LMFDB - Embedded newform 418.2.e.c.353.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [418,2,Mod(45,418)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(418, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("418.45");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 418 = 2 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	418.e (of order $3$, degree $2$, 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:	$3.33774680449$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			353.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	418.353
Dual form			418.2.e.c.45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.00000 - 3.46410i) q^{5} -4.00000 q^{7} +1.00000 q^{8} +(1.50000 + 2.59808i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.00000 - 3.46410i) q^{5} -4.00000 q^{7} +1.00000 q^{8} +(1.50000 + 2.59808i) q^{9} +(2.00000 + 3.46410i) q^{10} +1.00000 q^{11} +(-3.50000 - 6.06218i) q^{13} +(2.00000 - 3.46410i) q^{14} +(-0.500000 + 0.866025i) q^{16} -3.00000 q^{18} +(-0.500000 - 4.33013i) q^{19} -4.00000 q^{20} +(-0.500000 + 0.866025i) q^{22} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 - 9.52628i) q^{25} +7.00000 q^{26} +(2.00000 + 3.46410i) q^{28} +(1.50000 + 2.59808i) q^{29} +(-0.500000 - 0.866025i) q^{32} +(-8.00000 + 13.8564i) q^{35} +(1.50000 - 2.59808i) q^{36} +8.00000 q^{37} +(4.00000 + 1.73205i) q^{38} +(2.00000 - 3.46410i) q^{40} +(6.00000 - 10.3923i) q^{41} +(-3.50000 + 6.06218i) q^{43} +(-0.500000 - 0.866025i) q^{44} +12.0000 q^{45} +4.00000 q^{46} +(2.50000 + 4.33013i) q^{47} +9.00000 q^{49} +11.0000 q^{50} +(-3.50000 + 6.06218i) q^{52} +(-3.00000 - 5.19615i) q^{53} +(2.00000 - 3.46410i) q^{55} -4.00000 q^{56} -3.00000 q^{58} +(-3.00000 + 5.19615i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-6.00000 - 10.3923i) q^{63} +1.00000 q^{64} -28.0000 q^{65} +(4.00000 + 6.92820i) q^{67} +(-8.00000 - 13.8564i) q^{70} +(4.50000 - 7.79423i) q^{71} +(1.50000 + 2.59808i) q^{72} +(-5.00000 + 8.66025i) q^{73} +(-4.00000 + 6.92820i) q^{74} +(-3.50000 + 2.59808i) q^{76} -4.00000 q^{77} +(3.00000 - 5.19615i) q^{79} +(2.00000 + 3.46410i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(6.00000 + 10.3923i) q^{82} -3.00000 q^{83} +(-3.50000 - 6.06218i) q^{86} +1.00000 q^{88} +(0.500000 + 0.866025i) q^{89} +(-6.00000 + 10.3923i) q^{90} +(14.0000 + 24.2487i) q^{91} +(-2.00000 + 3.46410i) q^{92} -5.00000 q^{94} +(-16.0000 - 6.92820i) q^{95} +(2.50000 - 4.33013i) q^{97} +(-4.50000 + 7.79423i) q^{98} +(1.50000 + 2.59808i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 4 q^{5} - 8 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) $ 2 * q - q^2 - q^4 + 4 * q^5 - 8 * q^7 + 2 * q^8 + 3 * q^9	$ 2 q - q^{2} - q^{4} + 4 q^{5} - 8 q^{7} + 2 q^{8} + 3 q^{9} + 4 q^{10} + 2 q^{11} - 7 q^{13} + 4 q^{14} - q^{16} - 6 q^{18} - q^{19} - 8 q^{20} - q^{22} - 4 q^{23} - 11 q^{25} + 14 q^{26} + 4 q^{28} + 3 q^{29} - q^{32} - 16 q^{35} + 3 q^{36} + 16 q^{37} + 8 q^{38} + 4 q^{40} + 12 q^{41} - 7 q^{43} - q^{44} + 24 q^{45} + 8 q^{46} + 5 q^{47} + 18 q^{49} + 22 q^{50} - 7 q^{52} - 6 q^{53} + 4 q^{55} - 8 q^{56} - 6 q^{58} - 6 q^{59} + 5 q^{61} - 12 q^{63} + 2 q^{64} - 56 q^{65} + 8 q^{67} - 16 q^{70} + 9 q^{71} + 3 q^{72} - 10 q^{73} - 8 q^{74} - 7 q^{76} - 8 q^{77} + 6 q^{79} + 4 q^{80} - 9 q^{81} + 12 q^{82} - 6 q^{83} - 7 q^{86} + 2 q^{88} + q^{89} - 12 q^{90} + 28 q^{91} - 4 q^{92} - 10 q^{94} - 32 q^{95} + 5 q^{97} - 9 q^{98} + 3 q^{99}+O(q^{100}) $ 2 * q - q^2 - q^4 + 4 * q^5 - 8 * q^7 + 2 * q^8 + 3 * q^9 + 4 * q^10 + 2 * q^11 - 7 * q^13 + 4 * q^14 - q^16 - 6 * q^18 - q^19 - 8 * q^20 - q^22 - 4 * q^23 - 11 * q^25 + 14 * q^26 + 4 * q^28 + 3 * q^29 - q^32 - 16 * q^35 + 3 * q^36 + 16 * q^37 + 8 * q^38 + 4 * q^40 + 12 * q^41 - 7 * q^43 - q^44 + 24 * q^45 + 8 * q^46 + 5 * q^47 + 18 * q^49 + 22 * q^50 - 7 * q^52 - 6 * q^53 + 4 * q^55 - 8 * q^56 - 6 * q^58 - 6 * q^59 + 5 * q^61 - 12 * q^63 + 2 * q^64 - 56 * q^65 + 8 * q^67 - 16 * q^70 + 9 * q^71 + 3 * q^72 - 10 * q^73 - 8 * q^74 - 7 * q^76 - 8 * q^77 + 6 * q^79 + 4 * q^80 - 9 * q^81 + 12 * q^82 - 6 * q^83 - 7 * q^86 + 2 * q^88 + q^89 - 12 * q^90 + 28 * q^91 - 4 * q^92 - 10 * q^94 - 32 * q^95 + 5 * q^97 - 9 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/418\mathbb{Z}\right)^\times$.

$n$	$287$	$343$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$3$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	2.00000	−	3.46410i	0.894427	−	1.54919i	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	2.00000	−	3.46410i	0.894427	−	1.54919i	0.834512	−	0.550990i	$-0.185750\pi$
$6$		0			0
$7$	−4.00000			−1.51186			−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000			−1.51186			−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000			0.353553
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$	2.00000	+	3.46410i	0.632456	+	1.09545i
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$	−3.50000	−	6.06218i	−0.970725	−	1.68135i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−3.50000	−	6.06218i	−0.970725	−	1.68135i	−0.277350	−	0.960769i	$-0.589456\pi$
$14$	2.00000	−	3.46410i	0.534522	−	0.925820i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$	−3.00000			−0.707107
$19$	−0.500000	−	4.33013i	−0.114708	−	0.993399i
$19$	−0.500000	−	4.33013i	−0.114708	−	0.993399i
$20$	−4.00000			−0.894427
$21$		0			0
$22$	−0.500000	+	0.866025i	−0.106600	+	0.184637i
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	−5.50000	−	9.52628i	−1.10000	−	1.90526i
$26$	7.00000			1.37281
$27$		0			0
$28$	2.00000	+	3.46410i	0.377964	+	0.654654i
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$		0			0
$35$	−8.00000	+	13.8564i	−1.35225	+	2.34216i
$36$	1.50000	−	2.59808i	0.250000	−	0.433013i
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$	4.00000	+	1.73205i	0.648886	+	0.280976i
$39$		0			0
$40$	2.00000	−	3.46410i	0.316228	−	0.547723i
$41$	6.00000	−	10.3923i	0.937043	−	1.62301i	0.166092	−	0.986110i	$-0.446885\pi$
$41$	6.00000	−	10.3923i	0.937043	−	1.62301i	0.770950	−	0.636895i	$-0.219782\pi$
$42$		0			0
$43$	−3.50000	+	6.06218i	−0.533745	+	0.924473i	0.465478	+	0.885059i	$0.345882\pi$
$43$	−3.50000	+	6.06218i	−0.533745	+	0.924473i	−0.999223	+	0.0394140i	$0.987451\pi$
$44$	−0.500000	−	0.866025i	−0.0753778	−	0.130558i
$45$	12.0000			1.78885
$46$	4.00000			0.589768
$47$	2.50000	+	4.33013i	0.364662	+	0.631614i	0.988722	−	0.149763i	$-0.0478510\pi$
$47$	2.50000	+	4.33013i	0.364662	+	0.631614i	−0.624059	+	0.781377i	$0.714518\pi$
$48$		0			0
$49$	9.00000			1.28571
$50$	11.0000			1.55563
$51$		0			0
$52$	−3.50000	+	6.06218i	−0.485363	+	0.840673i
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$	2.00000	−	3.46410i	0.269680	−	0.467099i
$56$	−4.00000			−0.534522
$57$		0			0
$58$	−3.00000			−0.393919
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	−0.992524	−	0.122047i	$-0.961054\pi$
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	0.601958	+	0.798528i	$0.294388\pi$
$60$		0			0
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	0.980507	−	0.196485i	$-0.0629528\pi$
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	−0.660415	+	0.750901i	$0.729619\pi$
$62$		0			0
$63$	−6.00000	−	10.3923i	−0.755929	−	1.30931i
$64$	1.00000			0.125000
$65$	−28.0000			−3.47297
$66$		0			0
$67$	4.00000	+	6.92820i	0.488678	+	0.846415i	0.999915	−	0.0130248i	$-0.00414604\pi$
$67$	4.00000	+	6.92820i	0.488678	+	0.846415i	−0.511237	+	0.859440i	$0.670813\pi$
$68$		0			0
$69$		0			0
$70$	−8.00000	−	13.8564i	−0.956183	−	1.65616i
$71$	4.50000	−	7.79423i	0.534052	−	0.925005i	−0.465157	−	0.885228i	$-0.654002\pi$
$71$	4.50000	−	7.79423i	0.534052	−	0.925005i	0.999209	−	0.0397765i	$-0.0126646\pi$
$72$	1.50000	+	2.59808i	0.176777	+	0.306186i
$73$	−5.00000	+	8.66025i	−0.585206	+	1.01361i	0.409644	+	0.912245i	$0.365653\pi$
$73$	−5.00000	+	8.66025i	−0.585206	+	1.01361i	−0.994850	+	0.101361i	$0.967680\pi$
$74$	−4.00000	+	6.92820i	−0.464991	+	0.805387i
$75$		0			0
$76$	−3.50000	+	2.59808i	−0.401478	+	0.298020i
$77$	−4.00000			−0.455842
$78$		0			0
$79$	3.00000	−	5.19615i	0.337526	−	0.584613i	−0.646440	−	0.762964i	$-0.723743\pi$
$79$	3.00000	−	5.19615i	0.337526	−	0.584613i	0.983967	+	0.178352i	$0.0570765\pi$
$80$	2.00000	+	3.46410i	0.223607	+	0.387298i
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$	6.00000	+	10.3923i	0.662589	+	1.14764i
$83$	−3.00000			−0.329293			−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000			−0.329293			−0.164646	+	0.986353i	$0.552648\pi$
$84$		0			0
$85$		0			0
$86$	−3.50000	−	6.06218i	−0.377415	−	0.653701i
$87$		0			0
$88$	1.00000			0.106600
$89$	0.500000	+	0.866025i	0.0529999	+	0.0917985i	0.891308	−	0.453398i	$-0.149788\pi$
$89$	0.500000	+	0.866025i	0.0529999	+	0.0917985i	−0.838308	+	0.545197i	$0.816455\pi$
$90$	−6.00000	+	10.3923i	−0.632456	+	1.09545i
$91$	14.0000	+	24.2487i	1.46760	+	2.54196i
$92$	−2.00000	+	3.46410i	−0.208514	+	0.361158i
$93$		0			0
$94$	−5.00000			−0.515711
$95$	−16.0000	−	6.92820i	−1.64157	−	0.710819i
$96$		0			0
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	0.964579	+	0.263795i	$0.0849741\pi$
$98$	−4.50000	+	7.79423i	−0.454569	+	0.787336i
$99$	1.50000	+	2.59808i	0.150756	+	0.261116i
$100$	−5.50000	+	9.52628i	−0.550000	+	0.952628i
$101$	7.50000	+	12.9904i	0.746278	+	1.29259i	0.949595	+	0.313478i	$0.101494\pi$
$101$	7.50000	+	12.9904i	0.746278	+	1.29259i	−0.203317	+	0.979113i	$0.565172\pi$
$102$		0			0
$103$	11.0000			1.08386			0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000			1.08386			0.541931	+	0.840423i	$0.317693\pi$
$104$	−3.50000	−	6.06218i	−0.343203	−	0.594445i
$105$		0			0
$106$	6.00000			0.582772
$107$	−3.00000			−0.290021			−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000			−0.290021			−0.145010	+	0.989430i	$0.546322\pi$
$108$		0			0
$109$	2.50000	−	4.33013i	0.239457	−	0.414751i	−0.721102	−	0.692829i	$-0.756364\pi$
$109$	2.50000	−	4.33013i	0.239457	−	0.414751i	0.960558	+	0.278078i	$0.0896974\pi$
$110$	2.00000	+	3.46410i	0.190693	+	0.330289i
$111$		0			0
$112$	2.00000	−	3.46410i	0.188982	−	0.327327i
$113$	−3.00000			−0.282216			−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000			−0.282216			−0.141108	+	0.989994i	$0.545067\pi$
$114$		0			0
$115$	−16.0000			−1.49201
$116$	1.50000	−	2.59808i	0.139272	−	0.241225i
$117$	10.5000	−	18.1865i	0.970725	−	1.68135i
$118$	−3.00000	−	5.19615i	−0.276172	−	0.478345i
$119$		0			0
$120$		0			0
$121$	1.00000			0.0909091
$122$	−5.00000			−0.452679
$123$		0			0
$124$		0			0
$125$	−24.0000			−2.14663
$126$	12.0000			1.06904
$127$	−5.00000	−	8.66025i	−0.443678	−	0.768473i	0.554281	−	0.832330i	$-0.312993\pi$
$127$	−5.00000	−	8.66025i	−0.443678	−	0.768473i	−0.997959	+	0.0638564i	$0.979660\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	14.0000	−	24.2487i	1.22788	−	2.12675i
$131$	2.50000	−	4.33013i	0.218426	−	0.378325i	−0.735901	−	0.677089i	$-0.763241\pi$
$131$	2.50000	−	4.33013i	0.218426	−	0.378325i	0.954327	+	0.298764i	$0.0965744\pi$
$132$		0			0
$133$	2.00000	+	17.3205i	0.173422	+	1.50188i
$134$	−8.00000			−0.691095
$135$		0			0
$136$		0			0
$137$	−3.50000	−	6.06218i	−0.299025	−	0.517927i	0.676888	−	0.736086i	$-0.263328\pi$
$137$	−3.50000	−	6.06218i	−0.299025	−	0.517927i	−0.975913	+	0.218159i	$0.929995\pi$
$138$		0			0
$139$	10.0000	+	17.3205i	0.848189	+	1.46911i	0.882823	+	0.469706i	$0.155640\pi$
$139$	10.0000	+	17.3205i	0.848189	+	1.46911i	−0.0346338	+	0.999400i	$0.511026\pi$
$140$	16.0000			1.35225
$141$		0			0
$142$	4.50000	+	7.79423i	0.377632	+	0.654077i
$143$	−3.50000	−	6.06218i	−0.292685	−	0.506945i
$144$	−3.00000			−0.250000
$145$	12.0000			0.996546
$146$	−5.00000	−	8.66025i	−0.413803	−	0.716728i
$147$		0			0
$148$	−4.00000	−	6.92820i	−0.328798	−	0.569495i
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	6.00000			0.488273			0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000			0.488273			0.244137	+	0.969741i	$0.421495\pi$
$152$	−0.500000	−	4.33013i	−0.0405554	−	0.351220i
$153$		0			0
$154$	2.00000	−	3.46410i	0.161165	−	0.279145i
$155$		0			0
$156$		0			0
$157$	7.00000	−	12.1244i	0.558661	−	0.967629i	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	7.00000	−	12.1244i	0.558661	−	0.967629i	0.997609	−	0.0691164i	$-0.0220180\pi$
$158$	3.00000	+	5.19615i	0.238667	+	0.413384i
$159$		0			0
$160$	−4.00000			−0.316228
$161$	8.00000	+	13.8564i	0.630488	+	1.09204i
$162$	−4.50000	−	7.79423i	−0.353553	−	0.612372i
$163$	6.00000			0.469956			0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000			0.469956			0.234978	+	0.972001i	$0.424498\pi$
$164$	−12.0000			−0.937043
$165$		0			0
$166$	1.50000	−	2.59808i	0.116423	−	0.201650i
$167$	8.00000	+	13.8564i	0.619059	+	1.07224i	0.989658	+	0.143448i	$0.0458190\pi$
$167$	8.00000	+	13.8564i	0.619059	+	1.07224i	−0.370599	+	0.928793i	$0.620848\pi$
$168$		0			0
$169$	−18.0000	+	31.1769i	−1.38462	+	2.39822i
$170$		0			0
$171$	10.5000	−	7.79423i	0.802955	−	0.596040i
$172$	7.00000			0.533745
$173$	−1.00000	+	1.73205i	−0.0760286	+	0.131685i	−0.901533	−	0.432710i	$-0.857557\pi$
$173$	−1.00000	+	1.73205i	−0.0760286	+	0.131685i	0.825505	+	0.564396i	$0.190891\pi$
$174$		0			0
$175$	22.0000	+	38.1051i	1.66304	+	2.88048i
$176$	−0.500000	+	0.866025i	−0.0376889	+	0.0652791i
$177$		0			0
$178$	−1.00000			−0.0749532
$179$	−16.0000			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$180$	−6.00000	−	10.3923i	−0.447214	−	0.774597i
$181$	5.00000	+	8.66025i	0.371647	+	0.643712i	0.989819	−	0.142331i	$-0.0454598\pi$
$181$	5.00000	+	8.66025i	0.371647	+	0.643712i	−0.618172	+	0.786043i	$0.712126\pi$
$182$	−28.0000			−2.07550
$183$		0			0
$184$	−2.00000	−	3.46410i	−0.147442	−	0.255377i
$185$	16.0000	−	27.7128i	1.17634	−	2.03749i
$186$		0			0
$187$		0			0
$188$	2.50000	−	4.33013i	0.182331	−	0.315807i
$189$		0			0
$190$	14.0000	−	10.3923i	1.01567	−	0.753937i
$191$	−3.00000			−0.217072			−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000			−0.217072			−0.108536	+	0.994092i	$0.534616\pi$
$192$		0			0
$193$	3.00000	−	5.19615i	0.215945	−	0.374027i	−0.737620	−	0.675216i	$-0.764050\pi$
$193$	3.00000	−	5.19615i	0.215945	−	0.374027i	0.953564	+	0.301189i	$0.0973836\pi$
$194$	2.50000	+	4.33013i	0.179490	+	0.310885i
$195$		0			0
$196$	−4.50000	−	7.79423i	−0.321429	−	0.556731i
$197$	−3.00000			−0.213741			−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000			−0.213741			−0.106871	+	0.994273i	$0.534083\pi$
$198$	−3.00000			−0.213201
$199$	−7.50000	−	12.9904i	−0.531661	−	0.920864i	−0.999317	−	0.0369532i	$-0.988235\pi$
$199$	−7.50000	−	12.9904i	−0.531661	−	0.920864i	0.467656	−	0.883911i	$-0.345099\pi$
$200$	−5.50000	−	9.52628i	−0.388909	−	0.673610i
$201$		0			0
$202$	−15.0000			−1.05540
$203$	−6.00000	−	10.3923i	−0.421117	−	0.729397i
$204$		0			0
$205$	−24.0000	−	41.5692i	−1.67623	−	2.90332i
$206$	−5.50000	+	9.52628i	−0.383203	+	0.663727i
$207$	6.00000	−	10.3923i	0.417029	−	0.722315i
$208$	7.00000			0.485363
$209$	−0.500000	−	4.33013i	−0.0345857	−	0.299521i
$210$		0			0
$211$	6.00000	−	10.3923i	0.413057	−	0.715436i	−0.582165	−	0.813070i	$-0.697794\pi$
$211$	6.00000	−	10.3923i	0.413057	−	0.715436i	0.995222	+	0.0976347i	$0.0311277\pi$
$212$	−3.00000	+	5.19615i	−0.206041	+	0.356873i
$213$		0			0
$214$	1.50000	−	2.59808i	0.102538	−	0.177601i
$215$	14.0000	+	24.2487i	0.954792	+	1.65375i
$216$		0			0
$217$		0			0
$218$	2.50000	+	4.33013i	0.169321	+	0.293273i
$219$		0			0
$220$	−4.00000			−0.269680
$221$		0			0
$222$		0			0
$223$	0.500000	−	0.866025i	0.0334825	−	0.0579934i	−0.848799	−	0.528716i	$-0.822674\pi$
$223$	0.500000	−	0.866025i	0.0334825	−	0.0579934i	0.882281	+	0.470723i	$0.156007\pi$
$224$	2.00000	+	3.46410i	0.133631	+	0.231455i
$225$	16.5000	−	28.5788i	1.10000	−	1.90526i
$226$	1.50000	−	2.59808i	0.0997785	−	0.172821i
$227$	20.0000			1.32745			0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000			1.32745			0.663723	+	0.747978i	$0.268975\pi$
$228$		0			0
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$	8.00000	−	13.8564i	0.527504	−	0.913664i
$231$		0			0
$232$	1.50000	+	2.59808i	0.0984798	+	0.170572i
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	−0.404674	−	0.914461i	$-0.632615\pi$
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	0.994283	−	0.106773i	$-0.0340517\pi$
$234$	10.5000	+	18.1865i	0.686406	+	1.18889i
$235$	20.0000			1.30466
$236$	6.00000			0.390567
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	0.980917	−	0.194429i	$-0.0622852\pi$
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	−0.658838	+	0.752285i	$0.728952\pi$
$242$	−0.500000	+	0.866025i	−0.0321412	+	0.0556702i
$243$		0			0
$244$	2.50000	−	4.33013i	0.160046	−	0.277208i
$245$	18.0000	−	31.1769i	1.14998	−	1.99182i
$246$		0			0
$247$	−24.5000	+	18.1865i	−1.55890	+	1.15718i
$248$		0			0
$249$		0			0
$250$	12.0000	−	20.7846i	0.758947	−	1.31453i
$251$	6.00000	+	10.3923i	0.378717	+	0.655956i	0.990876	−	0.134778i	$-0.0430322\pi$
$251$	6.00000	+	10.3923i	0.378717	+	0.655956i	−0.612159	+	0.790735i	$0.709699\pi$
$252$	−6.00000	+	10.3923i	−0.377964	+	0.654654i
$253$	−2.00000	−	3.46410i	−0.125739	−	0.217786i
$254$	10.0000			0.627456
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−7.00000	−	12.1244i	−0.436648	−	0.756297i	0.560781	−	0.827964i	$-0.310501\pi$
$257$	−7.00000	−	12.1244i	−0.436648	−	0.756297i	−0.997429	+	0.0716680i	$0.977168\pi$
$258$		0			0
$259$	−32.0000			−1.98838
$260$	14.0000	+	24.2487i	0.868243	+	1.50384i
$261$	−4.50000	+	7.79423i	−0.278543	+	0.482451i
$262$	2.50000	+	4.33013i	0.154451	+	0.267516i
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	−24.0000			−1.47431
$266$	−16.0000	−	6.92820i	−0.981023	−	0.424795i
$267$		0			0
$268$	4.00000	−	6.92820i	0.244339	−	0.423207i
$269$	8.00000	−	13.8564i	0.487769	−	0.844840i	−0.512132	−	0.858906i	$-0.671144\pi$
$269$	8.00000	−	13.8564i	0.487769	−	0.844840i	0.999901	+	0.0140665i	$0.00447764\pi$
$270$		0			0
$271$	15.0000	−	25.9808i	0.911185	−	1.57822i	0.0987925	−	0.995108i	$-0.468502\pi$
$271$	15.0000	−	25.9808i	0.911185	−	1.57822i	0.812393	−	0.583111i	$-0.198165\pi$
$272$		0			0
$273$		0			0
$274$	7.00000			0.422885
$275$	−5.50000	−	9.52628i	−0.331662	−	0.574456i
$276$		0			0
$277$	−7.00000			−0.420589			−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000			−0.420589			−0.210295	+	0.977638i	$0.567442\pi$
$278$	−20.0000			−1.19952
$279$		0			0
$280$	−8.00000	+	13.8564i	−0.478091	+	0.828079i
$281$	5.00000	+	8.66025i	0.298275	+	0.516627i	0.975741	−	0.218926i	$-0.0702554\pi$
$281$	5.00000	+	8.66025i	0.298275	+	0.516627i	−0.677466	+	0.735554i	$0.736922\pi$
$282$		0			0
$283$	16.0000	−	27.7128i	0.951101	−	1.64736i	0.208053	−	0.978117i	$-0.433287\pi$
$283$	16.0000	−	27.7128i	0.951101	−	1.64736i	0.743048	−	0.669238i	$-0.233379\pi$
$284$	−9.00000			−0.534052
$285$		0			0
$286$	7.00000			0.413919
$287$	−24.0000	+	41.5692i	−1.41668	+	2.45375i
$288$	1.50000	−	2.59808i	0.0883883	−	0.153093i
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$	−6.00000	+	10.3923i	−0.352332	+	0.610257i
$291$		0			0
$292$	10.0000			0.585206
$293$	−5.00000			−0.292103			−0.146052	−	0.989277i	$-0.546657\pi$
$293$	−5.00000			−0.292103			−0.146052	+	0.989277i	$0.546657\pi$
$294$		0			0
$295$	12.0000	+	20.7846i	0.698667	+	1.21013i
$296$	8.00000			0.464991
$297$		0			0
$298$	−3.00000	−	5.19615i	−0.173785	−	0.301005i
$299$	−14.0000	+	24.2487i	−0.809641	+	1.40234i
$300$		0			0
$301$	14.0000	−	24.2487i	0.806947	−	1.39767i
$302$	−3.00000	+	5.19615i	−0.172631	+	0.299005i
$303$		0			0
$304$	4.00000	+	1.73205i	0.229416	+	0.0993399i
$305$	20.0000			1.14520
$306$		0			0
$307$	−3.50000	+	6.06218i	−0.199756	+	0.345987i	−0.948449	−	0.316929i	$-0.897348\pi$
$307$	−3.50000	+	6.06218i	−0.199756	+	0.345987i	0.748694	+	0.662916i	$0.230681\pi$
$308$	2.00000	+	3.46410i	0.113961	+	0.197386i
$309$		0			0
$310$		0			0
$311$	−25.0000			−1.41762			−0.708810	−	0.705399i	$-0.750768\pi$
$311$	−25.0000			−1.41762			−0.708810	+	0.705399i	$0.750768\pi$
$312$		0			0
$313$	−4.50000	−	7.79423i	−0.254355	−	0.440556i	0.710365	−	0.703833i	$-0.248530\pi$
$313$	−4.50000	−	7.79423i	−0.254355	−	0.440556i	−0.964720	+	0.263278i	$0.915197\pi$
$314$	7.00000	+	12.1244i	0.395033	+	0.684217i
$315$	−48.0000			−2.70449
$316$	−6.00000			−0.337526
$317$	−6.00000	−	10.3923i	−0.336994	−	0.583690i	0.646872	−	0.762598i	$-0.276077\pi$
$317$	−6.00000	−	10.3923i	−0.336994	−	0.583690i	−0.983866	+	0.178908i	$0.942743\pi$
$318$		0			0
$319$	1.50000	+	2.59808i	0.0839839	+	0.145464i
$320$	2.00000	−	3.46410i	0.111803	−	0.193649i
$321$		0			0
$322$	−16.0000			−0.891645
$323$		0			0
$324$	9.00000			0.500000
$325$	−38.5000	+	66.6840i	−2.13560	+	3.69896i
$326$	−3.00000	+	5.19615i	−0.166155	+	0.287788i
$327$		0			0
$328$	6.00000	−	10.3923i	0.331295	−	0.573819i
$329$	−10.0000	−	17.3205i	−0.551318	−	0.954911i
$330$		0			0
$331$	6.00000			0.329790			0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000			0.329790			0.164895	+	0.986311i	$0.447272\pi$
$332$	1.50000	+	2.59808i	0.0823232	+	0.142588i
$333$	12.0000	+	20.7846i	0.657596	+	1.13899i
$334$	−16.0000			−0.875481
$335$	32.0000			1.74835
$336$		0			0
$337$	−11.0000	+	19.0526i	−0.599208	+	1.03786i	0.393730	+	0.919226i	$0.371184\pi$
$337$	−11.0000	+	19.0526i	−0.599208	+	1.03786i	−0.992938	+	0.118633i	$0.962149\pi$
$338$	−18.0000	−	31.1769i	−0.979071	−	1.69580i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	1.50000	+	12.9904i	0.0811107	+	0.702439i
$343$	−8.00000			−0.431959
$344$	−3.50000	+	6.06218i	−0.188707	+	0.326851i
$345$		0			0
$346$	−1.00000	−	1.73205i	−0.0537603	−	0.0931156i
$347$	1.50000	−	2.59808i	0.0805242	−	0.139472i	−0.822951	−	0.568112i	$-0.807674\pi$
$347$	1.50000	−	2.59808i	0.0805242	−	0.139472i	0.903475	+	0.428640i	$0.141007\pi$
$348$		0			0
$349$	13.0000			0.695874			0.347937	−	0.937518i	$-0.386882\pi$
$349$	13.0000			0.695874			0.347937	+	0.937518i	$0.386882\pi$
$350$	−44.0000			−2.35190
$351$		0			0
$352$	−0.500000	−	0.866025i	−0.0266501	−	0.0461593i
$353$	31.0000			1.64996			0.824982	−	0.565159i	$-0.191185\pi$
$353$	31.0000			1.64996			0.824982	+	0.565159i	$0.191185\pi$
$354$		0			0
$355$	−18.0000	−	31.1769i	−0.955341	−	1.65470i
$356$	0.500000	−	0.866025i	0.0264999	−	0.0458993i
$357$		0			0
$358$	8.00000	−	13.8564i	0.422813	−	0.732334i
$359$	1.00000	−	1.73205i	0.0527780	−	0.0914141i	−0.838429	−	0.545010i	$-0.816526\pi$
$359$	1.00000	−	1.73205i	0.0527780	−	0.0914141i	0.891207	+	0.453596i	$0.149859\pi$
$360$	12.0000			0.632456
$361$	−18.5000	+	4.33013i	−0.973684	+	0.227901i
$362$	−10.0000			−0.525588
$363$		0			0
$364$	14.0000	−	24.2487i	0.733799	−	1.27098i
$365$	20.0000	+	34.6410i	1.04685	+	1.81319i
$366$		0			0
$367$	−17.5000	−	30.3109i	−0.913493	−	1.58222i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	−17.5000	−	30.3109i	−0.913493	−	1.58222i	−0.104399	−	0.994535i	$-0.533292\pi$
$368$	4.00000			0.208514
$369$	36.0000			1.87409
$370$	16.0000	+	27.7128i	0.831800	+	1.44072i
$371$	12.0000	+	20.7846i	0.623009	+	1.07908i
$372$		0			0
$373$	−34.0000			−1.76045			−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000			−1.76045			−0.880227	+	0.474554i	$0.842610\pi$
$374$		0			0
$375$		0			0
$376$	2.50000	+	4.33013i	0.128928	+	0.223309i
$377$	10.5000	−	18.1865i	0.540778	−	0.936654i
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$	2.00000	+	17.3205i	0.102598	+	0.888523i
$381$		0			0
$382$	1.50000	−	2.59808i	0.0767467	−	0.132929i
$383$	6.00000	−	10.3923i	0.306586	−	0.531022i	−0.671027	−	0.741433i	$-0.734147\pi$
$383$	6.00000	−	10.3923i	0.306586	−	0.531022i	0.977613	+	0.210411i	$0.0674801\pi$
$384$		0			0
$385$	−8.00000	+	13.8564i	−0.407718	+	0.706188i
$386$	3.00000	+	5.19615i	0.152696	+	0.264477i
$387$	−21.0000			−1.06749
$388$	−5.00000			−0.253837
$389$	9.00000	+	15.5885i	0.456318	+	0.790366i	0.998763	−	0.0497253i	$-0.0158346\pi$
$389$	9.00000	+	15.5885i	0.456318	+	0.790366i	−0.542445	+	0.840091i	$0.682501\pi$
$390$		0			0
$391$		0			0
$392$	9.00000			0.454569
$393$		0			0
$394$	1.50000	−	2.59808i	0.0755689	−	0.130889i
$395$	−12.0000	−	20.7846i	−0.603786	−	1.04579i
$396$	1.50000	−	2.59808i	0.0753778	−	0.130558i
$397$	−4.00000	+	6.92820i	−0.200754	+	0.347717i	−0.948772	−	0.315963i	$-0.897673\pi$
$397$	−4.00000	+	6.92820i	−0.200754	+	0.347717i	0.748017	+	0.663679i	$0.231006\pi$
$398$	15.0000			0.751882
$399$		0			0
$400$	11.0000			0.550000
$401$	5.00000	−	8.66025i	0.249688	−	0.432472i	−0.713751	−	0.700399i	$-0.753005\pi$
$401$	5.00000	−	8.66025i	0.249688	−	0.432472i	0.963439	+	0.267927i	$0.0863386\pi$
$402$		0			0
$403$		0			0
$404$	7.50000	−	12.9904i	0.373139	−	0.646296i
$405$	18.0000	+	31.1769i	0.894427	+	1.54919i
$406$	12.0000			0.595550
$407$	8.00000			0.396545
$408$		0			0
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	0.962757	−	0.270367i	$-0.0871450\pi$
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	−0.715523	+	0.698589i	$0.753812\pi$
$410$	48.0000			2.37055
$411$		0			0
$412$	−5.50000	−	9.52628i	−0.270966	−	0.469326i
$413$	12.0000	−	20.7846i	0.590481	−	1.02274i
$414$	6.00000	+	10.3923i	0.294884	+	0.510754i
$415$	−6.00000	+	10.3923i	−0.294528	+	0.510138i
$416$	−3.50000	+	6.06218i	−0.171602	+	0.297223i
$417$		0			0
$418$	4.00000	+	1.73205i	0.195646	+	0.0847174i
$419$	−30.0000			−1.46560			−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000			−1.46560			−0.732798	+	0.680446i	$0.761786\pi$
$420$		0			0
$421$	5.00000	−	8.66025i	0.243685	−	0.422075i	−0.718076	−	0.695965i	$-0.754977\pi$
$421$	5.00000	−	8.66025i	0.243685	−	0.422075i	0.961761	+	0.273890i	$0.0883103\pi$
$422$	6.00000	+	10.3923i	0.292075	+	0.505889i
$423$	−7.50000	+	12.9904i	−0.364662	+	0.631614i
$424$	−3.00000	−	5.19615i	−0.145693	−	0.252347i
$425$		0			0
$426$		0			0
$427$	−10.0000	−	17.3205i	−0.483934	−	0.838198i
$428$	1.50000	+	2.59808i	0.0725052	+	0.125583i
$429$		0			0
$430$	−28.0000			−1.35028
$431$	−5.00000	−	8.66025i	−0.240842	−	0.417150i	0.720113	−	0.693857i	$-0.244090\pi$
$431$	−5.00000	−	8.66025i	−0.240842	−	0.417150i	−0.960954	+	0.276707i	$0.910757\pi$
$432$		0			0
$433$	2.50000	+	4.33013i	0.120142	+	0.208093i	0.919824	−	0.392332i	$-0.128332\pi$
$433$	2.50000	+	4.33013i	0.120142	+	0.208093i	−0.799681	+	0.600425i	$0.794998\pi$
$434$		0			0
$435$		0			0
$436$	−5.00000			−0.239457
$437$	−14.0000	+	10.3923i	−0.669711	+	0.497131i
$438$		0			0
$439$	18.0000	−	31.1769i	0.859093	−	1.48799i	−0.0137020	−	0.999906i	$-0.504362\pi$
$439$	18.0000	−	31.1769i	0.859093	−	1.48799i	0.872795	−	0.488087i	$-0.162305\pi$
$440$	2.00000	−	3.46410i	0.0953463	−	0.165145i
$441$	13.5000	+	23.3827i	0.642857	+	1.11346i
$442$		0			0
$443$	9.00000	+	15.5885i	0.427603	+	0.740630i	0.996660	−	0.0816684i	$-0.0260248\pi$
$443$	9.00000	+	15.5885i	0.427603	+	0.740630i	−0.569057	+	0.822298i	$0.692691\pi$
$444$		0			0
$445$	4.00000			0.189618
$446$	0.500000	+	0.866025i	0.0236757	+	0.0410075i
$447$		0			0
$448$	−4.00000			−0.188982
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	16.5000	+	28.5788i	0.777817	+	1.34722i
$451$	6.00000	−	10.3923i	0.282529	−	0.489355i
$452$	1.50000	+	2.59808i	0.0705541	+	0.122203i
$453$		0			0
$454$	−10.0000	+	17.3205i	−0.469323	+	0.812892i
$455$	112.000			5.25064
$456$		0			0
$457$	−2.00000			−0.0935561			−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000			−0.0935561			−0.0467780	+	0.998905i	$0.514895\pi$
$458$	−7.00000	+	12.1244i	−0.327089	+	0.566534i
$459$		0			0
$460$	8.00000	+	13.8564i	0.373002	+	0.646058i
$461$	−20.5000	+	35.5070i	−0.954780	+	1.65373i	−0.219910	+	0.975520i	$0.570576\pi$
$461$	−20.5000	+	35.5070i	−0.954780	+	1.65373i	−0.734870	+	0.678208i	$0.762757\pi$
$462$		0			0
$463$	11.0000			0.511213			0.255607	−	0.966781i	$-0.417725\pi$
$463$	11.0000			0.511213			0.255607	+	0.966781i	$0.417725\pi$
$464$	−3.00000			−0.139272
$465$		0			0
$466$	9.00000	+	15.5885i	0.416917	+	0.722121i
$467$	−34.0000			−1.57333			−0.786666	−	0.617379i	$-0.788195\pi$
$467$	−34.0000			−1.57333			−0.786666	+	0.617379i	$0.788195\pi$
$468$	−21.0000			−0.970725
$469$	−16.0000	−	27.7128i	−0.738811	−	1.27966i
$470$	−10.0000	+	17.3205i	−0.461266	+	0.798935i
$471$		0			0
$472$	−3.00000	+	5.19615i	−0.138086	+	0.239172i
$473$	−3.50000	+	6.06218i	−0.160930	+	0.278739i
$474$		0			0
$475$	−38.5000	+	28.5788i	−1.76650	+	1.31129i
$476$		0			0
$477$	9.00000	−	15.5885i	0.412082	−	0.713746i
$478$		0			0
$479$	−19.0000	−	32.9090i	−0.868132	−	1.50365i	−0.863903	−	0.503658i	$-0.831987\pi$
$479$	−19.0000	−	32.9090i	−0.868132	−	1.50365i	−0.00422900	−	0.999991i	$-0.501346\pi$
$480$		0			0
$481$	−28.0000	−	48.4974i	−1.27669	−	2.21129i
$482$	−10.0000			−0.455488
$483$		0			0
$484$	−0.500000	−	0.866025i	−0.0227273	−	0.0393648i
$485$	−10.0000	−	17.3205i	−0.454077	−	0.786484i
$486$		0			0
$487$	3.00000			0.135943			0.0679715	−	0.997687i	$-0.478347\pi$
$487$	3.00000			0.135943			0.0679715	+	0.997687i	$0.478347\pi$
$488$	2.50000	+	4.33013i	0.113170	+	0.196016i
$489$		0			0
$490$	18.0000	+	31.1769i	0.813157	+	1.40843i
$491$	10.0000	−	17.3205i	0.451294	−	0.781664i	−0.547173	−	0.837020i	$-0.684296\pi$
$491$	10.0000	−	17.3205i	0.451294	−	0.781664i	0.998467	+	0.0553560i	$0.0176294\pi$
$492$		0			0
$493$		0			0
$494$	−3.50000	−	30.3109i	−0.157472	−	1.36375i
$495$	12.0000			0.539360
$496$		0			0
$497$	−18.0000	+	31.1769i	−0.807410	+	1.39848i
$498$		0			0
$499$	−9.00000	+	15.5885i	−0.402895	+	0.697835i	−0.994074	−	0.108705i	$-0.965329\pi$
$499$	−9.00000	+	15.5885i	−0.402895	+	0.697835i	0.591179	+	0.806541i	$0.298663\pi$
$500$	12.0000	+	20.7846i	0.536656	+	0.929516i
$501$		0			0
$502$	−12.0000			−0.535586
$503$	7.00000	+	12.1244i	0.312115	+	0.540598i	0.978820	−	0.204723i	$-0.0656294\pi$
$503$	7.00000	+	12.1244i	0.312115	+	0.540598i	−0.666705	+	0.745321i	$0.732296\pi$
$504$	−6.00000	−	10.3923i	−0.267261	−	0.462910i
$505$	60.0000			2.66996
$506$	4.00000			0.177822
$507$		0			0
$508$	−5.00000	+	8.66025i	−0.221839	+	0.384237i
$509$	−12.0000	−	20.7846i	−0.531891	−	0.921262i	−0.999307	−	0.0372243i	$-0.988148\pi$
$509$	−12.0000	−	20.7846i	−0.531891	−	0.921262i	0.467416	−	0.884037i	$-0.345185\pi$
$510$		0			0
$511$	20.0000	−	34.6410i	0.884748	−	1.53243i
$512$	1.00000			0.0441942
$513$		0			0
$514$	14.0000			0.617514
$515$	22.0000	−	38.1051i	0.969436	−	1.67911i
$516$		0			0
$517$	2.50000	+	4.33013i	0.109950	+	0.190439i
$518$	16.0000	−	27.7128i	0.703000	−	1.21763i
$519$		0			0
$520$	−28.0000			−1.22788
$521$	−1.00000			−0.0438108			−0.0219054	−	0.999760i	$-0.506973\pi$
$521$	−1.00000			−0.0438108			−0.0219054	+	0.999760i	$0.506973\pi$
$522$	−4.50000	−	7.79423i	−0.196960	−	0.341144i
$523$	−6.50000	−	11.2583i	−0.284225	−	0.492292i	0.688196	−	0.725525i	$-0.258403\pi$
$523$	−6.50000	−	11.2583i	−0.284225	−	0.492292i	−0.972421	+	0.233233i	$0.925070\pi$
$524$	−5.00000			−0.218426
$525$		0			0
$526$	−6.00000	−	10.3923i	−0.261612	−	0.453126i
$527$		0			0
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$	12.0000	−	20.7846i	0.521247	−	0.902826i
$531$	−18.0000			−0.781133
$532$	14.0000	−	10.3923i	0.606977	−	0.450564i
$533$	−84.0000			−3.63844
$534$		0			0
$535$	−6.00000	+	10.3923i	−0.259403	+	0.449299i
$536$	4.00000	+	6.92820i	0.172774	+	0.299253i
$537$		0			0
$538$	8.00000	+	13.8564i	0.344904	+	0.597392i
$539$	9.00000			0.387657
$540$		0			0
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	0.886721	−	0.462304i	$-0.152977\pi$
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	−0.843728	+	0.536771i	$0.819644\pi$
$542$	15.0000	+	25.9808i	0.644305	+	1.11597i
$543$		0			0
$544$		0			0
$545$	−10.0000	−	17.3205i	−0.428353	−	0.741929i
$546$		0			0
$547$	20.5000	+	35.5070i	0.876517	+	1.51817i	0.855138	+	0.518400i	$0.173472\pi$
$547$	20.5000	+	35.5070i	0.876517	+	1.51817i	0.0213785	+	0.999771i	$0.493195\pi$
$548$	−3.50000	+	6.06218i	−0.149513	+	0.258963i
$549$	−7.50000	+	12.9904i	−0.320092	+	0.554416i
$550$	11.0000			0.469042
$551$	10.5000	−	7.79423i	0.447315	−	0.332045i
$552$		0			0
$553$	−12.0000	+	20.7846i	−0.510292	+	0.883852i
$554$	3.50000	−	6.06218i	0.148701	−	0.257557i
$555$		0			0
$556$	10.0000	−	17.3205i	0.424094	−	0.734553i
$557$	−12.5000	−	21.6506i	−0.529642	−	0.917367i	−0.999402	−	0.0345728i	$-0.988993\pi$
$557$	−12.5000	−	21.6506i	−0.529642	−	0.917367i	0.469760	−	0.882794i	$-0.344340\pi$
$558$		0			0
$559$	49.0000			2.07248
$560$	−8.00000	−	13.8564i	−0.338062	−	0.585540i
$561$		0			0
$562$	−10.0000			−0.421825
$563$	28.0000			1.18006			0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000			1.18006			0.590030	+	0.807382i	$0.299116\pi$
$564$		0			0
$565$	−6.00000	+	10.3923i	−0.252422	+	0.437208i
$566$	16.0000	+	27.7128i	0.672530	+	1.16486i
$567$	18.0000	−	31.1769i	0.755929	−	1.30931i
$568$	4.50000	−	7.79423i	0.188816	−	0.327039i
$569$	−10.0000			−0.419222			−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000			−0.419222			−0.209611	+	0.977785i	$0.567220\pi$
$570$		0			0
$571$	33.0000			1.38101			0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000			1.38101			0.690504	+	0.723329i	$0.257389\pi$
$572$	−3.50000	+	6.06218i	−0.146342	+	0.253472i
$573$		0			0
$574$	−24.0000	−	41.5692i	−1.00174	−	1.73507i
$575$	−22.0000	+	38.1051i	−0.917463	+	1.58909i
$576$	1.50000	+	2.59808i	0.0625000	+	0.108253i
$577$	10.0000			0.416305			0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000			0.416305			0.208153	+	0.978096i	$0.433255\pi$
$578$	−17.0000			−0.707107
$579$		0			0
$580$	−6.00000	−	10.3923i	−0.249136	−	0.431517i
$581$	12.0000			0.497844
$582$		0			0
$583$	−3.00000	−	5.19615i	−0.124247	−	0.215203i
$584$	−5.00000	+	8.66025i	−0.206901	+	0.358364i
$585$	−42.0000	−	72.7461i	−1.73649	−	3.00768i
$586$	2.50000	−	4.33013i	0.103274	−	0.178876i
$587$	−8.00000	+	13.8564i	−0.330195	+	0.571915i	−0.982550	−	0.185999i	$-0.940448\pi$
$587$	−8.00000	+	13.8564i	−0.330195	+	0.571915i	0.652355	+	0.757914i	$0.273781\pi$
$588$		0			0
$589$		0			0
$590$	−24.0000			−0.988064
$591$		0			0
$592$	−4.00000	+	6.92820i	−0.164399	+	0.284747i
$593$	18.0000	+	31.1769i	0.739171	+	1.28028i	0.952869	+	0.303383i	$0.0981160\pi$
$593$	18.0000	+	31.1769i	0.739171	+	1.28028i	−0.213697	+	0.976900i	$0.568551\pi$
$594$		0			0
$595$		0			0
$596$	6.00000			0.245770
$597$		0			0
$598$	−14.0000	−	24.2487i	−0.572503	−	0.991604i
$599$	−4.50000	−	7.79423i	−0.183865	−	0.318464i	0.759328	−	0.650708i	$-0.225528\pi$
$599$	−4.50000	−	7.79423i	−0.183865	−	0.318464i	−0.943193	+	0.332244i	$0.892194\pi$
$600$		0			0
$601$	32.0000			1.30531			0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000			1.30531			0.652654	+	0.757656i	$0.273656\pi$
$602$	14.0000	+	24.2487i	0.570597	+	0.988304i
$603$	−12.0000	+	20.7846i	−0.488678	+	0.846415i
$604$	−3.00000	−	5.19615i	−0.122068	−	0.211428i
$605$	2.00000	−	3.46410i	0.0813116	−	0.140836i
$606$		0			0
$607$	22.0000			0.892952			0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000			0.892952			0.446476	+	0.894795i	$0.352679\pi$
$608$	−3.50000	+	2.59808i	−0.141944	+	0.105366i
$609$		0			0
$610$	−10.0000	+	17.3205i	−0.404888	+	0.701287i
$611$	17.5000	−	30.3109i	0.707974	−	1.22625i
$612$		0			0
$613$	−9.50000	+	16.4545i	−0.383701	+	0.664590i	−0.991588	−	0.129433i	$-0.958684\pi$
$613$	−9.50000	+	16.4545i	−0.383701	+	0.664590i	0.607887	+	0.794024i	$0.292017\pi$
$614$	−3.50000	−	6.06218i	−0.141249	−	0.244650i
$615$		0			0
$616$	−4.00000			−0.161165
$617$	7.50000	+	12.9904i	0.301939	+	0.522973i	0.976575	−	0.215177i	$-0.0690329\pi$
$617$	7.50000	+	12.9904i	0.301939	+	0.522973i	−0.674636	+	0.738150i	$0.735700\pi$
$618$		0			0
$619$	26.0000			1.04503			0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000			1.04503			0.522514	+	0.852631i	$0.324994\pi$
$620$		0			0
$621$		0			0
$622$	12.5000	−	21.6506i	0.501204	−	0.868111i
$623$	−2.00000	−	3.46410i	−0.0801283	−	0.138786i
$624$		0			0
$625$	−20.5000	+	35.5070i	−0.820000	+	1.42028i
$626$	9.00000			0.359712
$627$		0			0
$628$	−14.0000			−0.558661
$629$		0			0
$630$	24.0000	−	41.5692i	0.956183	−	1.65616i
$631$	−6.50000	−	11.2583i	−0.258761	−	0.448187i	0.707149	−	0.707064i	$-0.249981\pi$
$631$	−6.50000	−	11.2583i	−0.258761	−	0.448187i	−0.965910	+	0.258877i	$0.916648\pi$
$632$	3.00000	−	5.19615i	0.119334	−	0.206692i
$633$		0			0
$634$	12.0000			0.476581
$635$	−40.0000			−1.58735
$636$		0			0
$637$	−31.5000	−	54.5596i	−1.24808	−	2.16173i
$638$	−3.00000			−0.118771
$639$	27.0000			1.06810
$640$	2.00000	+	3.46410i	0.0790569	+	0.136931i
$641$	−4.50000	+	7.79423i	−0.177739	+	0.307854i	−0.941106	−	0.338112i	$-0.890212\pi$
$641$	−4.50000	+	7.79423i	−0.177739	+	0.307854i	0.763367	+	0.645966i	$0.223545\pi$
$642$		0			0
$643$	−12.0000	+	20.7846i	−0.473234	+	0.819665i	−0.999531	−	0.0306359i	$-0.990247\pi$
$643$	−12.0000	+	20.7846i	−0.473234	+	0.819665i	0.526297	+	0.850301i	$0.323580\pi$
$644$	8.00000	−	13.8564i	0.315244	−	0.546019i
$645$		0			0
$646$		0			0
$647$	−17.0000			−0.668339			−0.334169	−	0.942513i	$-0.608456\pi$
$647$	−17.0000			−0.668339			−0.334169	+	0.942513i	$0.608456\pi$
$648$	−4.50000	+	7.79423i	−0.176777	+	0.306186i
$649$	−3.00000	+	5.19615i	−0.117760	+	0.203967i
$650$	−38.5000	−	66.6840i	−1.51009	−	2.61556i
$651$		0			0
$652$	−3.00000	−	5.19615i	−0.117489	−	0.203497i
$653$	42.0000			1.64359			0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000			1.64359			0.821794	+	0.569785i	$0.192974\pi$
$654$		0			0
$655$	−10.0000	−	17.3205i	−0.390732	−	0.676768i
$656$	6.00000	+	10.3923i	0.234261	+	0.405751i
$657$	−30.0000			−1.17041
$658$	20.0000			0.779681
$659$	−17.5000	−	30.3109i	−0.681703	−	1.18074i	−0.974461	−	0.224558i	$-0.927906\pi$
$659$	−17.5000	−	30.3109i	−0.681703	−	1.18074i	0.292758	−	0.956187i	$-0.405427\pi$
$660$		0			0
$661$	23.0000	+	39.8372i	0.894596	+	1.54949i	0.834303	+	0.551306i	$0.185870\pi$
$661$	23.0000	+	39.8372i	0.894596	+	1.54949i	0.0602929	+	0.998181i	$0.480797\pi$
$662$	−3.00000	+	5.19615i	−0.116598	+	0.201954i
$663$		0			0
$664$	−3.00000			−0.116423
$665$	64.0000	+	27.7128i	2.48181	+	1.07466i
$666$	−24.0000			−0.929981
$667$	6.00000	−	10.3923i	0.232321	−	0.402392i
$668$	8.00000	−	13.8564i	0.309529	−	0.536120i
$669$		0			0
$670$	−16.0000	+	27.7128i	−0.618134	+	1.07064i
$671$	2.50000	+	4.33013i	0.0965114	+	0.167163i
$672$		0			0
$673$	−16.0000			−0.616755			−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000			−0.616755			−0.308377	+	0.951264i	$0.599786\pi$
$674$	−11.0000	−	19.0526i	−0.423704	−	0.733877i
$675$		0			0
$676$	36.0000			1.38462
$677$	39.0000			1.49889			0.749446	−	0.662066i	$-0.230320\pi$
$677$	39.0000			1.49889			0.749446	+	0.662066i	$0.230320\pi$
$678$		0			0
$679$	−10.0000	+	17.3205i	−0.383765	+	0.664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−14.0000			−0.535695			−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000			−0.535695			−0.267848	+	0.963461i	$0.586312\pi$
$684$	−12.0000	−	5.19615i	−0.458831	−	0.198680i
$685$	−28.0000			−1.06983
$686$	4.00000	−	6.92820i	0.152721	−	0.264520i
$687$		0			0
$688$	−3.50000	−	6.06218i	−0.133436	−	0.231118i
$689$	−21.0000	+	36.3731i	−0.800036	+	1.38570i
$690$		0			0
$691$	−10.0000			−0.380418			−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000			−0.380418			−0.190209	+	0.981744i	$0.560917\pi$
$692$	2.00000			0.0760286
$693$	−6.00000	−	10.3923i	−0.227921	−	0.394771i
$694$	1.50000	+	2.59808i	0.0569392	+	0.0986216i
$695$	80.0000			3.03457
$696$		0			0
$697$		0			0
$698$	−6.50000	+	11.2583i	−0.246029	+	0.426134i
$699$		0			0
$700$	22.0000	−	38.1051i	0.831522	−	1.44024i
$701$	20.5000	−	35.5070i	0.774274	−	1.34108i	−0.160927	−	0.986966i	$-0.551448\pi$
$701$	20.5000	−	35.5070i	0.774274	−	1.34108i	0.935201	−	0.354116i	$-0.115218\pi$
$702$		0			0
$703$	−4.00000	−	34.6410i	−0.150863	−	1.30651i
$704$	1.00000			0.0376889
$705$		0			0
$706$	−15.5000	+	26.8468i	−0.583350	+	1.01039i
$707$	−30.0000	−	51.9615i	−1.12827	−	1.95421i
$708$		0			0
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	0.995228	−	0.0975728i	$-0.0311079\pi$
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	−0.582115	+	0.813107i	$0.697775\pi$
$710$	36.0000			1.35106
$711$	18.0000			0.675053
$712$	0.500000	+	0.866025i	0.0187383	+	0.0324557i
$713$		0			0
$714$		0			0
$715$	−28.0000			−1.04714
$716$	8.00000	+	13.8564i	0.298974	+	0.517838i
$717$		0			0
$718$	1.00000	+	1.73205i	0.0373197	+	0.0646396i
$719$	−21.5000	+	37.2391i	−0.801815	+	1.38878i	0.116606	+	0.993178i	$0.462799\pi$
$719$	−21.5000	+	37.2391i	−0.801815	+	1.38878i	−0.918421	+	0.395606i	$0.870535\pi$
$720$	−6.00000	+	10.3923i	−0.223607	+	0.387298i
$721$	−44.0000			−1.63865
$722$	5.50000	−	18.1865i	0.204689	−	0.676833i
$723$		0			0
$724$	5.00000	−	8.66025i	0.185824	−	0.321856i
$725$	16.5000	−	28.5788i	0.612795	−	1.06139i
$726$		0			0
$727$	9.50000	−	16.4545i	0.352335	−	0.610263i	−0.634323	−	0.773068i	$-0.718721\pi$
$727$	9.50000	−	16.4545i	0.352335	−	0.610263i	0.986658	+	0.162805i	$0.0520543\pi$
$728$	14.0000	+	24.2487i	0.518875	+	0.898717i
$729$	−27.0000			−1.00000
$730$	−40.0000			−1.48047
$731$		0			0
$732$		0			0
$733$	−6.00000			−0.221615			−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000			−0.221615			−0.110808	+	0.993842i	$0.535344\pi$
$734$	35.0000			1.29187
$735$		0			0
$736$	−2.00000	+	3.46410i	−0.0737210	+	0.127688i
$737$	4.00000	+	6.92820i	0.147342	+	0.255204i
$738$	−18.0000	+	31.1769i	−0.662589	+	1.14764i
$739$	5.50000	−	9.52628i	0.202321	−	0.350430i	−0.746955	−	0.664875i	$-0.768485\pi$
$739$	5.50000	−	9.52628i	0.202321	−	0.350430i	0.949276	+	0.314445i	$0.101818\pi$
$740$	−32.0000			−1.17634
$741$		0			0
$742$	−24.0000			−0.881068
$743$	−18.0000	+	31.1769i	−0.660356	+	1.14377i	0.320166	+	0.947361i	$0.396261\pi$
$743$	−18.0000	+	31.1769i	−0.660356	+	1.14377i	−0.980522	+	0.196409i	$0.937072\pi$
$744$		0			0
$745$	12.0000	+	20.7846i	0.439646	+	0.761489i
$746$	17.0000	−	29.4449i	0.622414	−	1.07805i
$747$	−4.50000	−	7.79423i	−0.164646	−	0.285176i
$748$		0			0
$749$	12.0000			0.438470
$750$		0			0
$751$	13.5000	+	23.3827i	0.492622	+	0.853246i	0.999964	−	0.00849853i	$-0.00270520\pi$
$751$	13.5000	+	23.3827i	0.492622	+	0.853246i	−0.507342	+	0.861745i	$0.669372\pi$
$752$	−5.00000			−0.182331
$753$		0			0
$754$	10.5000	+	18.1865i	0.382387	+	0.662314i
$755$	12.0000	−	20.7846i	0.436725	−	0.756429i
$756$		0			0
$757$	−15.0000	+	25.9808i	−0.545184	+	0.944287i	0.453411	+	0.891302i	$0.350207\pi$
$757$	−15.0000	+	25.9808i	−0.545184	+	0.944287i	−0.998595	+	0.0529853i	$0.983126\pi$
$758$	14.0000	−	24.2487i	0.508503	−	0.880753i
$759$		0			0
$760$	−16.0000	−	6.92820i	−0.580381	−	0.251312i
$761$	4.00000			0.145000			0.0724999	−	0.997368i	$-0.476902\pi$
$761$	4.00000			0.145000			0.0724999	+	0.997368i	$0.476902\pi$
$762$		0			0
$763$	−10.0000	+	17.3205i	−0.362024	+	0.627044i
$764$	1.50000	+	2.59808i	0.0542681	+	0.0939951i
$765$		0			0
$766$	6.00000	+	10.3923i	0.216789	+	0.375489i
$767$	42.0000			1.51653
$768$		0			0
$769$	9.00000	+	15.5885i	0.324548	+	0.562134i	0.981421	−	0.191867i	$-0.0614544\pi$
$769$	9.00000	+	15.5885i	0.324548	+	0.562134i	−0.656873	+	0.754002i	$0.728121\pi$
$770$	−8.00000	−	13.8564i	−0.288300	−	0.499350i
$771$		0			0
$772$	−6.00000			−0.215945
$773$	16.0000	+	27.7128i	0.575480	+	0.996761i	0.995989	+	0.0894724i	$0.0285181\pi$
$773$	16.0000	+	27.7128i	0.575480	+	0.996761i	−0.420509	+	0.907288i	$0.638149\pi$
$774$	10.5000	−	18.1865i	0.377415	−	0.653701i
$775$		0			0
$776$	2.50000	−	4.33013i	0.0897448	−	0.155443i
$777$		0			0
$778$	−18.0000			−0.645331
$779$	−48.0000	−	20.7846i	−1.71978	−	0.744686i
$780$		0			0
$781$	4.50000	−	7.79423i	0.161023	−	0.278899i
$782$		0			0

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:	\( N \)	\(=\)	\( 418 = 2 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	418.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.00000 - 3.46410i) q^{5} -4.00000 q^{7} +1.00000 q^{8} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.00000 - 3.46410i) q^{5} -4.00000 q^{7} +1.00000 q^{8} +(1.50000 + 2.59808i) q^{9} +(2.00000 + 3.46410i) q^{10} +1.00000 q^{11} +(-3.50000 - 6.06218i) q^{13} +(2.00000 - 3.46410i) q^{14} +(-0.500000 + 0.866025i) q^{16} -3.00000 q^{18} +(-0.500000 - 4.33013i) q^{19} -4.00000 q^{20} +(-0.500000 + 0.866025i) q^{22} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 - 9.52628i) q^{25} +7.00000 q^{26} +(2.00000 + 3.46410i) q^{28} +(1.50000 + 2.59808i) q^{29} +(-0.500000 - 0.866025i) q^{32} +(-8.00000 + 13.8564i) q^{35} +(1.50000 - 2.59808i) q^{36} +8.00000 q^{37} +(4.00000 + 1.73205i) q^{38} +(2.00000 - 3.46410i) q^{40} +(6.00000 - 10.3923i) q^{41} +(-3.50000 + 6.06218i) q^{43} +(-0.500000 - 0.866025i) q^{44} +12.0000 q^{45} +4.00000 q^{46} +(2.50000 + 4.33013i) q^{47} +9.00000 q^{49} +11.0000 q^{50} +(-3.50000 + 6.06218i) q^{52} +(-3.00000 - 5.19615i) q^{53} +(2.00000 - 3.46410i) q^{55} -4.00000 q^{56} -3.00000 q^{58} +(-3.00000 + 5.19615i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-6.00000 - 10.3923i) q^{63} +1.00000 q^{64} -28.0000 q^{65} +(4.00000 + 6.92820i) q^{67} +(-8.00000 - 13.8564i) q^{70} +(4.50000 - 7.79423i) q^{71} +(1.50000 + 2.59808i) q^{72} +(-5.00000 + 8.66025i) q^{73} +(-4.00000 + 6.92820i) q^{74} +(-3.50000 + 2.59808i) q^{76} -4.00000 q^{77} +(3.00000 - 5.19615i) q^{79} +(2.00000 + 3.46410i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(6.00000 + 10.3923i) q^{82} -3.00000 q^{83} +(-3.50000 - 6.06218i) q^{86} +1.00000 q^{88} +(0.500000 + 0.866025i) q^{89} +(-6.00000 + 10.3923i) q^{90} +(14.0000 + 24.2487i) q^{91} +(-2.00000 + 3.46410i) q^{92} -5.00000 q^{94} +(-16.0000 - 6.92820i) q^{95} +(2.50000 - 4.33013i) q^{97} +(-4.50000 + 7.79423i) q^{98} +(1.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 4 q^{5} - 8 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) 2 * q - q^2 - q^4 + 4 * q^5 - 8 * q^7 + 2 * q^8 + 3 * q^9	\( 2 q - q^{2} - q^{4} + 4 q^{5} - 8 q^{7} + 2 q^{8} + 3 q^{9} + 4 q^{10} + 2 q^{11} - 7 q^{13} + 4 q^{14} - q^{16} - 6 q^{18} - q^{19} - 8 q^{20} - q^{22} - 4 q^{23} - 11 q^{25} + 14 q^{26} + 4 q^{28} + 3 q^{29} - q^{32} - 16 q^{35} + 3 q^{36} + 16 q^{37} + 8 q^{38} + 4 q^{40} + 12 q^{41} - 7 q^{43} - q^{44} + 24 q^{45} + 8 q^{46} + 5 q^{47} + 18 q^{49} + 22 q^{50} - 7 q^{52} - 6 q^{53} + 4 q^{55} - 8 q^{56} - 6 q^{58} - 6 q^{59} + 5 q^{61} - 12 q^{63} + 2 q^{64} - 56 q^{65} + 8 q^{67} - 16 q^{70} + 9 q^{71} + 3 q^{72} - 10 q^{73} - 8 q^{74} - 7 q^{76} - 8 q^{77} + 6 q^{79} + 4 q^{80} - 9 q^{81} + 12 q^{82} - 6 q^{83} - 7 q^{86} + 2 q^{88} + q^{89} - 12 q^{90} + 28 q^{91} - 4 q^{92} - 10 q^{94} - 32 q^{95} + 5 q^{97} - 9 q^{98} + 3 q^{99}+O(q^{100}) \) 2 * q - q^2 - q^4 + 4 * q^5 - 8 * q^7 + 2 * q^8 + 3 * q^9 + 4 * q^10 + 2 * q^11 - 7 * q^13 + 4 * q^14 - q^16 - 6 * q^18 - q^19 - 8 * q^20 - q^22 - 4 * q^23 - 11 * q^25 + 14 * q^26 + 4 * q^28 + 3 * q^29 - q^32 - 16 * q^35 + 3 * q^36 + 16 * q^37 + 8 * q^38 + 4 * q^40 + 12 * q^41 - 7 * q^43 - q^44 + 24 * q^45 + 8 * q^46 + 5 * q^47 + 18 * q^49 + 22 * q^50 - 7 * q^52 - 6 * q^53 + 4 * q^55 - 8 * q^56 - 6 * q^58 - 6 * q^59 + 5 * q^61 - 12 * q^63 + 2 * q^64 - 56 * q^65 + 8 * q^67 - 16 * q^70 + 9 * q^71 + 3 * q^72 - 10 * q^73 - 8 * q^74 - 7 * q^76 - 8 * q^77 + 6 * q^79 + 4 * q^80 - 9 * q^81 + 12 * q^82 - 6 * q^83 - 7 * q^86 + 2 * q^88 + q^89 - 12 * q^90 + 28 * q^91 - 4 * q^92 - 10 * q^94 - 32 * q^95 + 5 * q^97 - 9 * q^98 + 3 * q^99

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