LMFDB - Embedded newform 820.2.bm.b.523.1

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			523.1
Root			$-0.587785 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	820.523
Dual form			820.2.bm.b.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.39680 - 0.221232i) q^{2} +(1.90211 - 0.618034i) q^{4} +(1.56909 - 1.59310i) q^{5} +(2.52015 - 1.28408i) q^{8} -3.00000i q^{9} +O(q^{10})$	\(q+(1.39680 - 0.221232i) q^{2} +(1.90211 - 0.618034i) q^{4} +(1.56909 - 1.59310i) q^{5} +(2.52015 - 1.28408i) q^{8} -3.00000i q^{9} +(1.83927 - 2.57237i) q^{10} +(0.829740 - 0.131418i) q^{13} +(3.23607 - 2.35114i) q^{16} +(-7.31906 + 3.72925i) q^{17} +(-0.663695 - 4.19041i) q^{18} +(2.00000 - 4.00000i) q^{20} +(-0.0759100 - 4.99942i) q^{25} +(1.12991 - 0.367130i) q^{26} +(3.09017 + 9.51057i) q^{29} +(4.00000 - 4.00000i) q^{32} +(-9.39825 + 6.82823i) q^{34} +(-1.85410 - 5.70634i) q^{36} +(-3.26632 - 1.66427i) q^{37} +(1.90868 - 6.02967i) q^{40} +(6.17499 - 1.69394i) q^{41} +(-4.77929 - 4.70727i) q^{45} +(6.65740 + 2.16312i) q^{49} +(-1.21206 - 6.96641i) q^{50} +(1.49704 - 0.762779i) q^{52} +(11.7630 + 5.99354i) q^{53} +(6.42040 + 12.6007i) q^{58} +(-10.6942 + 7.76980i) q^{61} +(4.70228 - 6.47214i) q^{64} +(1.09258 - 1.52806i) q^{65} +(-11.6169 + 11.6169i) q^{68} +(-3.85224 - 7.56044i) q^{72} +(-1.35610 - 1.35610i) q^{73} +(-4.93059 - 1.60205i) q^{74} +(1.33209 - 8.84452i) q^{80} -9.00000 q^{81} +(8.25049 - 3.73221i) q^{82} +(-5.54322 + 17.5115i) q^{85} +(8.09017 - 5.87785i) q^{89} +(-7.71712 - 5.51780i) q^{90} +(-8.93001 + 17.5261i) q^{97} +(9.77762 + 1.54862i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 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 $\left(\mathbb{Z}/820\mathbb{Z}\right)^\times$.

$n$	$411$	$621$	$657$
$\chi(n)$	$-1$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{4}\right)$

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 $

$2$	1.39680	−	0.221232i	0.987688	−	0.156434i
$2$	1.39680	−	0.221232i	0.987688	−	0.156434i
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$	1.90211	−	0.618034i	0.951057	−	0.309017i
$5$	1.56909	−	1.59310i	0.701719	−	0.712454i
$5$	1.56909	−	1.59310i	0.701719	−	0.712454i
$6$		0			0
$7$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$7$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$8$	2.52015	−	1.28408i	0.891007	−	0.453990i
$9$		−	3.00000i		−	1.00000i
$10$	1.83927	−	2.57237i	0.581627	−	0.813456i
$11$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$11$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$12$		0			0
$13$	0.829740	−	0.131418i	0.230129	−	0.0364488i	−0.0403050	−	0.999187i	$-0.512833\pi$
$13$	0.829740	−	0.131418i	0.230129	−	0.0364488i	0.270434	+	0.962739i	$0.412833\pi$
$14$		0			0
$15$		0			0
$16$	3.23607	−	2.35114i	0.809017	−	0.587785i
$17$	−7.31906	+	3.72925i	−1.77513	+	0.904476i	−0.847713	+	0.530456i	$0.822021\pi$
$17$	−7.31906	+	3.72925i	−1.77513	+	0.904476i	−0.927421	+	0.374020i	$0.877979\pi$
$18$	−0.663695	−	4.19041i	−0.156434	−	0.987688i
$19$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$19$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$20$	2.00000	−	4.00000i	0.447214	−	0.894427i
$21$		0			0
$22$		0			0
$23$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$23$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$24$		0			0
$25$	−0.0759100	−	4.99942i	−0.0151820	−	0.999885i
$26$	1.12991	−	0.367130i	0.221593	−	0.0720001i
$27$		0			0
$28$		0			0
$29$	3.09017	+	9.51057i	0.573830	+	1.76607i	0.640129	+	0.768268i	$0.278881\pi$
$29$	3.09017	+	9.51057i	0.573830	+	1.76607i	−0.0662984	+	0.997800i	$0.521119\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$	4.00000	−	4.00000i	0.707107	−	0.707107i
$33$		0			0
$34$	−9.39825	+	6.82823i	−1.61179	+	1.17103i
$35$		0			0
$36$	−1.85410	−	5.70634i	−0.309017	−	0.951057i
$37$	−3.26632	−	1.66427i	−0.536979	−	0.273605i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−3.26632	−	1.66427i	−0.536979	−	0.273605i	−0.701378	+	0.712789i	$0.747432\pi$
$38$		0			0
$39$		0			0
$40$	1.90868	−	6.02967i	0.301788	−	0.953375i
$41$	6.17499	−	1.69394i	0.964372	−	0.264550i
$41$	6.17499	−	1.69394i	0.964372	−	0.264550i
$42$		0			0
$43$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$43$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$44$		0			0
$45$	−4.77929	−	4.70727i	−0.712454	−	0.701719i
$46$		0			0
$47$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$47$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$48$		0			0
$49$	6.65740	+	2.16312i	0.951057	+	0.309017i
$50$	−1.21206	−	6.96641i	−0.171412	−	0.985200i
$51$		0			0
$52$	1.49704	−	0.762779i	0.207602	−	0.105778i
$53$	11.7630	+	5.99354i	1.61577	+	0.823276i	0.999357	+	0.0358519i	$0.0114145\pi$
$53$	11.7630	+	5.99354i	1.61577	+	0.823276i	0.616412	+	0.787424i	$0.288586\pi$
$54$		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	$-0.300000\pi$
$59$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$60$		0			0
$61$	−10.6942	+	7.76980i	−1.36925	+	0.994821i	−0.371458	+	0.928450i	$0.621142\pi$
$61$	−10.6942	+	7.76980i	−1.36925	+	0.994821i	−0.997795	+	0.0663709i	$0.978858\pi$
$62$		0			0
$63$		0			0
$64$	4.70228	−	6.47214i	0.587785	−	0.809017i
$65$	1.09258	−	1.52806i	0.135517	−	0.189533i
$66$		0			0
$67$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$67$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$68$	−11.6169	+	11.6169i	−1.40875	+	1.40875i
$69$		0			0
$70$		0			0
$71$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$71$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$72$	−3.85224	−	7.56044i	−0.453990	−	0.891007i
$73$	−1.35610	−	1.35610i	−0.158719	−	0.158719i	0.623280	−	0.781999i	$-0.285800\pi$
$73$	−1.35610	−	1.35610i	−0.158719	−	0.158719i	−0.781999	+	0.623280i	$0.785800\pi$
$74$	−4.93059	−	1.60205i	−0.573169	−	0.186234i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	1.33209	−	8.84452i	0.148932	−	0.988847i
$81$	−9.00000			−1.00000
$82$	8.25049	−	3.73221i	0.911114	−	0.412154i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−5.54322	+	17.5115i	−0.601247	+	1.89939i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.09017	−	5.87785i	0.857556	−	0.623051i	−0.0696627	−	0.997571i	$-0.522192\pi$
$89$	8.09017	−	5.87785i	0.857556	−	0.623051i	0.927219	+	0.374519i	$0.122192\pi$
$90$	−7.71712	−	5.51780i	−0.813456	−	0.581627i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.93001	+	17.5261i	−0.906705	+	1.77951i	−0.406138	+	0.913812i	$0.633125\pi$
$97$	−8.93001	+	17.5261i	−0.906705	+	1.77951i	−0.500567	+	0.865698i	$0.666875\pi$
$98$	9.77762	+	1.54862i	0.987688	+	0.156434i
$99$		0			0
$100$	−3.23420	−	9.46255i	−0.323420	−	0.946255i
$101$	−8.81958	−	12.1391i	−0.877581	−	1.20789i	−0.977085	−	0.212850i	$-0.931726\pi$
$101$	−8.81958	−	12.1391i	−0.877581	−	1.20789i	0.0995037	−	0.995037i	$-0.468274\pi$
$102$		0			0
$103$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$103$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$104$	1.92232	−	1.39664i	0.188499	−	0.136952i
$105$		0			0
$106$	17.7565	+	5.76944i	1.72467	+	0.560378i
$107$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$107$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$108$		0			0
$109$	12.6536			1.21200			0.605999	−	0.795465i	$-0.292774\pi$
$109$	12.6536			1.21200			0.605999	+	0.795465i	$0.292774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.5866	+	8.96084i	−1.65441	+	0.842965i	−0.658505	+	0.752577i	$0.728811\pi$
$113$	−17.5866	+	8.96084i	−1.65441	+	0.842965i	−0.995907	+	0.0903879i	$0.971189\pi$
$114$		0			0
$115$		0			0
$116$	11.7557	+	16.1803i	1.09149	+	1.50231i
$117$	−0.394254	−	2.48922i	−0.0364488	−	0.230129i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.89919	+	6.46564i	0.809017	+	0.587785i
$122$	−13.2188	+	13.2188i	−1.19677	+	1.19677i
$123$		0			0
$124$		0			0
$125$	−8.08367	−	7.72362i	−0.723026	−	0.690821i
$126$		0			0
$127$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$127$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$128$	5.13632	−	10.0806i	0.453990	−	0.891007i
$129$		0			0
$130$	1.18806	−	2.37611i	0.104199	−	0.208399i
$131$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$131$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−13.6565	+	18.7965i	−1.17103	+	1.61179i
$137$	2.02214	+	2.02214i	0.172763	+	0.172763i	0.788192	−	0.615429i	$-0.211017\pi$
$137$	2.02214	+	2.02214i	0.172763	+	0.172763i	−0.615429	+	0.788192i	$0.711017\pi$
$138$		0			0
$139$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$139$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−7.05342	−	9.70820i	−0.587785	−	0.809017i
$145$	20.0000	+	10.0000i	1.66091	+	0.830455i
$146$	−2.19421	−	1.59419i	−0.181594	−	0.131936i
$147$		0			0
$148$	−7.24148	−	1.14694i	−0.595246	−	0.0942777i
$149$	−2.20467	−	6.78527i	−0.180613	−	0.555871i	0.819232	−	0.573462i	$-0.194400\pi$
$149$	−2.20467	−	6.78527i	−0.180613	−	0.555871i	−0.999845	+	0.0175917i	$0.994400\pi$
$150$		0			0
$151$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$151$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$152$		0			0
$153$	11.1877	+	21.9572i	0.904476	+	1.77513i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−24.7416	−	3.91869i	−1.97460	−	0.312745i	−0.991695	−	0.128615i	$-0.958947\pi$
$157$	−24.7416	−	3.91869i	−1.97460	−	0.312745i	−0.982902	−	0.184131i	$-0.941053\pi$
$158$		0			0
$159$		0			0
$160$	−0.0960221	−	12.6487i	−0.00759122	−	0.999971i
$161$		0			0
$162$	−12.5712	+	1.99109i	−0.987688	+	0.156434i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$	10.6986	−	7.03843i	0.835422	−	0.549609i
$165$		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
$169$	−11.6925	+	3.79914i	−0.899426	+	0.292241i
$170$	−3.86868	+	25.6864i	−0.296714	+	1.97006i
$171$		0			0
$172$		0			0
$173$	17.7160	+	17.7160i	1.34692	+	1.34692i	0.888986	+	0.457933i	$0.151410\pi$
$173$	17.7160	+	17.7160i	1.34692	+	1.34692i	0.457933	+	0.888986i	$0.348590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	10.0000	−	10.0000i	0.749532	−	0.749532i
$179$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$179$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$180$	−12.0000	−	6.00000i	−0.894427	−	0.447214i
$181$	−22.1590	−	7.19990i	−1.64707	−	0.535164i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−22.1590	−	7.19990i	−1.64707	−	0.535164i	−0.978101	+	0.208130i	$0.933262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.77649	+	2.59216i	−0.571739	+	0.190580i
$186$		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
$193$	−19.1695	−	9.76735i	−1.37985	−	0.703069i	−0.402644	−	0.915357i	$-0.631909\pi$
$193$	−19.1695	−	9.76735i	−1.37985	−	0.703069i	−0.977207	+	0.212287i	$0.931909\pi$
$194$	−8.59612	+	26.4561i	−0.617166	+	1.89944i
$195$		0			0
$196$	14.0000			1.00000
$197$	1.09175	+	2.14268i	0.0777840	+	0.152660i	0.926623	−	0.375992i	$-0.122698\pi$
$197$	1.09175	+	2.14268i	0.0777840	+	0.152660i	−0.848839	+	0.528651i	$0.822698\pi$
$198$		0			0
$199$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$199$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$200$	−6.61096	−	12.5018i	−0.467465	−	0.884011i
$201$		0			0
$202$	−15.0048	−	15.0048i	−1.05573	−	1.05573i
$203$		0			0
$204$		0			0
$205$	6.99051	−	12.4953i	0.488238	−	0.872710i
$206$		0			0
$207$		0			0
$208$	2.37611	−	2.37611i	0.164754	−	0.164754i
$209$		0			0
$210$		0			0
$211$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$211$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$212$	26.0787	+	4.13046i	1.79109	+	0.283681i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	17.6746	−	2.79938i	1.19708	−	0.189598i
$219$		0			0
$220$		0			0
$221$	−5.58283	+	4.05616i	−0.375542	+	0.272847i
$222$		0			0
$223$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$223$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$224$		0			0
$225$	−14.9983	+	0.227730i	−0.999885	+	0.0151820i
$226$	−22.5826	+	16.4072i	−1.50217	+	1.09139i
$227$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$227$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$228$		0			0
$229$	1.68917	−	5.19874i	0.111624	−	0.343543i	−0.879604	−	0.475706i	$-0.842192\pi$
$229$	1.68917	−	5.19874i	0.111624	−	0.343543i	0.991228	+	0.132164i	$0.0421925\pi$
$230$		0			0
$231$		0			0
$232$	20.0000	+	20.0000i	1.31306	+	1.31306i
$233$	11.5912	−	1.83587i	0.759367	−	0.120272i	0.235269	−	0.971930i	$-0.424403\pi$
$233$	11.5912	−	1.83587i	0.759367	−	0.120272i	0.524097	+	0.851658i	$0.324403\pi$
$234$	−1.10139	−	3.38973i	−0.0720001	−	0.221593i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$239$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$240$		0			0
$241$	7.44907	+	22.9259i	0.479837	+	1.47679i	0.839322	+	0.543635i	$0.182952\pi$
$241$	7.44907	+	22.9259i	0.479837	+	1.47679i	−0.359485	+	0.933151i	$0.617048\pi$
$242$	13.8608	+	7.06243i	0.891007	+	0.453990i
$243$		0			0
$244$	−15.5396	+	21.3884i	−0.994821	+	1.36925i
$245$	13.8921	−	7.21174i	0.887535	−	0.460741i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−13.0000	−	9.00000i	−0.822192	−	0.569210i
$251$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$251$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	−	15.2169i	0.309017	−	0.951057i
$257$	−18.9011	+	9.63059i	−1.17902	+	0.600740i	−0.929928	−	0.367740i	$-0.880131\pi$
$257$	−18.9011	+	9.63059i	−1.17902	+	0.600740i	−0.249090	+	0.968480i	$0.580131\pi$
$258$		0			0
$259$		0			0
$260$	1.13381	−	3.58180i	0.0703158	−	0.222134i
$261$	28.5317	−	9.27051i	1.76607	−	0.573830i
$262$		0			0
$263$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$263$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$264$		0			0
$265$	28.0055	−	9.33515i	1.72036	−	0.573454i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.9029	−	21.8884i	0.969615	−	1.33456i	0.0273737	−	0.999625i	$-0.491286\pi$
$269$	15.9029	−	21.8884i	0.969615	−	1.33456i	0.942241	−	0.334935i	$-0.108714\pi$
$270$		0			0
$271$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$271$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$272$	−14.9170	+	29.2762i	−0.904476	+	1.77513i
$273$		0			0
$274$	3.27189	+	2.37717i	0.197662	+	0.143610i
$275$		0			0
$276$		0			0
$277$	−13.8336	−	27.1499i	−0.831180	−	1.63128i	−0.774229	−	0.632905i	$-0.781862\pi$
$277$	−13.8336	−	27.1499i	−0.831180	−	1.63128i	−0.0569502	−	0.998377i	$-0.518138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.0722	−	22.1215i	0.958786	−	1.31966i	0.0112742	−	0.999936i	$-0.496411\pi$
$281$	16.0722	−	22.1215i	0.958786	−	1.31966i	0.947512	−	0.319720i	$-0.103589\pi$
$282$		0			0
$283$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$283$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−12.0000	−	12.0000i	−0.707107	−	0.707107i
$289$	29.6690	−	40.8359i	1.74524	−	2.40211i
$290$	30.1484	+	9.54339i	1.77037	+	0.560407i
$291$		0			0
$292$	−3.41756	−	1.74133i	−0.199998	−	0.101904i
$293$	32.5499	−	5.15540i	1.90159	−	0.301182i	0.908436	−	0.418023i	$-0.137277\pi$
$293$	32.5499	−	5.15540i	1.90159	−	0.301182i	0.993151	+	0.116841i	$0.0372769\pi$
$294$		0			0
$295$		0			0
$296$	−10.3687			−0.602666
$297$		0			0
$298$	−4.58060	−	8.98993i	−0.265347	−	0.520773i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.40215	+	29.2284i	−0.252066	+	1.67361i
$306$	20.4847	+	28.1948i	1.17103	+	1.61179i
$307$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$307$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$311$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$312$		0			0
$313$	−31.5018	−	16.0510i	−1.78059	−	0.907255i	−0.908440	−	0.418016i	$-0.862726\pi$
$313$	−31.5018	−	16.0510i	−1.78059	−	0.907255i	−0.872149	−	0.489240i	$-0.837274\pi$
$314$	−35.4261			−1.99921
$315$		0			0
$316$		0			0
$317$	−13.3298	+	6.79188i	−0.748677	+	0.381470i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−13.3298	+	6.79188i	−0.748677	+	0.381470i	0.0376418	+	0.999291i	$0.488015\pi$
$318$		0			0
$319$		0			0
$320$	−2.93243	−	17.6466i	−0.163928	−	0.986472i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−17.1190	+	5.56231i	−0.951057	+	0.309017i
$325$	−0.720000	−	4.13825i	−0.0399384	−	0.229549i
$326$		0			0
$327$		0			0
$328$	13.3867	−	12.1982i	0.739159	−	0.673531i
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	−4.99281	+	9.79895i	−0.273605	+	0.536979i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.03151	−	9.03151i	0.491978	−	0.491978i	−0.416951	−	0.908929i	$-0.636901\pi$
$337$	9.03151	−	9.03151i	0.491978	−	0.491978i	0.908929	+	0.416951i	$0.136901\pi$
$338$	−15.4917	+	7.89340i	−0.842636	+	0.429344i
$339$		0			0
$340$	0.278869	+	36.7347i	0.0151238	+	1.99222i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	28.6650	+	20.8264i	1.54104	+	1.11963i
$347$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$347$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$348$		0			0
$349$	−13.6641	−	18.8070i	−0.731423	−	1.00672i	−0.999066	−	0.0431990i	$-0.986245\pi$
$349$	−13.6641	−	18.8070i	−0.731423	−	1.00672i	0.267644	−	0.963518i	$-0.413755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.3552	+	1.64010i	0.551150	+	0.0872935i	0.425797	−	0.904819i	$-0.359994\pi$
$353$	10.3552	+	1.64010i	0.551150	+	0.0872935i	0.125353	+	0.992112i	$0.459994\pi$
$354$		0			0
$355$		0			0
$356$	11.7557	−	16.1803i	0.623051	−	0.857556i
$357$		0			0
$358$		0			0
$359$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$359$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$360$	−18.0890	−	5.72603i	−0.953375	−	0.301788i
$361$	5.87132	−	18.0701i	0.309017	−	0.951057i
$362$	−32.5446	−	5.15456i	−1.71051	−	0.270918i
$363$		0			0
$364$		0			0
$365$	−4.28823	+	0.0325538i	−0.224456	+	0.00170394i
$366$		0			0
$367$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$367$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$368$		0			0
$369$	−5.08183	−	18.5250i	−0.264550	−	0.964372i
$370$	−10.2888	+	5.34115i	−0.534887	+	0.277673i
$371$		0			0
$372$		0			0
$373$	−13.0830	+	25.6768i	−0.677411	+	1.32949i	0.254593	+	0.967048i	$0.418058\pi$
$373$	−13.0830	+	25.6768i	−0.677411	+	1.32949i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.81390	+	7.48520i	0.196426	+	0.385507i
$378$		0			0
$379$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$379$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	−28.9369	−	9.40215i	−1.47285	−	0.478557i
$387$		0			0
$388$	−6.15414	+	38.8557i	−0.312429	+	1.97260i
$389$	−9.25813	−	12.7427i	−0.469406	−	0.646082i	0.507020	−	0.861934i	$-0.330747\pi$
$389$	−9.25813	−	12.7427i	−0.469406	−	0.646082i	−0.976426	+	0.215852i	$0.930747\pi$
$390$		0			0
$391$		0			0
$392$	19.5552	−	3.09724i	0.987688	−	0.156434i
$393$		0			0
$394$	1.99899	+	2.75137i	0.100708	+	0.138612i
$395$		0			0
$396$		0			0
$397$	−6.16498	−	38.9242i	−0.309412	−	1.95355i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	−6.16498	−	38.9242i	−0.309412	−	1.95355i	−0.00828030	−	0.999966i	$-0.502636\pi$
$398$		0			0
$399$		0			0
$400$	−12.0000	−	16.0000i	−0.600000	−	0.800000i
$401$	25.1294			1.25490			0.627452	−	0.778655i	$-0.284098\pi$
$401$	25.1294			1.25490			0.627452	+	0.778655i	$0.284098\pi$
$402$		0			0
$403$		0			0
$404$	−24.2782	−	17.6392i	−1.20789	−	0.877581i
$405$	−14.1218	+	14.3379i	−0.701719	+	0.712454i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			36.1882i			1.78939i	0.446678	+	0.894695i	$0.352607\pi$
$409$			36.1882i			1.78939i	−0.446678	+	0.894695i	$0.647393\pi$
$410$	7.00000	−	19.0000i	0.345705	−	0.938343i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	2.79329	−	3.84463i	0.136952	−	0.188499i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	7.43015	+	2.41420i	0.362123	+	0.117661i	0.484427	−	0.874832i	$-0.339028\pi$
$421$	7.43015	+	2.41420i	0.362123	+	0.117661i	−0.122304	+	0.992493i	$0.539028\pi$
$422$		0			0
$423$		0			0
$424$	37.3406			1.81342
$425$	19.1997	+	36.3080i	0.931321	+	1.76120i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$		0			0
$433$	4.54024	+	28.6660i	0.218190	+	1.37760i	0.816968	+	0.576683i	$0.195653\pi$
$433$	4.54024	+	28.6660i	0.218190	+	1.37760i	−0.598778	+	0.800915i	$0.704347\pi$
$434$		0			0
$435$		0			0
$436$	24.0686	−	7.82037i	1.15268	−	0.374528i
$437$		0			0
$438$		0			0
$439$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$439$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$440$		0			0
$441$	6.48936	−	19.9722i	0.309017	−	0.951057i
$442$	−6.90076	+	6.90076i	−0.328236	+	0.328236i
$443$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$443$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$444$		0			0
$445$	3.33023	−	22.1113i	0.157868	−	1.04818i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	20.4771	−	28.1842i	0.966372	−	1.33010i	0.0225137	−	0.999747i	$-0.492833\pi$
$449$	20.4771	−	28.1842i	0.966372	−	1.33010i	0.943858	−	0.330350i	$-0.107167\pi$
$450$	−20.8992	+	3.63619i	−0.985200	+	0.171412i
$451$		0			0
$452$	−27.9137	+	27.9137i	−1.31295	+	1.31295i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.86776	+	11.7926i	0.0873700	+	0.551632i	0.992080	+	0.125605i	$0.0400872\pi$
$457$	1.86776	+	11.7926i	0.0873700	+	0.551632i	−0.904710	+	0.426027i	$0.859913\pi$
$458$	1.20932	−	7.63532i	0.0565076	−	0.356775i
$459$		0			0
$460$		0			0
$461$	9.50653	−	29.2581i	0.442763	−	1.36269i	−0.442155	−	0.896939i	$-0.645786\pi$
$461$	9.50653	−	29.2581i	0.442763	−	1.36269i	0.884918	−	0.465746i	$-0.154214\pi$
$462$		0			0
$463$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$463$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$464$	32.3607	+	23.5114i	1.50231	+	1.09149i
$465$		0			0
$466$	15.7845	−	5.12870i	0.731203	−	0.237582i
$467$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$467$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$468$	−2.28834	−	4.49112i	−0.105778	−	0.207602i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	17.9806	−	35.2889i	0.823276	−	1.61577i
$478$		0			0
$479$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$479$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$480$		0			0
$481$	−2.92891	−	0.951660i	−0.133547	−	0.0433920i
$482$	15.4768	+	30.3749i	0.704949	+	1.38354i
$483$		0			0
$484$	20.9232	+	6.79837i	0.951057	+	0.309017i
$485$	13.9088	+	41.7265i	0.631567	+	1.89470i
$486$		0			0
$487$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$487$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$488$	−16.9739	+	33.3132i	−0.768374	+	1.50802i
$489$		0			0
$490$	17.8091	−	13.1468i	0.804532	−	0.593910i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−58.0844	−	58.0844i	−2.61599	−	2.61599i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$499$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$500$	−20.1495	−	9.69521i	−0.901114	−	0.433583i
$501$		0			0
$502$		0			0
$503$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$503$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$504$		0			0
$505$	−33.1775	−	4.99693i	−1.47638	−	0.222360i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.49197	−	16.9025i	0.243427	−	0.749192i	−0.752464	−	0.658633i	$-0.771135\pi$
$509$	5.49197	−	16.9025i	0.243427	−	0.749192i	0.995891	−	0.0905584i	$-0.0288652\pi$
$510$		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
$520$	0.791299	−	5.25390i	0.0347008	−	0.230399i
$521$	−31.6549	+	10.2853i	−1.38683	+	0.450607i	−0.904907	−	0.425609i	$-0.860060\pi$
$521$	−31.6549	+	10.2853i	−1.38683	+	0.450607i	−0.481919	+	0.876216i	$0.660060\pi$
$522$	37.8022	−	19.2612i	1.65456	−	0.843039i
$523$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$523$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−21.8743	+	7.10739i	−0.951057	+	0.309017i
$530$	37.0529	−	19.2351i	1.60947	−	0.835518i
$531$		0			0
$532$		0			0
$533$	4.90103	−	2.21704i	0.212287	−	0.0960306i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	17.3707	−	34.0920i	0.748906	−	1.46981i
$539$		0			0
$540$		0			0
$541$	14.1327	−	43.4960i	0.607613	−	1.87004i	0.129892	−	0.991528i	$-0.458537\pi$
$541$	14.1327	−	43.4960i	0.607613	−	1.87004i	0.477721	−	0.878512i	$-0.341463\pi$
$542$		0			0
$543$		0			0
$544$	−14.3593	+	44.1932i	−0.615648	+	1.89477i
$545$	19.8547	−	20.1584i	0.850481	−	0.863493i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	5.09609	+	2.59659i	0.217694	+	0.110921i
$549$	23.3094	+	32.0826i	0.994821	+	1.36925i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−25.3292	−	34.8627i	−1.07614	−	1.48117i
$555$		0			0
$556$		0			0
$557$	−6.30037	+	3.21020i	−0.266955	+	0.136020i	−0.582345	−	0.812942i	$-0.697865\pi$
$557$	−6.30037	+	3.21020i	−0.266955	+	0.136020i	0.315390	+	0.948962i	$0.397865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	17.5557	−	34.4550i	0.740542	−	1.45340i
$563$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$563$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$564$		0			0
$565$	−13.3195	+	42.0776i	−0.560357	+	1.77022i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	43.8215	−	14.2385i	1.83709	−	0.596908i	0.838444	−	0.544988i	$-0.183466\pi$
$569$	43.8215	−	14.2385i	1.83709	−	0.596908i	0.998651	−	0.0519200i	$-0.0165341\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−19.4164	−	14.1068i	−0.809017	−	0.587785i
$577$	6.70636	+	6.70636i	0.279190	+	0.279190i	0.832785	−	0.553596i	$-0.186745\pi$
$577$	6.70636	+	6.70636i	0.279190	+	0.279190i	−0.553596	+	0.832785i	$0.686745\pi$
$578$	32.4076	−	63.6034i	1.34798	−	2.64555i
$579$		0			0
$580$	44.2226	+	6.66045i	1.83624	+	0.276560i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−5.15889	−	1.67623i	−0.213477	−	0.0693627i
$585$	−4.58419	−	3.27773i	−0.189533	−	0.135517i
$586$	44.3253	−	14.4022i	1.83106	−	0.594947i
$587$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$587$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−14.4830	+	2.29388i	−0.595246	+	0.0942777i
$593$	−5.49706	+	34.7070i	−0.225737	+	1.42525i	0.571014	+	0.820940i	$0.306550\pi$
$593$	−5.49706	+	34.7070i	−0.225737	+	1.42525i	−0.796751	+	0.604307i	$0.793450\pi$
$594$		0			0
$595$		0			0
$596$	−8.38705	−	11.5438i	−0.343547	−	0.472852i
$597$		0			0
$598$		0			0
$599$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$599$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$600$		0			0
$601$			42.5605i			1.73608i	0.496494	+	0.868040i	$0.334621\pi$
$601$			42.5605i			1.73608i	−0.496494	+	0.868040i	$0.665379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.2640	−	4.03209i	0.986472	−	0.163928i
$606$		0			0
$607$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$607$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$608$		0			0
$609$		0			0
$610$	0.317324	+	41.8002i	0.0128481	+	1.69244i
$611$		0			0
$612$	34.8506	+	34.8506i	1.40875	+	1.40875i
$613$	−7.43250	−	46.9270i	−0.300196	−	1.89536i	−0.428358	−	0.903609i	$-0.640908\pi$
$613$	−7.43250	−	46.9270i	−0.300196	−	1.89536i	0.128162	−	0.991753i	$-0.459092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−25.3456	+	4.01435i	−1.02037	+	0.161611i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	−25.3456	+	4.01435i	−1.02037	+	0.161611i	−0.376239	+	0.926523i	$0.622783\pi$
$618$		0			0
$619$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$619$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.9885	+	0.759012i	−0.999539	+	0.0303605i
$626$	−47.5528	−	15.4508i	−1.90059	−	0.617540i
$627$		0			0
$628$	−49.4832	+	7.83738i	−1.97460	+	0.312745i
$629$	30.1129			1.20068
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$		0			0
$633$		0			0
$634$	−17.1165	+	12.4359i	−0.679784	+	0.493892i
$635$		0			0
$636$		0			0
$637$	5.80818	+	0.919926i	0.230129	+	0.0364488i
$638$		0			0
$639$		0			0
$640$	−8.00000	−	24.0000i	−0.316228	−	0.948683i
$641$	−42.9432	+	13.9531i	−1.69615	+	0.551114i	−0.987934	−	0.154878i	$-0.950502\pi$
$641$	−42.9432	+	13.9531i	−1.69615	+	0.551114i	−0.708220	+	0.705992i	$0.750502\pi$
$642$		0			0
$643$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$643$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$644$		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	$0.750000\pi$
$648$	−22.6813	+	11.5567i	−0.891007	+	0.453990i
$649$		0			0
$650$	−1.92121	−	5.62103i	−0.0753560	−	0.220475i
$651$		0			0
$652$		0			0
$653$	19.3751	−	19.3751i	0.758206	−	0.758206i	−0.217789	−	0.975996i	$-0.569885\pi$
$653$	19.3751	−	19.3751i	0.758206	−	0.758206i	0.975996	+	0.217789i	$0.0698846\pi$
$654$		0			0
$655$		0			0
$656$	16.0000	−	20.0000i	0.624695	−	0.780869i
$657$	−4.06829	+	4.06829i	−0.158719	+	0.158719i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−15.8405	−	48.7521i	−0.616125	−	1.89624i	−0.382752	−	0.923851i	$-0.625024\pi$
$661$	−15.8405	−	48.7521i	−0.616125	−	1.89624i	−0.233373	−	0.972387i	$-0.574976\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−4.80614	+	14.7918i	−0.186234	+	0.573169i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	13.6608	+	26.8109i	0.526586	+	1.03348i	0.989152	+	0.146898i	$0.0469288\pi$
$673$	13.6608	+	26.8109i	0.526586	+	1.03348i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	10.6172	−	14.6133i	0.408958	−	0.562883i
$675$		0			0
$676$	−19.8925	+	14.4528i	−0.765097	+	0.555876i
$677$	−7.10594	+	44.8651i	−0.273103	+	1.72431i	0.345341	+	0.938477i	$0.387763\pi$
$677$	−7.10594	+	44.8651i	−0.273103	+	1.72431i	−0.618444	+	0.785829i	$0.712237\pi$
$678$		0			0
$679$		0			0
$680$	8.51642	+	51.2495i	0.326590	+	1.96533i
$681$		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
$685$	6.39439	−	0.0485426i	0.244317	−	0.00185472i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.5479	+	3.42721i	0.401842	+	0.130566i
$690$		0			0
$691$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$691$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$692$	44.6468	+	22.7487i	1.69722	+	0.864776i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−38.8780	+	35.4262i	−1.47261	+	1.34186i
$698$	−23.2468	−	23.2468i	−0.879903	−	0.879903i
$699$		0			0
$700$		0			0
$701$	11.1836	+	34.4197i	0.422400	+	1.30001i	0.905462	+	0.424428i	$0.139525\pi$
$701$	11.1836	+	34.4197i	0.422400	+	1.30001i	−0.483061	+	0.875587i	$0.660475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	14.8270			0.558020
$707$		0			0
$708$		0			0
$709$	0.491968	+	1.51412i	0.0184762	+	0.0568640i	0.959870	−	0.280447i	$-0.0904826\pi$
$709$	0.491968	+	1.51412i	0.0184762	+	0.0568640i	−0.941393	+	0.337311i	$0.890483\pi$
$710$		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	$-0.200000\pi$
$719$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$720$	−26.5336	−	3.99627i	−0.988847	−	0.148932i
$721$		0			0
$722$	4.20340	−	26.5392i	0.156434	−	0.987688i
$723$		0			0
$724$	−46.5987			−1.73183
$725$	47.3128	−	16.1710i	1.75715	−	0.600576i
$726$		0			0
$727$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$727$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$	−5.98260	+	0.994163i	−0.221426	+	0.0367956i
$731$		0			0
$732$		0			0
$733$	9.51176	−	18.6679i	0.351325	−	0.689514i	−0.645943	−	0.763386i	$-0.723536\pi$
$733$	9.51176	−	18.6679i	0.351325	−	0.689514i	0.997268	+	0.0738717i	$0.0235355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−11.1966	−	24.7515i	−0.412154	−	0.911114i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−13.1897	+	9.73672i	−0.484864	+	0.357929i
$741$		0			0
$742$		0			0
$743$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$743$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$744$		0			0
$745$	−14.2689	−	7.13445i	−0.522772	−	0.261386i
$746$	−12.5938	+	38.7598i	−0.461092	+	1.41910i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$751$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$752$		0			0
$753$		0			0
$754$	6.98322	+	9.61158i	0.254314	+	0.350033i
$755$		0			0
$756$		0			0
$757$	53.5086	+	8.47493i	1.94480	+	0.308027i	0.999818	−	0.0190839i	$-0.00607496\pi$
$757$	53.5086	+	8.47493i	1.94480	+	0.308027i	0.944986	+	0.327111i	$0.106075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	43.8925	+	31.8897i	1.59110	+	1.15600i	0.902351	+	0.431003i	$0.141840\pi$
$761$	43.8925	+	31.8897i	1.59110	+	1.15600i	0.688749	+	0.724999i	$0.258160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	52.5345	+	16.6297i	1.89939	+	0.601247i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−52.2788	−	16.9864i	−1.88522	−	0.612546i	−0.983700	−	0.179815i	$-0.942450\pi$
$769$	−52.2788	−	16.9864i	−1.88522	−	0.612546i	−0.901523	−	0.432731i	$-0.857550\pi$
$770$		0			0
$771$		0			0
$772$	−42.4991	−	6.73120i	−1.52958	−	0.242261i
$773$	−6.98401	+	1.10616i	−0.251197	+	0.0397858i	−0.280763	−	0.959777i	$-0.590588\pi$
$773$	−6.98401	+	1.10616i	−0.251197	+	0.0397858i	0.0295658	+	0.999563i	$0.490588\pi$
$774$		0			0
$775$		0			0
$776$			55.6353i			1.99719i
$777$		0			0
$778$	−15.7509	−	15.7509i	−0.564696	−	0.564696i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	26.6296	−	8.65248i	0.951057	−	0.309017i
$785$	−45.0647	+	33.2670i	−1.60843	+	1.18735i
$786$		0			0
$787$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$787$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$788$	3.40088	+	3.40088i	0.121151	+	0.121151i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7.85232	+	7.85232i	−0.278844	+	0.278844i
$794$	−17.2225	−	53.0055i	−0.611205	−	1.88109i
$795$		0			0
$796$		0			0
$797$	−29.4961	+	4.67172i	−1.04480	+	0.165481i	−0.655164	−	0.755487i	$-0.727401\pi$
$797$	−29.4961	+	4.67172i	−1.04480	+	0.165481i	−0.389640	+	0.920967i	$0.627401\pi$
$798$		0			0
$799$		0			0
$800$	−20.3013	−	19.6941i	−0.717761	−	0.696290i
$801$	−17.6336	−	24.2705i	−0.623051	−	0.857556i
$802$	35.1009	−	5.55943i	1.23945	−	0.196310i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−37.8142	−	19.2673i	−1.33030	−	0.677822i
$809$	−15.5031	+	47.7135i	−0.545059	+	1.67752i	0.175791	+	0.984428i	$0.443752\pi$
$809$	−15.5031	+	47.7135i	−0.545059	+	1.67752i	−0.720850	+	0.693091i	$0.756248\pi$
$810$	−16.5534	+	23.1514i	−0.581627	+	0.813456i
$811$		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
$818$	8.00597	+	50.5477i	0.279922	+	1.76736i
$819$		0			0
$820$	5.57421	−	28.0879i	0.194660	−	0.980871i
$821$	−38.9003			−1.35763			−0.678816	−	0.734309i	$-0.737507\pi$
$821$	−38.9003			−1.35763			−0.678816	+	0.734309i	$0.737507\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$827$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$828$		0			0
$829$			35.7080i			1.24019i	0.784526	+	0.620096i	$0.212906\pi$
$829$			35.7080i			1.24019i	−0.784526	+	0.620096i	$0.787094\pi$
$830$		0			0
$831$		0			0
$832$	3.05112	−	5.98816i	0.105778	−	0.207602i
$833$	−56.7927	+	8.99508i	−1.96775	+	0.311661i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$839$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$840$		0			0
$841$	−57.4402	+	41.7328i	−1.98070	+	1.43906i
$842$	10.9125	+	1.72838i	0.376071	+	0.0595638i
$843$		0			0
$844$		0			0
$845$	−12.2943	+	24.5885i	−0.422935	+	0.845871i
$846$		0			0
$847$		0			0
$848$	52.1574	−	8.26093i	1.79109	−	0.283681i
$849$		0			0
$850$	34.8506	+	46.4675i	1.19537	+	1.59382i
$851$		0			0
$852$		0			0
$853$	−51.6630	+	26.3236i	−1.76891	+	0.901304i	−0.829496	+	0.558512i	$0.811372\pi$
$853$	−51.6630	+	26.3236i	−1.76891	+	0.901304i	−0.939411	+	0.342792i	$0.888628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	52.1572	+	26.5754i	1.78166	+	0.907799i	0.899913	+	0.436069i	$0.143630\pi$
$857$	52.1572	+	26.5754i	1.78166	+	0.907799i	0.881743	+	0.471731i	$0.156370\pi$
$858$		0			0
$859$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$859$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$863$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$864$		0			0
$865$	56.0212	−	0.425281i	1.90478	−	0.0144600i
$866$	12.6836	+	39.0362i	0.431008	+	1.32651i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	31.8890	−	16.2483i	1.07990	−	0.550235i
$873$	52.5784	+	26.7900i	1.77951	+	0.906705i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	8.66783	−	54.7265i	0.292692	−	1.84798i	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	8.66783	−	54.7265i	0.292692	−	1.84798i	0.495297	−	0.868723i	$-0.335059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−25.8885	+	18.8091i	−0.872207	+	0.633696i	−0.931178	−	0.364564i	$-0.881218\pi$
$881$	−25.8885	+	18.8091i	−0.872207	+	0.633696i	0.0589711	+	0.998260i	$0.481218\pi$
$882$	4.64587	−	29.3328i	0.156434	−	0.987688i
$883$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$883$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$884$	−8.11233	+	11.1657i	−0.272847	+	0.375542i
$885$		0			0
$886$		0			0
$887$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$887$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$888$		0			0
$889$		0			0
$890$	−0.240055	−	31.6219i	−0.00804667	−	1.05997i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	22.3671	−	43.8980i	0.746401	−	1.46490i
$899$		0			0
$900$	−28.3877	+	9.70261i	−0.946255	+	0.323420i
$901$	−108.445			−3.61284
$902$		0			0
$903$		0			0
$904$	−32.8145	+	45.1653i	−1.09139	+	1.50217i
$905$	−46.2396	+	24.0041i	−1.53706	+	0.797924i
$906$		0			0
$907$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$907$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$908$		0			0
$909$	−36.4173	+	26.4587i	−1.20789	+	0.877581i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	5.21777	+	16.0587i	0.172589	+	0.531173i
$915$		0			0
$916$		−	10.9326i		−	0.361222i
$917$		0			0
$918$		0			0
$919$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$919$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$920$		0			0
$921$		0			0
$922$	6.80592	−	42.9709i	0.224141	−	1.41517i
$923$		0			0
$924$		0			0
$925$	−8.07245	+	16.4560i	−0.265421	+	0.541071i
$926$		0			0
$927$		0			0
$928$	50.4029	+	25.6816i	1.65456	+	0.843039i
$929$	−31.3879			−1.02980			−0.514902	−	0.857249i	$-0.672172\pi$
$929$	−31.3879			−1.02980			−0.514902	+	0.857249i	$0.672172\pi$
$930$		0			0
$931$		0			0
$932$	20.9132	−	10.6558i	0.685035	−	0.349043i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−4.18993	−	5.76695i	−0.136952	−	0.188499i
$937$	−1.10616	−	6.98401i	−0.0361366	−	0.228158i	0.963010	−	0.269466i	$-0.0868473\pi$
$937$	−1.10616	−	6.98401i	−0.0361366	−	0.228158i	−0.999146	+	0.0413087i	$0.986847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.888418	−	0.645473i	−0.0289616	−	0.0210418i	0.573210	−	0.819408i	$-0.305698\pi$
$941$	−0.888418	−	0.645473i	−0.0289616	−	0.0210418i	−0.602172	+	0.798366i	$0.705698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$947$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$948$		0			0
$949$	−1.30342	−	0.946992i	−0.0423109	−	0.0307407i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	51.6630	−	26.3236i	1.67353	−	0.852706i	0.680782	−	0.732486i	$-0.261640\pi$
$953$	51.6630	−	26.3236i	1.67353	−	0.852706i	0.992747	−	0.120219i	$-0.0383598\pi$
$954$	17.3083	−	53.2695i	0.560378	−	1.72467i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−25.0795	−	18.2213i	−0.809017	−	0.587785i
$962$	−4.30165	−	0.681314i	−0.138691	−	0.0219664i
$963$		0			0
$964$	28.3379	+	39.0038i	0.912704	+	1.25623i
$965$	−45.6390	+	15.2130i	−1.46917	+	0.489724i
$966$		0			0
$967$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$967$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$968$	30.7296	+	4.86710i	0.987688	+	0.156434i
$969$		0			0
$970$	28.6591	+	55.2065i	0.920188	+	1.77258i
$971$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$971$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−16.3393	+	50.2872i	−0.523008	+	1.60965i
$977$	48.8881	+	7.74311i	1.56407	+	0.247724i	0.877585	−	0.479421i	$-0.159153\pi$
$977$	48.8881	+	7.74311i	1.56407	+	0.247724i	0.686483	+	0.727145i	$0.259153\pi$
$978$		0			0
$979$		0			0
$980$	21.9673	−	22.3033i	0.701719	−	0.712454i
$981$		−	37.9609i		−	1.21200i
$982$		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
$985$	5.12655	+	1.62280i	0.163346	+	0.0517066i
$986$	−93.9825	−	68.2823i	−2.99301	−	2.17455i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$991$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.336854	−	2.12681i	−0.0106683	−	0.0673568i	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	−0.336854	−	2.12681i	−0.0106683	−	0.0673568i	−0.992448	+	0.122665i	$0.960856\pi$
$998$		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.523.1	yes	8
4.3	odd	2	CM	820.2.bm.b.523.1	yes	8
5.2	odd	4		820.2.bm.a.687.1	✓	8
20.7	even	4		820.2.bm.a.687.1	✓	8
41.4	even	10		820.2.bm.a.783.1	yes	8
164.127	odd	10		820.2.bm.a.783.1	yes	8
205.127	odd	20	inner	820.2.bm.b.127.1	yes	8
820.127	even	20	inner	820.2.bm.b.127.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
820.2.bm.a.687.1	✓	8	5.2	odd	4
820.2.bm.a.687.1	✓	8	20.7	even	4
820.2.bm.a.783.1	yes	8	41.4	even	10
820.2.bm.a.783.1	yes	8	164.127	odd	10
820.2.bm.b.127.1	yes	8	205.127	odd	20	inner
820.2.bm.b.127.1	yes	8	820.127	even	20	inner
820.2.bm.b.523.1	yes	8	1.1	even	1	trivial
820.2.bm.b.523.1	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)

\(f(q)\)	\(=\)	\(q+(1.39680 - 0.221232i) q^{2} +(1.90211 - 0.618034i) q^{4} +(1.56909 - 1.59310i) q^{5} +(2.52015 - 1.28408i) q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.39680 - 0.221232i) q^{2} +(1.90211 - 0.618034i) q^{4} +(1.56909 - 1.59310i) q^{5} +(2.52015 - 1.28408i) q^{8} -3.00000i q^{9} +(1.83927 - 2.57237i) q^{10} +(0.829740 - 0.131418i) q^{13} +(3.23607 - 2.35114i) q^{16} +(-7.31906 + 3.72925i) q^{17} +(-0.663695 - 4.19041i) q^{18} +(2.00000 - 4.00000i) q^{20} +(-0.0759100 - 4.99942i) q^{25} +(1.12991 - 0.367130i) q^{26} +(3.09017 + 9.51057i) q^{29} +(4.00000 - 4.00000i) q^{32} +(-9.39825 + 6.82823i) q^{34} +(-1.85410 - 5.70634i) q^{36} +(-3.26632 - 1.66427i) q^{37} +(1.90868 - 6.02967i) q^{40} +(6.17499 - 1.69394i) q^{41} +(-4.77929 - 4.70727i) q^{45} +(6.65740 + 2.16312i) q^{49} +(-1.21206 - 6.96641i) q^{50} +(1.49704 - 0.762779i) q^{52} +(11.7630 + 5.99354i) q^{53} +(6.42040 + 12.6007i) q^{58} +(-10.6942 + 7.76980i) q^{61} +(4.70228 - 6.47214i) q^{64} +(1.09258 - 1.52806i) q^{65} +(-11.6169 + 11.6169i) q^{68} +(-3.85224 - 7.56044i) q^{72} +(-1.35610 - 1.35610i) q^{73} +(-4.93059 - 1.60205i) q^{74} +(1.33209 - 8.84452i) q^{80} -9.00000 q^{81} +(8.25049 - 3.73221i) q^{82} +(-5.54322 + 17.5115i) q^{85} +(8.09017 - 5.87785i) q^{89} +(-7.71712 - 5.51780i) q^{90} +(-8.93001 + 17.5261i) q^{97} +(9.77762 + 1.54862i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 4 q^{5} - 4 q^{8}+O(q^{10}) \) 8 * q + 2 * q^2 - 4 * q^5 - 4 * q^8	\( 8 q + 2 q^{2} - 4 q^{5} - 4 q^{8} + 6 q^{10} + 10 q^{13} + 8 q^{16} - 10 q^{17} - 6 q^{18} + 16 q^{20} - 6 q^{25} - 10 q^{26} - 20 q^{29} + 32 q^{32} - 30 q^{34} + 12 q^{36} - 4 q^{37} - 4 q^{40} + 8 q^{41} - 6 q^{45} + 14 q^{50} + 20 q^{52} + 10 q^{53} + 20 q^{58} - 24 q^{61} + 30 q^{65} - 20 q^{68} - 12 q^{72} - 22 q^{73} - 50 q^{74} + 16 q^{80} - 72 q^{81} + 2 q^{82} - 30 q^{85} + 20 q^{89} - 6 q^{90} - 50 q^{97} + 14 q^{98}+O(q^{100}) \) 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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