right)^\times$.

$n$	$411$	$621$	$657$
$\chi(n)$	$-1$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{1}{4}\right)$

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 $

$2$	1.39680	+	0.221232i	0.987688	+	0.156434i
$2$	1.39680	+	0.221232i	0.987688	+	0.156434i
$3$		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
$5$	−1.03025	+	1.98459i	−0.460741	+	0.887535i
$5$	−1.03025	+	1.98459i	−0.460741	+	0.887535i
$6$		0			0
$7$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$7$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$8$	2.52015	+	1.28408i	0.891007	+	0.453990i
$9$			3.00000i			1.00000i
$10$	−1.87811	+	2.54415i	−0.593910	+	0.804532i
$11$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$11$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$12$		0			0
$13$	−3.48662	−	0.552226i	−0.967013	−	0.153160i	−0.347098	−	0.937829i	$-0.612833\pi$
$13$	−3.48662	−	0.552226i	−0.967013	−	0.153160i	−0.619915	+	0.784669i	$0.712833\pi$
$14$		0			0
$15$		0			0
$16$	3.23607	+	2.35114i	0.809017	+	0.587785i
$17$	2.87515	+	1.46496i	0.697325	+	0.355305i	0.766451	−	0.642303i	$-0.222021\pi$
$17$	2.87515	+	1.46496i	0.697325	+	0.355305i	−0.0691254	+	0.997608i	$0.522021\pi$
$18$	−0.663695	+	4.19041i	−0.156434	+	0.987688i
$19$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$19$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$20$	−3.18619	+	3.13818i	−0.712454	+	0.701719i
$21$		0			0
$22$		0			0
$23$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$23$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$24$		0			0
$25$	−2.87718	−	4.08924i	−0.575435	−	0.817848i
$26$	−4.74794	−	1.54270i	−0.931148	−	0.302548i
$27$		0			0
$28$		0			0
$29$	−3.09017	+	9.51057i	−0.573830	+	1.76607i	0.0662984	+	0.997800i	$0.478881\pi$
$29$	−3.09017	+	9.51057i	−0.573830	+	1.76607i	−0.640129	+	0.768268i	$0.721119\pi$
$30$		0			0
$31$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$31$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$		0			0
$34$	3.69192	+	2.68233i	0.633158	+	0.460016i
$35$		0			0
$36$	−1.85410	+	5.70634i	−0.309017	+	0.951057i
$37$	8.71769	−	4.44189i	1.43318	−	0.730242i	0.446786	−	0.894641i	$-0.352568\pi$
$37$	8.71769	−	4.44189i	1.43318	−	0.730242i	0.986394	+	0.164399i	$0.0525685\pi$
$38$		0			0
$39$		0			0
$40$	−5.14475	+	3.67853i	−0.813456	+	0.581627i
$41$	0.297142	−	6.39623i	0.0464057	−	0.998923i
$41$	0.297142	−	6.39623i	0.0464057	−	0.998923i
$42$		0			0
$43$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$43$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$44$		0			0
$45$	−5.95376	−	3.09075i	−0.887535	−	0.460741i
$46$		0			0
$47$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$47$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$48$		0			0
$49$	6.65740	−	2.16312i	0.951057	−	0.309017i
$50$	−3.11418	−	6.34838i	−0.440411	−	0.897796i
$51$		0			0
$52$	−6.29064	−	3.20524i	−0.872355	−	0.444487i
$53$	1.56877	−	0.799328i	0.215487	−	0.109796i	−0.342916	−	0.939366i	$-0.611414\pi$
$53$	1.56877	−	0.799328i	0.215487	−	0.109796i	0.558403	+	0.829570i	$0.311414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−6.42040	+	12.6007i	−0.843039	+	1.65456i
$59$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$59$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$60$		0			0
$61$	4.69421	+	3.41054i	0.601032	+	0.436675i	0.846245	−	0.532794i	$-0.178858\pi$
$61$	4.69421	+	3.41054i	0.601032	+	0.436675i	−0.245213	+	0.969469i	$0.578858\pi$
$62$		0			0
$63$		0			0
$64$	4.70228	+	6.47214i	0.587785	+	0.809017i
$65$	4.68802	−	6.35056i	0.581478	−	0.787691i
$66$		0			0
$67$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$67$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$68$	4.56346	+	4.56346i	0.553401	+	0.553401i
$69$		0			0
$70$		0			0
$71$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$71$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$72$	−3.85224	+	7.56044i	−0.453990	+	0.891007i
$73$	8.15447	−	8.15447i	0.954408	−	0.954408i	−0.0445966	−	0.999005i	$-0.514200\pi$
$73$	8.15447	−	8.15447i	0.954408	−	0.954408i	0.999005	+	0.0445966i	$0.0142003\pi$
$74$	13.1596	−	4.27581i	1.52977	−	0.497053i
$75$		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
$82$	1.83010	−	8.86853i	0.202100	−	0.979365i
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	−5.86946	+	4.19671i	−0.636632	+	0.455197i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.09017	−	5.87785i	−0.857556	−	0.623051i	0.0696627	−	0.997571i	$-0.477808\pi$
$89$	−8.09017	−	5.87785i	−0.857556	−	0.623051i	−0.927219	+	0.374519i	$0.877808\pi$
$90$	−7.63246	−	5.63432i	−0.804532	−	0.593910i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.94600	−	13.6323i	−0.705260	−	1.38415i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−6.94600	−	13.6323i	−0.705260	−	1.38415i	0.208552	−	0.978011i	$-0.433125\pi$
$98$	9.77762	−	1.54862i	0.987688	−	0.156434i
$99$		0			0
$100$	−2.94542	−	9.55638i	−0.294542	−	0.955638i
$101$	10.2015	−	14.0412i	1.01509	−	1.39715i	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	10.2015	−	14.0412i	1.01509	−	1.39715i	0.915588	−	0.402117i	$-0.131726\pi$
$102$		0			0
$103$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$103$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$104$	−8.07768	−	5.86878i	−0.792082	−	0.575481i
$105$		0			0
$106$	2.36810	−	0.769441i	0.230010	−	0.0747348i
$107$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$107$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$108$		0			0
$109$	−19.7071			−1.88759			−0.943797	−	0.330527i	$-0.892774\pi$
$109$	−19.7071			−1.88759			−0.943797	+	0.330527i	$0.892774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18.3654	+	9.35764i	1.72767	+	0.880293i	0.975095	+	0.221788i	$0.0711893\pi$
$113$	18.3654	+	9.35764i	1.72767	+	0.880293i	0.752577	+	0.658505i	$0.228811\pi$
$114$		0			0
$115$		0			0
$116$	−11.7557	+	16.1803i	−1.09149	+	1.50231i
$117$	1.65668	−	10.4598i	0.153160	−	0.967013i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.89919	−	6.46564i	0.809017	−	0.587785i
$122$	5.80236	+	5.80236i	0.525321	+	0.525321i
$123$		0			0
$124$		0			0
$125$	11.0797	−	1.49707i	0.990995	−	0.133902i
$126$		0			0
$127$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$127$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$128$	5.13632	+	10.0806i	0.453990	+	0.891007i
$129$		0			0
$130$	7.95319	−	7.83334i	0.697541	−	0.687030i
$131$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$131$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	5.36467	+	7.38383i	0.460016	+	0.633158i
$137$	11.2927	−	11.2927i	0.964796	−	0.964796i	−0.0346048	−	0.999401i	$-0.511017\pi$
$137$	11.2927	−	11.2927i	0.964796	−	0.964796i	0.999401	+	0.0346048i	$0.0110173\pi$
$138$		0			0
$139$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$139$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−7.05342	+	9.70820i	−0.587785	+	0.809017i
$145$	−15.6909	−	15.9310i	−1.30306	−	1.32299i
$146$	13.1942	−	9.58615i	1.09196	−	0.793356i
$147$		0			0
$148$	19.3273	−	3.06114i	1.58869	−	0.251624i
$149$	−6.02433	+	18.5410i	−0.493532	+	1.51894i	0.325700	+	0.945473i	$0.394400\pi$
$149$	−6.02433	+	18.5410i	−0.493532	+	1.51894i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$151$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$152$		0			0
$153$	−4.39488	+	8.62544i	−0.355305	+	0.697325i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−20.4253	+	3.23504i	−1.63011	+	0.258185i	−0.903414	−	0.428770i	$-0.858947\pi$
$157$	−20.4253	+	3.23504i	−1.63011	+	0.258185i	−0.726700	+	0.686955i	$0.758947\pi$
$158$		0			0
$159$		0			0
$160$	−12.0593	+	3.81736i	−0.953375	+	0.301788i
$161$		0			0
$162$	−12.5712	−	1.99109i	−0.987688	−	0.156434i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$	4.51828	−	11.9827i	0.352819	−	0.935692i
$165$		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
$169$	−0.512197	−	0.166423i	−0.0393997	−	0.0128018i
$170$	−9.12692	+	4.56346i	−0.700003	+	0.350001i
$171$		0			0
$172$		0			0
$173$	0.0824082	−	0.0824082i	0.00626538	−	0.00626538i	−0.703967	−	0.710233i	$-0.748590\pi$
$173$	0.0824082	−	0.0824082i	0.00626538	−	0.00626538i	0.710233	+	0.703967i	$0.248590\pi$
$174$		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	$-0.400000\pi$
$179$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$180$	−9.41454	−	9.55858i	−0.701719	−	0.712454i
$181$	−10.4033	+	3.38024i	−0.773271	+	0.251251i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−10.4033	+	3.38024i	−0.773271	+	0.251251i	−0.104306	+	0.994545i	$0.533262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.166080	+	21.8773i	−0.0122104	+	1.60845i
$186$		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
$193$	−8.97530	+	4.57314i	−0.646056	+	0.329182i	−0.746132	−	0.665798i	$-0.768091\pi$
$193$	−8.97530	+	4.57314i	−0.646056	+	0.329182i	0.100076	+	0.994980i	$0.468091\pi$
$194$	−6.68629	−	20.5783i	−0.480048	−	1.47744i
$195$		0			0
$196$	14.0000			1.00000
$197$	12.4132	−	24.3623i	0.884403	−	1.73574i	0.240472	−	0.970656i	$-0.422698\pi$
$197$	12.4132	−	24.3623i	0.884403	−	1.73574i	0.643932	−	0.765083i	$-0.277302\pi$
$198$		0			0
$199$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$199$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$200$	−2.00000	−	14.0000i	−0.141421	−	0.989949i
$201$		0			0
$202$	17.3559	−	17.3559i	1.22116	−	1.22116i
$203$		0			0
$204$		0			0
$205$	12.3877	+	7.17941i	0.865197	+	0.501432i
$206$		0			0
$207$		0			0
$208$	−9.98457	−	9.98457i	−0.692305	−	0.692305i
$209$		0			0
$210$		0			0
$211$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$211$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$212$	3.47799	−	0.550859i	0.238869	−	0.0378332i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−27.5269	−	4.35983i	−1.86435	−	0.295285i
$219$		0			0
$220$		0			0
$221$	−9.21554	−	6.69548i	−0.619905	−	0.450387i
$222$		0			0
$223$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$223$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$224$		0			0
$225$	12.2677	−	8.63153i	0.817848	−	0.575435i
$226$	23.5826	+	17.1338i	1.56869	+	1.13972i
$227$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$227$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$228$		0			0
$229$	−4.04032	−	12.4348i	−0.266992	−	0.821716i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−4.04032	−	12.4348i	−0.266992	−	0.821716i	0.724236	−	0.689552i	$-0.242192\pi$
$230$		0			0
$231$		0			0
$232$	−20.0000	+	20.0000i	−1.31306	+	1.31306i
$233$	22.8916	+	3.62567i	1.49968	+	0.237526i	0.851658	−	0.524097i	$-0.175597\pi$
$233$	22.8916	+	3.62567i	1.49968	+	0.237526i	0.648020	+	0.761623i	$0.275597\pi$
$234$	4.62810	−	14.2438i	0.302548	−	0.931148i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$239$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$240$		0			0
$241$	−3.44907	+	10.6151i	−0.222174	+	0.683781i	0.776392	+	0.630250i	$0.217048\pi$
$241$	−3.44907	+	10.6151i	−0.222174	+	0.683781i	−0.998566	+	0.0535313i	$0.982952\pi$
$242$	13.8608	−	7.06243i	0.891007	−	0.453990i
$243$		0			0
$244$	6.82108	+	9.38842i	0.436675	+	0.601032i
$245$	−2.56587	+	15.4407i	−0.163928	+	0.986472i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	15.8073	+	0.360055i	0.999741	+	0.0227719i
$251$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$251$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	+	15.2169i	0.309017	+	0.951057i
$257$	18.9011	+	9.63059i	1.17902	+	0.600740i	0.929928	−	0.367740i	$-0.119869\pi$
$257$	18.9011	+	9.63059i	1.17902	+	0.600740i	0.249090	+	0.968480i	$0.419869\pi$
$258$		0			0
$259$		0			0
$260$	12.8420	−	9.18213i	0.796428	−	0.569452i
$261$	−28.5317	−	9.27051i	−1.76607	−	0.573830i
$262$		0			0
$263$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$263$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$264$		0			0
$265$	−0.0298865	+	3.93687i	−0.00183591	+	0.241840i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.45781	−	8.88842i	−0.393740	−	0.541936i	0.565419	−	0.824804i	$-0.308714\pi$
$269$	−6.45781	−	8.88842i	−0.393740	−	0.541936i	−0.959159	+	0.282867i	$0.908714\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$	5.85984	+	11.5006i	0.355305	+	0.697325i
$273$		0			0
$274$	18.2719	−	13.2753i	1.10385	−	0.801991i
$275$		0			0
$276$		0			0
$277$	−10.0598	+	19.7434i	−0.604433	+	1.18627i	0.362678	+	0.931914i	$0.381862\pi$
$277$	−10.0598	+	19.7434i	−0.604433	+	1.18627i	−0.967112	+	0.254353i	$0.918138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	19.7049	+	27.1215i	1.17550	+	1.61793i	0.596869	+	0.802339i	$0.296411\pi$
$281$	19.7049	+	27.1215i	1.17550	+	1.61793i	0.578627	+	0.815592i	$0.303589\pi$
$282$		0			0
$283$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$283$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−12.0000	+	12.0000i	−0.707107	+	0.707107i
$289$	−3.87199	−	5.32934i	−0.227764	−	0.313490i
$290$	−18.3927	−	25.7237i	−1.08005	−	1.51055i
$291$		0			0
$292$	20.5505	−	10.4710i	1.20262	−	0.612768i
$293$	−1.35118	−	0.214005i	−0.0789365	−	0.0125023i	0.116841	−	0.993151i	$-0.462723\pi$
$293$	−1.35118	−	0.214005i	−0.0789365	−	0.0125023i	−0.195778	+	0.980648i	$0.562723\pi$
$294$		0			0
$295$		0			0
$296$	27.6736			1.60850
$297$		0			0
$298$	−12.5166	+	24.5653i	−0.725070	+	1.42303i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−11.6047	+	5.80236i	−0.664484	+	0.332242i
$306$	−8.04700	+	11.0757i	−0.460016	+	0.633158i
$307$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$307$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$311$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$312$		0			0
$313$	31.5018	−	16.0510i	1.78059	−	0.907255i	0.872149	−	0.489240i	$-0.162726\pi$
$313$	31.5018	−	16.0510i	1.78059	−	0.907255i	0.908440	−	0.418016i	$-0.137274\pi$
$314$	−29.2458			−1.65043
$315$		0			0
$316$		0			0
$317$	6.13940	+	3.12818i	0.344823	+	0.175696i	0.617822	−	0.786318i	$-0.288015\pi$
$317$	6.13940	+	3.12818i	0.344823	+	0.175696i	−0.272999	+	0.962014i	$0.588015\pi$
$318$		0			0
$319$		0			0
$320$	−17.6890	+	2.66418i	−0.988847	+	0.148932i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−17.1190	−	5.56231i	−0.951057	−	0.309017i
$325$	7.77342	+	15.8465i	0.431192	+	0.879003i
$326$		0			0
$327$		0			0
$328$	8.96210	−	15.7379i	0.494849	−	0.868979i
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	13.3257	+	26.1531i	0.730242	+	1.43318i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−20.3578	−	20.3578i	−1.10896	−	1.10896i	−0.993288	−	0.115670i	$-0.963099\pi$
$337$	−20.3578	−	20.3578i	−1.10896	−	1.10896i	−0.115670	−	0.993288i	$-0.536901\pi$
$338$	−0.678619	−	0.345774i	−0.0369120	−	0.0188076i
$339$		0			0
$340$	−13.7581	+	4.35509i	−0.746137	+	0.236188i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.133339	−	0.0968766i	0.00716836	−	0.00520812i
$347$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$347$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$348$		0			0
$349$	−20.5739	+	28.3176i	−1.10130	+	1.51581i	−0.267644	+	0.963518i	$0.586245\pi$
$349$	−20.5739	+	28.3176i	−1.10130	+	1.51581i	−0.833653	+	0.552288i	$0.813755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−30.6958	+	4.86174i	−1.63377	+	0.258764i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−30.6958	+	4.86174i	−1.63377	+	0.258764i	−0.728955	+	0.684561i	$0.759994\pi$
$354$		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	$-0.100000\pi$
$359$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$360$	−11.0356	−	15.4342i	−0.581627	−	0.813456i
$361$	5.87132	+	18.0701i	0.309017	+	0.951057i
$362$	−15.2792	+	2.41998i	−0.803055	+	0.127191i
$363$		0			0
$364$		0			0
$365$	7.78213	+	24.5844i	0.407335	+	1.28681i
$366$		0			0
$367$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$367$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$368$		0			0
$369$	19.1887	+	0.891425i	0.998923	+	0.0464057i
$370$	−5.07193	+	30.5215i	−0.263677	+	1.58674i
$371$		0			0
$372$		0			0
$373$	17.4478	+	34.2433i	0.903413	+	1.77305i	0.540967	+	0.841044i	$0.318058\pi$
$373$	17.4478	+	34.2433i	0.903413	+	1.77305i	0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	16.0262	−	31.4532i	0.825392	−	1.61992i
$378$		0			0
$379$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$379$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$380$		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
$386$	−13.5484	+	4.40215i	−0.689597	+	0.224064i
$387$		0			0
$388$	−4.78686	−	30.2230i	−0.243016	−	1.53434i
$389$	−23.0778	+	31.7639i	−1.17009	+	1.61049i	−0.507020	+	0.861934i	$0.669253\pi$
$389$	−23.0778	+	31.7639i	−1.17009	+	1.61049i	−0.663070	+	0.748557i	$0.730747\pi$
$390$		0			0
$391$		0			0
$392$	19.5552	+	3.09724i	0.987688	+	0.156434i
$393$		0			0
$394$	22.7285	−	31.2831i	1.14504	−	1.57602i
$395$		0			0
$396$		0			0
$397$	2.78403	−	17.5777i	0.139726	−	0.882198i	−0.813856	−	0.581066i	$-0.802636\pi$
$397$	2.78403	−	17.5777i	0.139726	−	0.882198i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$		0			0
$399$		0			0
$400$	0.303640	−	19.9977i	0.0151820	−	0.999885i
$401$	−21.8934			−1.09330			−0.546652	−	0.837360i	$-0.684098\pi$
$401$	−21.8934			−1.09330			−0.546652	+	0.837360i	$0.684098\pi$
$402$		0			0
$403$		0			0
$404$	28.0825	−	20.4031i	1.39715	−	1.01509i
$405$	9.27224	−	17.8613i	0.460741	−	0.887535i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			39.8964i			1.97275i	0.164518	+	0.986374i	$0.447393\pi$
$409$			39.8964i			1.97275i	−0.164518	+	0.986374i	$0.552607\pi$
$410$	15.7149	+	12.7688i	0.776104	+	0.630605i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−11.7376	−	16.1554i	−0.575481	−	0.792082i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−38.7351	+	12.5858i	−1.88783	+	0.613394i	−0.906123	+	0.423015i	$0.860972\pi$
$421$	−38.7351	+	12.5858i	−1.88783	+	0.613394i	−0.981711	+	0.190380i	$0.939028\pi$
$422$		0			0
$423$		0			0
$424$	4.97993			0.241847
$425$	−2.28173	−	15.9721i	−0.110680	−	0.774761i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$431$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$432$		0			0
$433$	5.84061	−	36.8762i	0.280682	−	1.77216i	−0.296001	−	0.955188i	$-0.595653\pi$
$433$	5.84061	−	36.8762i	0.280682	−	1.77216i	0.576683	−	0.816968i	$-0.304347\pi$
$434$		0			0
$435$		0			0
$436$	−37.4850	−	12.1796i	−1.79521	−	0.583298i
$437$		0			0
$438$		0			0
$439$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$439$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$440$		0			0
$441$	6.48936	+	19.9722i	0.309017	+	0.951057i
$442$	−11.3910	−	11.3910i	−0.541816	−	0.541816i
$443$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$443$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$444$		0			0
$445$	20.0000	−	10.0000i	0.948091	−	0.474045i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−7.16226	−	9.85801i	−0.338008	−	0.465228i	0.605850	−	0.795579i	$-0.292833\pi$
$449$	−7.16226	−	9.85801i	−0.338008	−	0.465228i	−0.943858	+	0.330350i	$0.892833\pi$
$450$	19.0451	−	9.34253i	0.897796	−	0.440411i
$451$		0			0
$452$	29.1497	+	29.1497i	1.37109	+	1.37109i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.28598	−	33.3743i	0.247267	−	1.56119i	−0.481514	−	0.876439i	$-0.659913\pi$
$457$	5.28598	−	33.3743i	0.247267	−	1.56119i	0.728781	−	0.684747i	$-0.240087\pi$
$458$	−2.89255	−	18.2628i	−0.135160	−	0.853366i
$459$		0			0
$460$		0			0
$461$	−2.24918	−	6.92225i	−0.104755	−	0.322401i	0.884918	−	0.465746i	$-0.154214\pi$
$461$	−2.24918	−	6.92225i	−0.104755	−	0.322401i	−0.989673	+	0.143345i	$0.954214\pi$
$462$		0			0
$463$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$463$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$464$	−32.3607	+	23.5114i	−1.50231	+	1.09149i
$465$		0			0
$466$	31.1729	+	10.1287i	1.44406	+	0.469203i
$467$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$467$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$468$	9.61573	−	18.8719i	0.444487	−	0.872355i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	2.39798	+	4.70631i	0.109796	+	0.215487i
$478$		0			0
$479$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$479$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$480$		0			0
$481$	−32.8482	+	10.6730i	−1.49775	+	0.486648i
$482$	−7.16607	+	14.0642i	−0.326406	+	0.640607i
$483$		0			0
$484$	20.9232	−	6.79837i	0.951057	−	0.309017i
$485$	34.2106	+	0.259707i	1.55342	+	0.0117927i
$486$		0			0
$487$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$487$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$488$	7.45069	+	14.6228i	0.337277	+	0.661943i
$489$		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
$493$	−22.8173	+	22.8173i	−1.02764	+	1.02764i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$499$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$500$	22.0000	+	4.00000i	0.983870	+	0.178885i
$501$		0			0
$502$		0			0
$503$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$503$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$504$		0			0
$505$	17.3559	+	34.7118i	0.772328	+	1.54466i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.4920	+	32.2910i	0.465048	+	1.43127i	0.858922	+	0.512107i	$0.171135\pi$
$509$	10.4920	+	32.2910i	0.465048	+	1.43127i	−0.393873	+	0.919165i	$0.628865\pi$
$510$		0			0
$511$		0			0
$512$	3.53971	+	22.3488i	0.156434	+	0.987688i
$513$		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
$520$	19.9691	−	9.98457i	0.875704	−	0.437852i
$521$	−8.14348	−	2.64598i	−0.356772	−	0.115922i	0.125146	−	0.992138i	$-0.460060\pi$
$521$	−8.14348	−	2.64598i	−0.356772	−	0.115922i	−0.481919	+	0.876216i	$0.660060\pi$
$522$	−37.8022	−	19.2612i	−1.65456	−	0.843039i
$523$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$523$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−21.8743	−	7.10739i	−0.951057	−	0.309017i
$530$	−0.912705	+	5.49241i	−0.0396454	+	0.238575i
$531$		0			0
$532$		0			0
$533$	−4.56818	+	22.1371i	−0.197870	+	0.958864i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−7.05389	−	13.8440i	−0.304115	−	0.596859i
$539$		0			0
$540$		0			0
$541$	6.86729	+	21.1353i	0.295248	+	0.908679i	0.983138	+	0.182865i	$0.0585370\pi$
$541$	6.86729	+	21.1353i	0.295248	+	0.908679i	−0.687890	+	0.725815i	$0.741463\pi$
$542$		0			0
$543$		0			0
$544$	5.64074	+	17.3604i	0.241845	+	0.744322i
$545$	20.3032	−	39.1104i	0.869692	−	1.67530i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	28.4591	−	14.5007i	1.21571	−	0.619437i
$549$	−10.2316	+	14.0826i	−0.436675	+	0.601032i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−18.4194	+	25.3521i	−0.782565	+	1.07711i
$555$		0			0
$556$		0			0
$557$	6.30037	+	3.21020i	0.266955	+	0.136020i	0.582345	−	0.812942i	$-0.302135\pi$
$557$	6.30037	+	3.21020i	0.266955	+	0.136020i	−0.315390	+	0.948962i	$0.602135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	21.5237	+	42.2427i	0.907923	+	1.78190i
$563$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$563$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$564$		0			0
$565$	−37.4920	+	26.8070i	−1.57730	+	1.12778i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.5391	−	9.27293i	−1.19642	−	0.388741i	−0.357979	−	0.933730i	$-0.616534\pi$
$569$	−28.5391	−	9.27293i	−1.19642	−	0.388741i	−0.838444	+	0.544988i	$0.816534\pi$
$570$		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
$577$	−33.7445	+	33.7445i	−1.40480	+	1.40480i	−0.620956	+	0.783846i	$0.713255\pi$
$577$	−33.7445	+	33.7445i	−1.40480	+	1.40480i	−0.783846	+	0.620956i	$0.786745\pi$
$578$	−4.22939	−	8.30064i	−0.175919	−	0.345261i
$579$		0			0
$580$	−20.0000	−	40.0000i	−0.830455	−	1.66091i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	31.0214	−	10.0795i	1.28368	−	0.417092i
$585$	19.0517	+	14.0641i	0.787691	+	0.581478i
$586$	−1.83998	−	0.597846i	−0.0760089	−	0.0246968i
$587$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$587$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	38.6546	+	6.12228i	1.58869	+	0.251624i
$593$	−7.54799	−	47.6561i	−0.309959	−	1.95700i	−0.289383	−	0.957214i	$-0.593450\pi$
$593$	−7.54799	−	47.6561i	−0.309959	−	1.95700i	−0.0205761	−	0.999788i	$-0.506550\pi$
$594$		0			0
$595$		0			0
$596$	−22.9179	+	31.5438i	−0.938754	+	1.29208i
$597$		0			0
$598$		0			0
$599$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$599$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$600$		0			0
$601$		−	48.7409i		−	1.98818i	−0.108550	−	0.994091i	$-0.534621\pi$
$601$		−	48.7409i		−	1.98818i	0.108550	−	0.994091i	$-0.465379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.66325	+	24.3224i	0.148932	+	0.988847i
$606$		0			0
$607$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$607$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$608$		0			0
$609$		0			0
$610$	−17.4932	+	5.53742i	−0.708278	+	0.224204i
$611$		0			0
$612$	−13.6904	+	13.6904i	−0.553401	+	0.553401i
$613$	7.29577	−	46.0637i	0.294673	−	1.86049i	−0.184579	−	0.982818i	$-0.559092\pi$
$613$	7.29577	−	46.0637i	0.294673	−	1.86049i	0.479252	−	0.877677i	$-0.340908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	32.1258	+	5.08823i	1.29334	+	0.204844i	0.764911	−	0.644136i	$-0.222783\pi$
$617$	32.1258	+	5.08823i	1.29334	+	0.204844i	0.528425	+	0.848980i	$0.322783\pi$
$618$		0			0
$619$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$619$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−8.44373	+	23.5309i	−0.337749	+	0.941236i
$626$	47.5528	−	15.4508i	1.90059	−	0.617540i
$627$		0			0
$628$	−40.8505	−	6.47009i	−1.63011	−	0.258185i
$629$	31.5718			1.25885
$630$		0			0
$631$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$631$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$632$		0			0
$633$		0			0
$634$	7.88347	+	5.72768i	0.313093	+	0.227475i
$635$		0			0
$636$		0			0
$637$	−24.4063	+	3.86558i	−0.967013	+	0.153160i
$638$		0			0
$639$		0			0
$640$	−25.2975	−	0.192044i	−0.999971	−	0.00759122i
$641$	33.9989	+	11.0469i	1.34288	+	0.436327i	0.890290	−	0.455394i	$-0.150502\pi$
$641$	33.9989	+	11.0469i	1.34288	+	0.436327i	0.452586	+	0.891721i	$0.350502\pi$
$642$		0			0
$643$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$643$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$644$		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	$0.250000\pi$
$648$	−22.6813	−	11.5567i	−0.891007	−	0.453990i
$649$		0			0
$650$	7.35220	+	23.8541i	0.288377	+	0.935634i
$651$		0			0
$652$		0			0
$653$	−2.25609	−	2.25609i	−0.0882875	−	0.0882875i	0.661584	−	0.749871i	$-0.269885\pi$
$653$	−2.25609	−	2.25609i	−0.0882875	−	0.0882875i	−0.749871	+	0.661584i	$0.769885\pi$
$654$		0			0
$655$		0			0
$656$	16.0000	−	20.0000i	0.624695	−	0.780869i
$657$	24.4634	+	24.4634i	0.954408	+	0.954408i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	13.5487	−	41.6987i	0.526985	−	1.62189i	−0.233373	−	0.972387i	$-0.574976\pi$
$661$	13.5487	−	41.6987i	0.526985	−	1.62189i	0.760358	−	0.649505i	$-0.225024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	12.8274	+	39.4787i	0.497053	+	1.52977i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.227265	+	0.446032i	−0.00876041	+	0.0171933i	−0.895345	−	0.445373i	$-0.853071\pi$
$673$	−0.227265	+	0.446032i	−0.00876041	+	0.0171933i	0.886585	+	0.462566i	$0.153071\pi$
$674$	−23.9320	−	32.9395i	−0.921825	−	1.26878i
$675$		0			0
$676$	−0.871401	−	0.633110i	−0.0335154	−	0.0243504i
$677$	3.41426	+	21.5568i	0.131221	+	0.828495i	0.962230	+	0.272237i	$0.0877633\pi$
$677$	3.41426	+	21.5568i	0.131221	+	0.828495i	−0.831010	+	0.556258i	$0.812237\pi$
$678$		0			0
$679$		0			0
$680$	−20.1808	+	3.03947i	−0.773898	+	0.116558i
$681$		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
$685$	10.7770	+	34.0455i	0.411769	+	1.30081i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.91111	+	1.92063i	−0.225195	+	0.0731703i
$690$		0			0
$691$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$691$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$692$	0.207681	−	0.105819i	0.00789483	−	0.00402262i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	10.2245	−	17.9548i	0.387282	−	0.680086i
$698$	−35.0025	+	35.0025i	−1.32486	+	1.32486i
$699$		0			0
$700$		0			0
$701$	14.8164	−	45.6001i	0.559606	−	1.72229i	−0.123852	−	0.992301i	$-0.539525\pi$
$701$	14.8164	−	45.6001i	0.559606	−	1.72229i	0.683458	−	0.729990i	$-0.260475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−43.9516			−1.65414
$707$		0			0
$708$		0			0
$709$	15.4920	−	47.6794i	0.581813	−	1.79064i	−0.0298952	−	0.999553i	$-0.509517\pi$
$709$	15.4920	−	47.6794i	0.581813	−	1.79064i	0.611708	−	0.791083i	$-0.290483\pi$
$710$		0			0
$711$		0			0
$712$	−12.8408	−	25.2015i	−0.481229	−	0.944465i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$719$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$720$	−12.0000	−	24.0000i	−0.447214	−	0.894427i
$721$		0			0
$722$	4.20340	+	26.5392i	0.156434	+	0.987688i
$723$		0			0
$724$	−21.8774			−0.813065
$725$	47.7819	−	14.7271i	1.77458	−	0.546952i
$726$		0			0
$727$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$727$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$	5.43124	+	36.0612i	0.201019	+	1.33468i
$731$		0			0
$732$		0			0
$733$	−21.0190	−	41.2522i	−0.776356	−	1.52368i	−0.850227	−	0.526416i	$-0.823536\pi$
$733$	−21.0190	−	41.2522i	−0.776356	−	1.52368i	0.0738717	−	0.997268i	$-0.476464\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	26.6056	+	5.49029i	0.979365	+	0.202100i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−13.8368	+	41.5104i	−0.508651	+	1.52595i
$741$		0			0
$742$		0			0
$743$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$743$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$744$		0			0
$745$	−30.5896	−	31.0576i	−1.12072	−	1.13786i
$746$	16.7954	+	51.6911i	0.614925	+	1.89254i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$751$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$752$		0			0
$753$		0			0
$754$	29.3439	−	40.3884i	1.06864	−	1.47086i
$755$		0			0
$756$		0			0
$757$	−25.5940	+	4.05368i	−0.930228	+	0.147334i	−0.603117	−	0.797652i	$-0.706075\pi$
$757$	−25.5940	+	4.05368i	−0.930228	+	0.147334i	−0.327111	+	0.944986i	$0.606075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.85019	−	4.25041i	0.212069	−	0.154077i	−0.476680	−	0.879077i	$-0.658160\pi$
$761$	5.85019	−	4.25041i	0.212069	−	0.154077i	0.688749	+	0.724999i	$0.258160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−12.5901	−	17.6084i	−0.455197	−	0.636632i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	38.1720	−	12.4028i	1.37652	−	0.447258i	0.474995	−	0.879989i	$-0.342450\pi$
$769$	38.1720	−	12.4028i	1.37652	−	0.447258i	0.901523	+	0.432731i	$0.142450\pi$
$770$		0			0
$771$		0			0
$772$	−19.8984	+	3.15159i	−0.716158	+	0.113428i
$773$	6.98401	+	1.10616i	0.251197	+	0.0397858i	0.280763	−	0.959777i	$-0.409412\pi$
$773$	6.98401	+	1.10616i	0.251197	+	0.0397858i	−0.0295658	+	0.999563i	$0.509412\pi$
$774$		0			0
$775$		0			0
$776$		−	43.2746i		−	1.55347i
$777$		0			0
$778$	−39.2623	+	39.2623i	−1.40762	+	1.40762i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	26.6296	+	8.65248i	0.951057	+	0.309017i
$785$	14.6229	−	43.8686i	0.521913	−	1.56574i
$786$		0			0
$787$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$787$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$788$	38.6680	−	38.6680i	1.37749	−	1.37749i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−14.4835	−	14.4835i	−0.514325	−	0.514325i
$794$	7.77747	−	23.9366i	0.276012	−	0.849478i
$795$		0			0
$796$		0			0
$797$	−54.1267	−	8.57282i	−1.91726	−	0.303665i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−54.1267	−	8.57282i	−1.91726	−	0.303665i	−0.996297	+	0.0859751i	$0.972599\pi$
$798$		0			0
$799$		0			0
$800$	4.84825	−	27.8657i	0.171412	−	0.985200i
$801$	17.6336	−	24.2705i	0.623051	−	0.857556i
$802$	−30.5807	−	4.84351i	−1.07984	−	0.171030i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	43.7394	−	22.2864i	1.53875	−	0.784031i
$809$	−17.4129	−	53.5914i	−0.612205	−	1.88417i	−0.436414	−	0.899746i	$-0.643752\pi$
$809$	−17.4129	−	53.5914i	−0.612205	−	1.88417i	−0.175791	−	0.984428i	$-0.556248\pi$
$810$	16.9030	−	22.8974i	0.593910	−	0.804532i
$811$		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
$818$	−8.82634	+	55.7273i	−0.308606	+	1.94846i
$819$		0			0
$820$	19.1258	+	21.3121i	0.667901	+	0.744250i
$821$	56.2053			1.96158			0.980789	−	0.195070i	$-0.0624935\pi$
$821$	56.2053			1.96158			0.980789	+	0.195070i	$0.0624935\pi$
$822$		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		0.987688	−	0.156434i	$-0.0500000\pi$
$827$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$828$		0			0
$829$			2.33421i			0.0810706i	0.999178	+	0.0405353i	$0.0129063\pi$
$829$			2.33421i			0.0810706i	−0.999178	+	0.0405353i	$0.987094\pi$
$830$		0			0
$831$		0			0
$832$	−12.8210	−	25.1626i	−0.444487	−	0.872355i
$833$	22.3099	+	3.53354i	0.772991	+	0.122430i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$839$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$840$		0			0
$841$	−57.4402	−	41.7328i	−1.98070	−	1.43906i
$842$	−56.8897	+	9.01044i	−1.96055	+	0.310520i
$843$		0			0
$844$		0			0
$845$	0.857970	−	0.845042i	0.0295151	−	0.0290703i
$846$		0			0
$847$		0			0
$848$	6.95598	+	1.10172i	0.238869	+	0.0378332i
$849$		0			0
$850$	0.346412	−	22.8147i	0.0118818	−	0.782536i
$851$		0			0
$852$		0			0
$853$	−51.6630	−	26.3236i	−1.76891	−	0.901304i	−0.939411	−	0.342792i	$-0.888628\pi$
$853$	−51.6630	−	26.3236i	−1.76891	−	0.901304i	−0.829496	−	0.558512i	$-0.811372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.1246	−	7.70636i	0.516646	−	0.263244i	−0.176169	−	0.984360i	$-0.556370\pi$
$857$	15.1246	−	7.70636i	0.516646	−	0.263244i	0.692814	+	0.721116i	$0.256370\pi$
$858$		0			0
$859$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$859$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$863$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$864$		0			0
$865$	0.0786453	+	0.248447i	0.00267402	+	0.00844745i
$866$	16.3164	−	50.2166i	0.554452	−	1.70643i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−49.6647	−	25.3054i	−1.68186	−	0.856949i
$873$	40.8969	−	20.8380i	1.38415	−	0.705260i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.434741	−	2.74485i	−0.0146802	−	0.0926869i	0.979260	−	0.202606i	$-0.0649409\pi$
$877$	−0.434741	−	2.74485i	−0.0146802	−	0.0926869i	−0.993941	+	0.109919i	$0.964941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−25.8885	−	18.8091i	−0.872207	−	0.633696i	0.0589711	−	0.998260i	$-0.481218\pi$
$881$	−25.8885	−	18.8091i	−0.872207	−	0.633696i	−0.931178	+	0.364564i	$0.881218\pi$
$882$	4.64587	+	29.3328i	0.156434	+	0.987688i
$883$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$883$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$884$	−13.3910	−	18.4311i	−0.450387	−	0.619905i
$885$		0			0
$886$		0			0
$887$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$887$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$888$		0			0
$889$		0			0
$890$	30.1484	−	9.54339i	1.01058	−	0.319895i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−7.82336	−	15.3542i	−0.261069	−	0.512377i
$899$		0			0
$900$	28.6692	−	8.83627i	0.955638	−	0.294542i
$901$	5.68142			0.189276
$902$		0			0
$903$		0			0
$904$	34.2676	+	47.1653i	1.13972	+	1.56869i
$905$	4.00961	−	24.1287i	0.133284	−	0.802067i
$906$		0			0
$907$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$907$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$908$		0			0
$909$	42.1237	+	30.6046i	1.39715	+	1.01509i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	14.7669	−	45.4479i	0.488446	−	1.50328i
$915$		0			0
$916$		−	26.1495i		−	0.864003i
$917$		0			0
$918$		0			0
$919$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$919$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$920$		0			0
$921$		0			0
$922$	−1.61023	−	10.1666i	−0.0530302	−	0.334819i
$923$		0			0
$924$		0			0
$925$	−43.2463	−	22.8686i	−1.42193	−	0.751916i
$926$		0			0
$927$		0			0
$928$	−50.4029	+	25.6816i	−1.65456	+	0.843039i
$929$	−56.1093			−1.84089			−0.920443	−	0.390877i	$-0.872172\pi$
$929$	−56.1093			−1.84089			−0.920443	+	0.390877i	$0.872172\pi$
$930$		0			0
$931$		0			0
$932$	41.3016	+	21.0442i	1.35288	+	0.689327i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	17.6063	−	24.2331i	0.575481	−	0.792082i
$937$	1.10616	−	6.98401i	0.0361366	−	0.228158i	−0.963010	−	0.269466i	$-0.913153\pi$
$937$	1.10616	−	6.98401i	0.0361366	−	0.228158i	0.999146	+	0.0413087i	$0.0131527\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	29.8884	−	21.7152i	0.974335	−	0.707896i	0.0178992	−	0.999840i	$-0.494302\pi$
$941$	29.8884	−	21.7152i	0.974335	−	0.707896i	0.956435	+	0.291944i	$0.0943022\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$947$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$948$		0			0
$949$	−32.9346	+	23.9284i	−1.06910	+	0.776749i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	51.6630	+	26.3236i	1.67353	+	0.852706i	0.992747	+	0.120219i	$0.0383598\pi$
$953$	51.6630	+	26.3236i	1.67353	+	0.852706i	0.680782	+	0.732486i	$0.261640\pi$
$954$	2.30832	+	7.10429i	0.0747348	+	0.230010i
$955$		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
$962$	−48.2436	+	7.64104i	−1.55544	+	0.246357i
$963$		0			0
$964$	−13.1210	+	18.0596i	−0.422600	+	0.581659i
$965$	0.170987	−	22.5237i	0.00550428	−	0.725065i
$966$		0			0
$967$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$967$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$968$	30.7296	−	4.86710i	0.987688	−	0.156434i
$969$		0			0
$970$	47.7280	+	7.93123i	1.53245	+	0.254656i
$971$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$971$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	7.17211	+	22.0735i	0.229574	+	0.706555i
$977$	−48.8881	+	7.74311i	−1.56407	+	0.247724i	−0.877585	−	0.479421i	$-0.840847\pi$
$977$	−48.8881	+	7.74311i	−1.56407	+	0.247724i	−0.686483	+	0.727145i	$0.740847\pi$
$978$		0			0
$979$		0			0
$980$	−14.4235	+	27.7842i	−0.460741	+	0.887535i
$981$		−	59.1212i		−	1.88759i
$982$		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
$985$	35.5604	+	49.7343i	1.13305	+	1.58467i
$986$	−36.9192	+	26.8233i	−1.17575	+	0.854229i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$991$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−9.28587	+	58.6287i	−0.294086	+	1.85679i	0.190022	+	0.981780i	$0.439144\pi$
$997$	−9.28587	+	58.6287i	−0.294086	+	1.85679i	−0.484108	+	0.875008i	$0.660856\pi$
$998$		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.127.1	✓	8
4.3	odd	2	CM	820.2.bm.a.127.1	✓	8
5.3	odd	4		820.2.bm.b.783.1	yes	8
20.3	even	4		820.2.bm.b.783.1	yes	8
41.31	even	10		820.2.bm.b.687.1	yes	8
164.31	odd	10		820.2.bm.b.687.1	yes	8
205.113	odd	20	inner	820.2.bm.a.523.1	yes	8
820.523	even	20	inner	820.2.bm.a.523.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
820.2.bm.a.127.1	✓	8	1.1	even	1	trivial
820.2.bm.a.127.1	✓	8	4.3	odd	2	CM
820.2.bm.a.523.1	yes	8	205.113	odd	20	inner
820.2.bm.a.523.1	yes	8	820.523	even	20	inner
820.2.bm.b.687.1	yes	8	41.31	even	10
820.2.bm.b.687.1	yes	8	164.31	odd	10
820.2.bm.b.783.1	yes	8	5.3	odd	4
820.2.bm.b.783.1	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)

\(f(q)\)	\(=\)	\(q+(1.39680 + 0.221232i) q^{2} +(1.90211 + 0.618034i) q^{4} +(-1.03025 + 1.98459i) q^{5} +(2.52015 + 1.28408i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.39680 + 0.221232i) q^{2} +(1.90211 + 0.618034i) q^{4} +(-1.03025 + 1.98459i) q^{5} +(2.52015 + 1.28408i) q^{8} +3.00000i q^{9} +(-1.87811 + 2.54415i) q^{10} +(-3.48662 - 0.552226i) q^{13} +(3.23607 + 2.35114i) q^{16} +(2.87515 + 1.46496i) q^{17} +(-0.663695 + 4.19041i) q^{18} +(-3.18619 + 3.13818i) q^{20} +(-2.87718 - 4.08924i) q^{25} +(-4.74794 - 1.54270i) q^{26} +(-3.09017 + 9.51057i) q^{29} +(4.00000 + 4.00000i) q^{32} +(3.69192 + 2.68233i) q^{34} +(-1.85410 + 5.70634i) q^{36} +(8.71769 - 4.44189i) q^{37} +(-5.14475 + 3.67853i) q^{40} +(0.297142 - 6.39623i) q^{41} +(-5.95376 - 3.09075i) q^{45} +(6.65740 - 2.16312i) q^{49} +(-3.11418 - 6.34838i) q^{50} +(-6.29064 - 3.20524i) q^{52} +(1.56877 - 0.799328i) q^{53} +(-6.42040 + 12.6007i) q^{58} +(4.69421 + 3.41054i) q^{61} +(4.70228 + 6.47214i) q^{64} +(4.68802 - 6.35056i) q^{65} +(4.56346 + 4.56346i) q^{68} +(-3.85224 + 7.56044i) q^{72} +(8.15447 - 8.15447i) q^{73} +(13.1596 - 4.27581i) q^{74} +(-8.00000 + 4.00000i) q^{80} -9.00000 q^{81} +(1.83010 - 8.86853i) q^{82} +(-5.86946 + 4.19671i) q^{85} +(-8.09017 - 5.87785i) q^{89} +(-7.63246 - 5.63432i) q^{90} +(-6.94600 - 13.6323i) q^{97} +(9.77762 - 1.54862i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 4 q^{5} - 4 q^{8}+O(q^{10}) \) 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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