LMFDB - Embedded newform 820.2.bm.b.783.1

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			783.1
Root			$0.587785 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	820.783
Dual form			820.2.bm.b.687.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.221232 - 1.39680i) q^{2} +(-1.90211 - 0.618034i) q^{4} +(-0.333023 + 2.21113i) q^{5} +(-1.28408 + 2.52015i) q^{8} -3.00000i q^{9} +O(q^{10})$	\(q+(0.221232 - 1.39680i) q^{2} +(-1.90211 - 0.618034i) q^{4} +(-0.333023 + 2.21113i) q^{5} +(-1.28408 + 2.52015i) q^{8} -3.00000i q^{9} +(3.01484 + 0.954339i) q^{10} +(0.552226 - 3.48662i) q^{13} +(3.23607 + 2.35114i) q^{16} +(1.46496 - 2.87515i) q^{17} +(-4.19041 - 0.663695i) q^{18} +(2.00000 - 4.00000i) q^{20} +(-4.77819 - 1.47271i) q^{25} +(-4.74794 - 1.54270i) q^{26} +(3.09017 - 9.51057i) q^{29} +(4.00000 - 4.00000i) q^{32} +(-3.69192 - 2.68233i) q^{34} +(-1.85410 + 5.70634i) q^{36} +(-4.44189 - 8.71769i) q^{37} +(-5.14475 - 3.67853i) q^{40} +(0.297142 - 6.39623i) q^{41} +(6.63339 + 0.999068i) q^{45} +(-6.65740 + 2.16312i) q^{49} +(-3.11418 + 6.34838i) q^{50} +(-3.20524 + 6.29064i) q^{52} +(0.799328 + 1.56877i) q^{53} +(-12.6007 - 6.42040i) q^{58} +(4.69421 + 3.41054i) q^{61} +(-4.70228 - 6.47214i) q^{64} +(7.52546 + 2.38216i) q^{65} +(-4.56346 + 4.56346i) q^{68} +(7.56044 + 3.85224i) q^{72} +(8.15447 + 8.15447i) q^{73} +(-13.1596 + 4.27581i) q^{74} +(-6.27636 + 6.37238i) q^{80} -9.00000 q^{81} +(-8.86853 - 1.83010i) q^{82} +(5.86946 + 4.19671i) q^{85} +(8.09017 + 5.87785i) q^{89} +(2.86302 - 9.04451i) q^{90} +(-13.6323 + 6.94600i) q^{97} +(1.54862 + 9.77762i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 16 q^{20} - 6 q^{25} - 10 q^{26} - 20 q^{29} + 32 q^{32} - 30 q^{34} + 12 q^{36} - 4 q^{37} - 4 q^{40} + 8 q^{41} - 6 q^{45} + 14 q^{50} + 20 q^{52} + 10 q^{53} + 20 q^{58} - 24 q^{61} + 30 q^{65} - 20 q^{68} - 12 q^{72} - 22 q^{73} - 50 q^{74} + 16 q^{80} - 72 q^{81} + 2 q^{82} - 30 q^{85} + 20 q^{89} - 6 q^{90} - 50 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 + 16 * q^20 - 6 * q^25 - 10 * q^26 - 20 * q^29 + 32 * q^32 - 30 * q^34 + 12 * q^36 - 4 * q^37 - 4 * q^40 + 8 * q^41 - 6 * q^45 + 14 * q^50 + 20 * q^52 + 10 * q^53 + 20 * q^58 - 24 * q^61 + 30 * q^65 - 20 * q^68 - 12 * q^72 - 22 * q^73 - 50 * q^74 + 16 * q^80 - 72 * q^81 + 2 * q^82 - 30 * q^85 + 20 * q^89 - 6 * q^90 - 50 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

Character values

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

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

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	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$	−1.90211	−	0.618034i	−0.951057	−	0.309017i
$5$	−0.333023	+	2.21113i	−0.148932	+	0.988847i
$5$	−0.333023	+	2.21113i	−0.148932	+	0.988847i
$6$		0			0
$7$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$7$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$8$	−1.28408	+	2.52015i	−0.453990	+	0.891007i
$9$		−	3.00000i		−	1.00000i
$10$	3.01484	+	0.954339i	0.953375	+	0.301788i
$11$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$11$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$12$		0			0
$13$	0.552226	−	3.48662i	0.153160	−	0.967013i	−0.784669	−	0.619915i	$-0.787167\pi$
$13$	0.552226	−	3.48662i	0.153160	−	0.967013i	0.937829	−	0.347098i	$-0.112833\pi$
$14$		0			0
$15$		0			0
$16$	3.23607	+	2.35114i	0.809017	+	0.587785i
$17$	1.46496	−	2.87515i	0.355305	−	0.697325i	−0.642303	−	0.766451i	$-0.722021\pi$
$17$	1.46496	−	2.87515i	0.355305	−	0.697325i	0.997608	+	0.0691254i	$0.0220209\pi$
$18$	−4.19041	−	0.663695i	−0.987688	−	0.156434i
$19$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$19$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$20$	2.00000	−	4.00000i	0.447214	−	0.894427i
$21$		0			0
$22$		0			0
$23$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$23$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$24$		0			0
$25$	−4.77819	−	1.47271i	−0.955638	−	0.294542i
$26$	−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	$-0.521119\pi$
$29$	3.09017	−	9.51057i	0.573830	−	1.76607i	0.640129	−	0.768268i	$-0.278881\pi$
$30$		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
$37$	−4.44189	−	8.71769i	−0.730242	−	1.43318i	−0.894641	−	0.446786i	$-0.852568\pi$
$37$	−4.44189	−	8.71769i	−0.730242	−	1.43318i	0.164399	−	0.986394i	$-0.447432\pi$
$38$		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		−0.987688	−	0.156434i	$-0.950000\pi$
$43$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$44$		0			0
$45$	6.63339	+	0.999068i	0.988847	+	0.148932i
$46$		0			0
$47$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$47$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$48$		0			0
$49$	−6.65740	+	2.16312i	−0.951057	+	0.309017i
$50$	−3.11418	+	6.34838i	−0.440411	+	0.897796i
$51$		0			0
$52$	−3.20524	+	6.29064i	−0.444487	+	0.872355i
$53$	0.799328	+	1.56877i	0.109796	+	0.215487i	0.939366	−	0.342916i	$-0.111414\pi$
$53$	0.799328	+	1.56877i	0.109796	+	0.215487i	−0.829570	+	0.558403i	$0.811414\pi$
$54$		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	$-0.700000\pi$
$59$		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
$65$	7.52546	+	2.38216i	0.933418	+	0.295471i
$66$		0			0
$67$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$67$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$68$	−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	$0.900000\pi$
$72$	7.56044	+	3.85224i	0.891007	+	0.453990i
$73$	8.15447	+	8.15447i	0.954408	+	0.954408i	0.999005	−	0.0445966i	$-0.0142003\pi$
$73$	8.15447	+	8.15447i	0.954408	+	0.954408i	−0.0445966	+	0.999005i	$0.514200\pi$
$74$	−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			$\pi$
$80$	−6.27636	+	6.37238i	−0.701719	+	0.712454i
$81$	−9.00000			−1.00000
$82$	−8.86853	−	1.83010i	−0.979365	−	0.202100i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		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	0.927219	−	0.374519i	$-0.122192\pi$
$89$	8.09017	+	5.87785i	0.857556	+	0.623051i	−0.0696627	+	0.997571i	$0.522192\pi$
$90$	2.86302	−	9.04451i	0.301788	−	0.953375i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.6323	+	6.94600i	−1.38415	+	0.705260i	−0.978011	−	0.208552i	$-0.933125\pi$
$97$	−13.6323	+	6.94600i	−1.38415	+	0.705260i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	1.54862	+	9.77762i	0.156434	+	0.987688i
$99$		0			0
$100$	8.17848	+	5.75435i	0.817848	+	0.575435i
$101$	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		−0.156434	−	0.987688i	$-0.550000\pi$
$103$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$104$	8.07768	+	5.86878i	0.792082	+	0.575481i
$105$		0			0
$106$	2.36810	−	0.769441i	0.230010	−	0.0747348i
$107$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$107$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$108$		0			0
$109$	19.7071			1.88759			0.943797	−	0.330527i	$-0.107226\pi$
$109$	19.7071			1.88759			0.943797	+	0.330527i	$0.107226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.35764	+	18.3654i	−0.880293	+	1.72767i	−0.221788	+	0.975095i	$0.571189\pi$
$113$	−9.35764	+	18.3654i	−0.880293	+	1.72767i	−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$		0			0
$116$	−11.7557	+	16.1803i	−1.09149	+	1.50231i
$117$	−10.4598	−	1.65668i	−0.967013	−	0.153160i
$118$		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
$125$	4.84760	−	10.0748i	0.433583	−	0.901114i
$126$		0			0
$127$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$127$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$128$	−10.0806	+	5.13632i	−0.891007	+	0.453990i
$129$		0			0
$130$	4.99228	−	9.98457i	0.437852	−	0.875704i
$131$		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	$-0.488983\pi$
$137$	−11.2927	−	11.2927i	−0.964796	−	0.964796i	−0.999401	+	0.0346048i	$0.988983\pi$
$138$		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
$145$	20.0000	+	10.0000i	1.66091	+	0.830455i
$146$	13.1942	−	9.58615i	1.09196	−	0.793356i
$147$		0			0
$148$	3.06114	+	19.3273i	0.251624	+	1.58869i
$149$	6.02433	−	18.5410i	0.493532	−	1.51894i	−0.325700	−	0.945473i	$-0.605600\pi$
$149$	6.02433	−	18.5410i	0.493532	−	1.51894i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		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
$153$	−8.62544	−	4.39488i	−0.697325	−	0.355305i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.23504	+	20.4253i	0.258185	+	1.63011i	0.686955	+	0.726700i	$0.258947\pi$
$157$	3.23504	+	20.4253i	0.258185	+	1.63011i	−0.428770	+	0.903414i	$0.641053\pi$
$158$		0			0
$159$		0			0
$160$	7.51243	+	10.1766i	0.593910	+	0.804532i
$161$		0			0
$162$	−1.99109	+	12.5712i	−0.156434	+	0.987688i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$	−4.51828	+	11.9827i	−0.352819	+	0.935692i
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$	0.512197	+	0.166423i	0.0393997	+	0.0128018i
$170$	7.16048	−	7.27003i	0.549184	−	0.557586i
$171$		0			0
$172$		0			0
$173$	0.0824082	+	0.0824082i	0.00626538	+	0.00626538i	0.710233	−	0.703967i	$-0.248590\pi$
$173$	0.0824082	+	0.0824082i	0.00626538	+	0.00626538i	−0.703967	+	0.710233i	$0.748590\pi$
$174$		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	$0.600000\pi$
$180$	−12.0000	−	6.00000i	−0.894427	−	0.447214i
$181$	−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
$185$	20.7552	−	6.91840i	1.52595	−	0.508651i
$186$		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
$193$	−4.57314	−	8.97530i	−0.329182	−	0.646056i	0.665798	−	0.746132i	$-0.268091\pi$
$193$	−4.57314	−	8.97530i	−0.329182	−	0.646056i	−0.994980	+	0.100076i	$0.968091\pi$
$194$	6.68629	+	20.5783i	0.480048	+	1.47744i
$195$		0			0
$196$	14.0000			1.00000
$197$	−24.3623	−	12.4132i	−1.73574	−	0.884403i	−0.970656	−	0.240472i	$-0.922698\pi$
$197$	−24.3623	−	12.4132i	−1.73574	−	0.884403i	−0.765083	−	0.643932i	$-0.777302\pi$
$198$		0			0
$199$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$199$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$200$	9.84703	−	10.1507i	0.696290	−	0.717761i
$201$		0			0
$202$	−17.3559	−	17.3559i	−1.22116	−	1.22116i
$203$		0			0
$204$		0			0
$205$	14.0439	+	2.78711i	0.980871	+	0.194660i
$206$		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	$0.300000\pi$
$212$	−0.550859	−	3.47799i	−0.0378332	−	0.238869i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	4.35983	−	27.5269i	0.295285	−	1.86435i
$219$		0			0
$220$		0			0
$221$	−9.21554	−	6.69548i	−0.619905	−	0.450387i
$222$		0			0
$223$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$223$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$224$		0			0
$225$	−4.41814	+	14.3346i	−0.294542	+	0.955638i
$226$	23.5826	+	17.1338i	1.56869	+	1.13972i
$227$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$227$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$228$		0			0
$229$	4.04032	+	12.4348i	0.266992	+	0.821716i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.04032	+	12.4348i	0.266992	+	0.821716i	−0.724236	+	0.689552i	$0.757808\pi$
$230$		0			0
$231$		0			0
$232$	20.0000	+	20.0000i	1.31306	+	1.31306i
$233$	−3.62567	+	22.8916i	−0.237526	+	1.49968i	0.524097	+	0.851658i	$0.324403\pi$
$233$	−3.62567	+	22.8916i	−0.237526	+	1.49968i	−0.761623	+	0.648020i	$0.775597\pi$
$234$	−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	$0.982952\pi$
$242$	−7.06243	−	13.8608i	−0.453990	−	0.891007i
$243$		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
$250$	−13.0000	−	9.00000i	−0.822192	−	0.569210i
$251$		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
$257$	9.63059	−	18.9011i	0.600740	−	1.17902i	−0.367740	−	0.929928i	$-0.619869\pi$
$257$	9.63059	−	18.9011i	0.600740	−	1.17902i	0.968480	−	0.249090i	$-0.0801315\pi$
$258$		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		−0.891007	−	0.453990i	$-0.850000\pi$
$263$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$264$		0			0
$265$	−3.73495	+	1.24498i	−0.229436	+	0.0764787i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.45781	+	8.88842i	0.393740	+	0.541936i	0.959159	−	0.282867i	$-0.0912856\pi$
$269$	6.45781	+	8.88842i	0.393740	+	0.541936i	−0.565419	+	0.824804i	$0.691286\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$	11.5006	−	5.85984i	0.697325	−	0.355305i
$273$		0			0
$274$	−18.2719	+	13.2753i	−1.10385	+	0.801991i
$275$		0			0
$276$		0			0
$277$	19.7434	+	10.0598i	1.18627	+	0.604433i	0.931914	−	0.362678i	$-0.118138\pi$
$277$	19.7434	+	10.0598i	1.18627	+	0.604433i	0.254353	+	0.967112i	$0.418138\pi$
$278$		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		0.891007	−	0.453990i	$-0.150000\pi$
$283$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$284$		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
$292$	−10.4710	−	20.5505i	−0.612768	−	1.20262i
$293$	0.214005	−	1.35118i	0.0125023	−	0.0789365i	−0.980648	−	0.195778i	$-0.937277\pi$
$293$	0.214005	−	1.35118i	0.0125023	−	0.0789365i	0.993151	+	0.116841i	$0.0372769\pi$
$294$		0			0
$295$		0			0
$296$	27.6736			1.60850
$297$		0			0
$298$	−24.5653	−	12.5166i	−1.42303	−	0.725070i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.10443	+	9.24372i	−0.521318	+	0.529294i
$306$	−8.04700	+	11.0757i	−0.460016	+	0.633158i
$307$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$307$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$308$		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
$313$	16.0510	+	31.5018i	0.907255	+	1.78059i	0.489240	+	0.872149i	$0.337274\pi$
$313$	16.0510	+	31.5018i	0.907255	+	1.78059i	0.418016	+	0.908440i	$0.362726\pi$
$314$	29.2458			1.65043
$315$		0			0
$316$		0			0
$317$	3.12818	−	6.13940i	0.175696	−	0.344823i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	3.12818	−	6.13940i	0.175696	−	0.344823i	0.962014	+	0.272999i	$0.0880154\pi$
$318$		0			0
$319$		0			0
$320$	15.8767	−	8.24199i	0.887535	−	0.460741i
$321$		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
$328$	15.7379	+	8.96210i	0.868979	+	0.494849i
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	−26.1531	+	13.3257i	−1.43318	+	0.730242i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−20.3578	+	20.3578i	−1.10896	+	1.10896i	−0.115670	+	0.993288i	$0.536901\pi$
$337$	−20.3578	+	20.3578i	−1.10896	+	1.10896i	−0.993288	+	0.115670i	$0.963099\pi$
$338$	0.345774	−	0.678619i	0.0188076	−	0.0369120i
$339$		0			0
$340$	−8.57067	−	11.6101i	−0.464810	−	0.629648i
$341$		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		−0.156434	−	0.987688i	$-0.550000\pi$
$347$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$348$		0			0
$349$	20.5739	−	28.3176i	1.10130	−	1.51581i	0.267644	−	0.963518i	$-0.413755\pi$
$349$	20.5739	−	28.3176i	1.10130	−	1.51581i	0.833653	−	0.552288i	$-0.186245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.86174	−	30.6958i	−0.258764	−	1.63377i	−0.684561	−	0.728955i	$-0.740006\pi$
$353$	−4.86174	−	30.6958i	−0.258764	−	1.63377i	0.425797	−	0.904819i	$-0.359994\pi$
$354$		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
$362$	2.41998	+	15.2792i	0.127191	+	0.803055i
$363$		0			0
$364$		0			0
$365$	−20.7462	+	15.3150i	−1.08591	+	0.801622i
$366$		0			0
$367$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$367$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$368$		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
$373$	−34.2433	+	17.4478i	−1.77305	+	0.903413i	−0.841044	+	0.540967i	$0.818058\pi$
$373$	−34.2433	+	17.4478i	−1.77305	+	0.903413i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−31.4532	−	16.0262i	−1.61992	−	0.825392i
$378$		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	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	−13.5484	+	4.40215i	−0.689597	+	0.224064i
$387$		0			0
$388$	30.2230	−	4.78686i	1.53434	−	0.243016i
$389$	23.0778	−	31.7639i	1.17009	−	1.61049i	0.507020	−	0.861934i	$-0.330747\pi$
$389$	23.0778	−	31.7639i	1.17009	−	1.61049i	0.663070	−	0.748557i	$-0.269253\pi$
$390$		0			0
$391$		0			0
$392$	3.09724	−	19.5552i	0.156434	−	0.987688i
$393$		0			0
$394$	−22.7285	+	31.2831i	−1.14504	+	1.57602i
$395$		0			0
$396$		0			0
$397$	−17.5777	−	2.78403i	−0.882198	−	0.139726i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	−17.5777	−	2.78403i	−0.882198	−	0.139726i	−0.581066	+	0.813856i	$0.697364\pi$
$398$		0			0
$399$		0			0
$400$	−12.0000	−	16.0000i	−0.600000	−	0.800000i
$401$	−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
$405$	2.99720	−	19.9002i	0.148932	−	0.988847i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	39.8964i		−	1.97275i	−0.164518	−	0.986374i	$-0.552607\pi$
$409$		−	39.8964i		−	1.97275i	0.164518	−	0.986374i	$-0.447393\pi$
$410$	7.00000	−	19.0000i	0.345705	−	0.938343i
$411$		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
$425$	−11.2341	+	11.5805i	−0.544935	+	0.561739i
$426$		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
$433$	36.8762	+	5.84061i	1.77216	+	0.280682i	0.955188	−	0.296001i	$-0.0956533\pi$
$433$	36.8762	+	5.84061i	1.77216	+	0.280682i	0.816968	+	0.576683i	$0.195653\pi$
$434$		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		0.453990	−	0.891007i	$-0.350000\pi$
$443$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$444$		0			0
$445$	−15.6909	+	15.9310i	−0.743820	+	0.755200i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.16226	+	9.85801i	0.338008	+	0.465228i	0.943858	−	0.330350i	$-0.107167\pi$
$449$	7.16226	+	9.85801i	0.338008	+	0.465228i	−0.605850	+	0.795579i	$0.707167\pi$
$450$	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
$457$	−33.3743	−	5.28598i	−1.56119	−	0.247267i	−0.684747	−	0.728781i	$-0.740087\pi$
$457$	−33.3743	−	5.28598i	−1.56119	−	0.247267i	−0.876439	+	0.481514i	$0.840087\pi$
$458$	18.2628	−	2.89255i	0.853366	−	0.135160i
$459$		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		−0.453990	−	0.891007i	$-0.650000\pi$
$463$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$464$	32.3607	−	23.5114i	1.50231	−	1.09149i
$465$		0			0
$466$	31.1729	+	10.1287i	1.44406	+	0.469203i
$467$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$467$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$468$	18.8719	+	9.61573i	0.872355	+	0.444487i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.70631	−	2.39798i	0.215487	−	0.109796i
$478$		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
$482$	14.0642	+	7.16607i	0.640607	+	0.326406i
$483$		0			0
$484$	−20.9232	+	6.79837i	−0.951057	+	0.309017i
$485$	−10.8186	−	32.4559i	−0.491250	−	1.47375i
$486$		0			0
$487$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$487$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$488$	−14.6228	+	7.45069i	−0.661943	+	0.337277i
$489$		0			0
$490$	−22.1353	+	0.168039i	−0.999971	+	0.00759122i
$491$		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	$0.200000\pi$
$500$	−15.4472	+	16.1673i	−0.690821	+	0.723026i
$501$		0			0
$502$		0			0
$503$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$503$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$504$		0			0
$505$	27.6496	+	27.2330i	1.23039	+	1.21185i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.4920	−	32.2910i	−0.465048	−	1.43127i	−0.858922	−	0.512107i	$-0.828865\pi$
$509$	−10.4920	−	32.2910i	−0.465048	−	1.43127i	0.393873	−	0.919165i	$-0.371135\pi$
$510$		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
$520$	−15.6667	+	15.9064i	−0.687030	+	0.697541i
$521$	−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	$0.660060\pi$
$522$	−19.2612	+	37.8022i	−0.843039	+	1.65456i
$523$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$523$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$524$		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
$533$	−22.1371	−	4.56818i	−0.958864	−	0.197870i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	13.8440	−	7.05389i	0.596859	−	0.304115i
$539$		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
$545$	−6.56289	+	43.5749i	−0.281123	+	1.86654i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	14.5007	+	28.4591i	0.619437	+	1.21571i
$549$	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
$557$	3.21020	−	6.30037i	0.136020	−	0.266955i	−0.812942	−	0.582345i	$-0.802135\pi$
$557$	3.21020	−	6.30037i	0.136020	−	0.266955i	0.948962	+	0.315390i	$0.102135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	42.2427	−	21.5237i	1.78190	−	0.907923i
$563$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$563$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$564$		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	0.838444	−	0.544988i	$-0.183466\pi$
$569$	28.5391	+	9.27293i	1.19642	+	0.388741i	0.357979	+	0.933730i	$0.383466\pi$
$570$		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	0.783846	+	0.620956i	$0.213255\pi$
$577$	33.7445	+	33.7445i	1.40480	+	1.40480i	0.620956	+	0.783846i	$0.286745\pi$
$578$	8.30064	−	4.22939i	0.345261	−	0.175919i
$579$		0			0
$580$	−31.8619	−	31.3818i	−1.32299	−	1.30306i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−31.0214	+	10.0795i	−1.28368	+	0.417092i
$585$	7.14649	−	22.5764i	0.295471	−	0.933418i
$586$	−1.83998	−	0.597846i	−0.0760089	−	0.0246968i
$587$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$587$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	6.12228	−	38.6546i	0.251624	−	1.58869i
$593$	47.6561	−	7.54799i	1.95700	−	0.309959i	0.957214	−	0.289383i	$-0.0934500\pi$
$593$	47.6561	−	7.54799i	1.95700	−	0.309959i	0.999788	−	0.0205761i	$-0.00655004\pi$
$594$		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
$605$	11.3327	+	21.8305i	0.460741	+	0.887535i
$606$		0			0
$607$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$607$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$608$		0			0
$609$		0			0
$610$	10.8975	+	14.7621i	0.441225	+	0.597700i
$611$		0			0
$612$	13.6904	+	13.6904i	0.553401	+	0.553401i
$613$	46.0637	+	7.29577i	1.86049	+	0.294673i	0.982818	−	0.184579i	$-0.0590921\pi$
$613$	46.0637	+	7.29577i	1.86049	+	0.294673i	0.877677	+	0.479252i	$0.159092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.08823	−	32.1258i	0.204844	−	1.29334i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	5.08823	−	32.1258i	0.204844	−	1.29334i	0.848980	−	0.528425i	$-0.177217\pi$
$618$		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
$625$	20.6622	+	14.0738i	0.826489	+	0.562952i
$626$	47.5528	−	15.4508i	1.90059	−	0.617540i
$627$		0			0
$628$	6.47009	−	40.8505i	0.258185	−	1.63011i
$629$	−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
$637$	3.86558	+	24.4063i	0.153160	+	0.967013i
$638$		0			0
$639$		0			0
$640$	−8.00000	−	24.0000i	−0.316228	−	0.948683i
$641$	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		0.156434	−	0.987688i	$-0.450000\pi$
$643$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	11.5567	−	22.6813i	0.453990	−	0.891007i
$649$		0			0
$650$	20.4146	+	14.3637i	0.800728	+	0.563390i
$651$		0			0
$652$		0			0
$653$	2.25609	−	2.25609i	0.0882875	−	0.0882875i	−0.661584	−	0.749871i	$-0.730115\pi$
$653$	2.25609	−	2.25609i	0.0882875	−	0.0882875i	0.749871	+	0.661584i	$0.230115\pi$
$654$		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
$673$	−0.446032	−	0.227265i	−0.0171933	−	0.00876041i	0.445373	−	0.895345i	$-0.353071\pi$
$673$	−0.446032	−	0.227265i	−0.0171933	−	0.00876041i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	23.9320	+	32.9395i	0.921825	+	1.26878i
$675$		0			0
$676$	−0.871401	−	0.633110i	−0.0335154	−	0.0243504i
$677$	21.5568	−	3.41426i	0.828495	−	0.131221i	0.272237	−	0.962230i	$-0.412237\pi$
$677$	21.5568	−	3.41426i	0.828495	−	0.131221i	0.556258	+	0.831010i	$0.312237\pi$
$678$		0			0
$679$		0			0
$680$	−18.1132	+	9.40299i	−0.694608	+	0.360588i
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	28.7302	−	21.2088i	1.09773	−	0.810347i
$686$		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	$0.300000\pi$
$692$	−0.105819	−	0.207681i	−0.00402262	−	0.00789483i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−17.9548	−	10.2245i	−0.680086	−	0.387282i
$698$	−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	$0.490483\pi$
$709$	−15.4920	+	47.6794i	−0.581813	+	1.79064i	−0.611708	+	0.791083i	$0.709517\pi$
$710$		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	$-0.800000\pi$
$719$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$720$	19.1172	+	18.8291i	0.712454	+	0.701719i
$721$		0			0
$722$	26.5392	−	4.20340i	0.987688	−	0.156434i
$723$		0			0
$724$	21.8774			0.813065
$725$	−28.7718	+	40.8924i	−1.06856	+	1.51870i
$726$		0			0
$727$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$727$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$	16.8023	+	32.3665i	0.621880	+	1.19794i
$731$		0			0
$732$		0			0
$733$	41.2522	−	21.0190i	1.52368	−	0.776356i	0.526416	−	0.850227i	$-0.323536\pi$
$733$	41.2522	−	21.0190i	1.52368	−	0.776356i	0.997268	+	0.0738717i	$0.0235355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−5.49029	+	26.6056i	−0.202100	+	0.979365i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−43.7546	+	0.332160i	−1.60845	+	0.0122104i
$741$		0			0
$742$		0			0
$743$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$743$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$744$		0			0
$745$	38.9903	+	19.4951i	1.42849	+	0.714246i
$746$	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
$757$	4.05368	+	25.5940i	0.147334	+	0.930228i	0.944986	+	0.327111i	$0.106075\pi$
$757$	4.05368	+	25.5940i	0.147334	+	0.930228i	−0.797652	+	0.603117i	$0.793925\pi$
$758$		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	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−38.1720	+	12.4028i	−1.37652	+	0.447258i	−0.474995	+	0.879989i	$0.657550\pi$
$770$		0			0
$771$		0			0
$772$	3.15159	+	19.8984i	0.113428	+	0.716158i
$773$	−1.10616	+	6.98401i	−0.0397858	+	0.251197i	−0.999563	−	0.0295658i	$-0.990588\pi$
$773$	−1.10616	+	6.98401i	−0.0397858	+	0.251197i	0.959777	+	0.280763i	$0.0905876\pi$
$774$		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
$785$	−46.2403	+	0.351030i	−1.65039	+	0.0125288i
$786$		0			0
$787$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$787$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$788$	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
$797$	−8.57282	+	54.1267i	−0.303665	+	1.91726i	0.0859751	+	0.996297i	$0.472599\pi$
$797$	−8.57282	+	54.1267i	−0.303665	+	1.91726i	−0.389640	+	0.920967i	$0.627401\pi$
$798$		0			0
$799$		0			0
$800$	−25.0036	+	13.2219i	−0.884011	+	0.467465i
$801$	17.6336	−	24.2705i	0.623051	−	0.857556i
$802$	−4.84351	+	30.5807i	−0.171030	+	1.07984i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	22.2864	+	43.7394i	0.784031	+	1.53875i
$809$	17.4129	+	53.5914i	0.612205	+	1.88417i	0.436414	+	0.899746i	$0.356248\pi$
$809$	17.4129	+	53.5914i	0.612205	+	1.88417i	0.175791	+	0.984428i	$0.443752\pi$
$810$	−27.1335	−	8.58905i	−0.953375	−	0.301788i
$811$		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
$818$	−55.7273	−	8.82634i	−1.94846	−	0.308606i
$819$		0			0
$820$	−24.9906	−	13.9810i	−0.872710	−	0.488238i
$821$	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	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$827$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$828$		0			0
$829$		−	2.33421i		−	0.0810706i	−0.999178	−	0.0405353i	$-0.987094\pi$
$829$		−	2.33421i		−	0.0810706i	0.999178	−	0.0405353i	$-0.0129063\pi$
$830$		0			0
$831$		0			0
$832$	−25.1626	+	12.8210i	−0.872355	+	0.444487i
$833$	−3.53354	+	22.3099i	−0.122430	+	0.772991i
$834$		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
$842$	9.01044	+	56.8897i	0.310520	+	1.96055i
$843$		0			0
$844$		0			0
$845$	−0.538555	+	1.07711i	−0.0185269	+	0.0370537i
$846$		0			0
$847$		0			0
$848$	−1.10172	+	6.95598i	−0.0378332	+	0.238869i
$849$		0			0
$850$	13.6904	+	18.2538i	0.469576	+	0.626101i
$851$		0			0
$852$		0			0
$853$	26.3236	−	51.6630i	0.901304	−	1.76891i	0.342792	−	0.939411i	$-0.388628\pi$
$853$	26.3236	−	51.6630i	0.901304	−	1.76891i	0.558512	−	0.829496i	$-0.311372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.70636	−	15.1246i	−0.263244	−	0.516646i	0.721116	−	0.692814i	$-0.243630\pi$
$857$	−7.70636	−	15.1246i	−0.263244	−	0.516646i	−0.984360	+	0.176169i	$0.943630\pi$
$858$		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		−0.987688	−	0.156434i	$-0.950000\pi$
$863$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$864$		0			0
$865$	−0.209659	+	0.154771i	−0.00712862	+	0.00526238i
$866$	16.3164	−	50.2166i	0.554452	−	1.70643i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−25.3054	+	49.6647i	−0.856949	+	1.68186i
$873$	20.8380	+	40.8969i	0.705260	+	1.38415i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−2.74485	+	0.434741i	−0.0926869	+	0.0146802i	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	−2.74485	+	0.434741i	−0.0926869	+	0.0146802i	0.109919	+	0.993941i	$0.464941\pi$
$878$		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	$0.881218\pi$
$882$	29.3328	−	4.64587i	0.987688	−	0.156434i
$883$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$883$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$884$	13.3910	+	18.4311i	0.450387	+	0.619905i
$885$		0			0
$886$		0			0
$887$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$887$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$888$		0			0
$889$		0			0
$890$	18.7811	+	25.4415i	0.629543	+	0.852802i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	15.3542	−	7.82336i	0.512377	−	0.261069i
$899$		0			0
$900$	17.2631	−	24.5354i	0.575435	−	0.817848i
$901$	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		−0.453990	−	0.891007i	$-0.650000\pi$
$907$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$908$		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
$922$	−10.1666	+	1.61023i	−0.334819	+	0.0530302i
$923$		0			0
$924$		0			0
$925$	8.38553	+	48.1964i	0.275715	+	1.58469i
$926$		0			0
$927$		0			0
$928$	−25.6816	−	50.4029i	−0.843039	−	1.65456i
$929$	56.1093			1.84089			0.920443	−	0.390877i	$-0.127828\pi$
$929$	56.1093			1.84089			0.920443	+	0.390877i	$0.127828\pi$
$930$		0			0
$931$		0			0
$932$	21.0442	−	41.3016i	0.689327	−	1.35288i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	17.6063	−	24.2331i	0.575481	−	0.792082i
$937$	−6.98401	−	1.10616i	−0.228158	−	0.0361366i	0.0413087	−	0.999146i	$-0.486847\pi$
$937$	−6.98401	−	1.10616i	−0.228158	−	0.0361366i	−0.269466	+	0.963010i	$0.586847\pi$
$938$		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		−0.453990	−	0.891007i	$-0.650000\pi$
$947$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$948$		0			0
$949$	32.9346	−	23.9284i	1.06910	−	0.776749i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.3236	+	51.6630i	−0.852706	+	1.67353i	−0.120219	+	0.992747i	$0.538360\pi$
$953$	−26.3236	+	51.6630i	−0.852706	+	1.67353i	−0.732486	+	0.680782i	$0.761640\pi$
$954$	−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
$962$	7.64104	+	48.2436i	0.246357	+	1.55544i
$963$		0			0
$964$	13.1210	−	18.0596i	0.422600	−	0.581659i
$965$	21.3685	−	7.12283i	0.687876	−	0.229292i
$966$		0			0
$967$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$967$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$968$	4.86710	+	30.7296i	0.156434	+	0.987688i
$969$		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
$977$	7.74311	+	48.8881i	0.247724	+	1.56407i	0.727145	+	0.686483i	$0.240847\pi$
$977$	7.74311	+	48.8881i	0.247724	+	1.56407i	−0.479421	+	0.877585i	$0.659153\pi$
$978$		0			0
$979$		0			0
$980$	−4.66232	+	30.9558i	−0.148932	+	0.988847i
$981$		−	59.1212i		−	1.88759i
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		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
$997$	58.6287	+	9.28587i	1.85679	+	0.294086i	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	58.6287	+	9.28587i	1.85679	+	0.294086i	0.875008	+	0.484108i	$0.160856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	820.2.bm.b.783.1	yes	8
4.3	odd	2	CM	820.2.bm.b.783.1	yes	8
5.2	odd	4		820.2.bm.a.127.1	✓	8
20.7	even	4		820.2.bm.a.127.1	✓	8
41.31	even	10		820.2.bm.a.523.1	yes	8
164.31	odd	10		820.2.bm.a.523.1	yes	8
205.72	odd	20	inner	820.2.bm.b.687.1	yes	8
820.687	even	20	inner	820.2.bm.b.687.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
820.2.bm.a.127.1	✓	8	5.2	odd	4
820.2.bm.a.127.1	✓	8	20.7	even	4
820.2.bm.a.523.1	yes	8	41.31	even	10
820.2.bm.a.523.1	yes	8	164.31	odd	10
820.2.bm.b.687.1	yes	8	205.72	odd	20	inner
820.2.bm.b.687.1	yes	8	820.687	even	20	inner
820.2.bm.b.783.1	yes	8	1.1	even	1	trivial
820.2.bm.b.783.1	yes	8	4.3	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(0.221232 - 1.39680i) q^{2} +(-1.90211 - 0.618034i) q^{4} +(-0.333023 + 2.21113i) q^{5} +(-1.28408 + 2.52015i) q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q+(0.221232 - 1.39680i) q^{2} +(-1.90211 - 0.618034i) q^{4} +(-0.333023 + 2.21113i) q^{5} +(-1.28408 + 2.52015i) q^{8} -3.00000i q^{9} +(3.01484 + 0.954339i) q^{10} +(0.552226 - 3.48662i) q^{13} +(3.23607 + 2.35114i) q^{16} +(1.46496 - 2.87515i) q^{17} +(-4.19041 - 0.663695i) q^{18} +(2.00000 - 4.00000i) q^{20} +(-4.77819 - 1.47271i) q^{25} +(-4.74794 - 1.54270i) q^{26} +(3.09017 - 9.51057i) q^{29} +(4.00000 - 4.00000i) q^{32} +(-3.69192 - 2.68233i) q^{34} +(-1.85410 + 5.70634i) q^{36} +(-4.44189 - 8.71769i) q^{37} +(-5.14475 - 3.67853i) q^{40} +(0.297142 - 6.39623i) q^{41} +(6.63339 + 0.999068i) q^{45} +(-6.65740 + 2.16312i) q^{49} +(-3.11418 + 6.34838i) q^{50} +(-3.20524 + 6.29064i) q^{52} +(0.799328 + 1.56877i) q^{53} +(-12.6007 - 6.42040i) q^{58} +(4.69421 + 3.41054i) q^{61} +(-4.70228 - 6.47214i) q^{64} +(7.52546 + 2.38216i) q^{65} +(-4.56346 + 4.56346i) q^{68} +(7.56044 + 3.85224i) q^{72} +(8.15447 + 8.15447i) q^{73} +(-13.1596 + 4.27581i) q^{74} +(-6.27636 + 6.37238i) q^{80} -9.00000 q^{81} +(-8.86853 - 1.83010i) q^{82} +(5.86946 + 4.19671i) q^{85} +(8.09017 + 5.87785i) q^{89} +(2.86302 - 9.04451i) q^{90} +(-13.6323 + 6.94600i) q^{97} +(1.54862 + 9.77762i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 4 q^{5} - 4 q^{8}+O(q^{10}) \) 8 * q + 2 * q^2 - 4 * q^5 - 4 * q^8	\( 8 q + 2 q^{2} - 4 q^{5} - 4 q^{8} + 6 q^{10} + 10 q^{13} + 8 q^{16} - 10 q^{17} - 6 q^{18} + 16 q^{20} - 6 q^{25} - 10 q^{26} - 20 q^{29} + 32 q^{32} - 30 q^{34} + 12 q^{36} - 4 q^{37} - 4 q^{40} + 8 q^{41} - 6 q^{45} + 14 q^{50} + 20 q^{52} + 10 q^{53} + 20 q^{58} - 24 q^{61} + 30 q^{65} - 20 q^{68} - 12 q^{72} - 22 q^{73} - 50 q^{74} + 16 q^{80} - 72 q^{81} + 2 q^{82} - 30 q^{85} + 20 q^{89} - 6 q^{90} - 50 q^{97} + 14 q^{98}+O(q^{100}) \) 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 + 16 * q^20 - 6 * q^25 - 10 * q^26 - 20 * q^29 + 32 * q^32 - 30 * q^34 + 12 * q^36 - 4 * q^37 - 4 * q^40 + 8 * q^41 - 6 * q^45 + 14 * q^50 + 20 * q^52 + 10 * q^53 + 20 * q^58 - 24 * q^61 + 30 * q^65 - 20 * q^68 - 12 * q^72 - 22 * q^73 - 50 * q^74 + 16 * q^80 - 72 * q^81 + 2 * q^82 - 30 * q^85 + 20 * q^89 - 6 * q^90 - 50 * q^97 + 14 * q^98

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