LMFDB - Embedded newform 2268.2.bm.a.1025.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(593,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.593"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.bm (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-6,0,1,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.1100711784$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1025.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2268.1025
Dual form			2268.2.bm.a.593.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{5} +(0.500000 - 2.59808i) q^{7} +5.19615i q^{11} +(1.50000 - 2.59808i) q^{17} +(1.50000 - 0.866025i) q^{19} +5.19615i q^{23} +4.00000 q^{25} +(1.50000 - 0.866025i) q^{31} +(-1.50000 + 7.79423i) q^{35} +(-3.50000 - 6.06218i) q^{37} +(3.00000 - 5.19615i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(1.50000 - 2.59808i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(4.50000 + 2.59808i) q^{53} -15.5885i q^{55} +(-1.50000 - 2.59808i) q^{59} +(10.5000 + 6.06218i) q^{61} +(-2.50000 - 4.33013i) q^{67} -10.3923i q^{71} +(-10.5000 - 6.06218i) q^{73} +(13.5000 + 2.59808i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(-4.50000 - 7.79423i) q^{89} +(-4.50000 + 2.59808i) q^{95} +(6.00000 - 3.46410i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + q^{7} + 3 q^{17} + 3 q^{19} + 8 q^{25} + 3 q^{31} - 3 q^{35} - 7 q^{37} + 6 q^{41} - 4 q^{43} + 3 q^{47} - 13 q^{49} + 9 q^{53} - 3 q^{59} + 21 q^{61} - 5 q^{67} - 21 q^{73} + 27 q^{77}+ \cdots + 12 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + q^7 + 3 * q^17 + 3 * q^19 + 8 * q^25 + 3 * q^31 - 3 * q^35 - 7 * q^37 + 6 * q^41 - 4 * q^43 + 3 * q^47 - 13 * q^49 + 9 * q^53 - 3 * q^59 + 21 * q^61 - 5 * q^67 - 21 * q^73 + 27 * q^77 + q^79 - 12 * q^83 - 9 * q^85 - 9 * q^89 - 9 * q^95 + 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.00000			−1.34164			−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000			−1.34164			−0.670820	+	0.741620i	$0.734058\pi$
$6$		0			0
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.19615i			1.56670i	0.621582	+	0.783349i	$0.286490\pi$
$11$			5.19615i			1.56670i	−0.621582	+	0.783349i	$0.713510\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	−0.624780	−	0.780801i	$-0.714811\pi$
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	0.988583	+	0.150675i	$0.0481447\pi$
$18$		0			0
$19$	1.50000	−	0.866025i	0.344124	−	0.198680i	−0.317970	−	0.948101i	$-0.603001\pi$
$19$	1.50000	−	0.866025i	0.344124	−	0.198680i	0.662094	+	0.749421i	$0.269668\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.19615i			1.08347i	0.840548	+	0.541736i	$0.182233\pi$
$23$			5.19615i			1.08347i	−0.840548	+	0.541736i	$0.817767\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	1.50000	−	0.866025i	0.269408	−	0.155543i	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	1.50000	−	0.866025i	0.269408	−	0.155543i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.50000	+	7.79423i	−0.253546	+	1.31747i
$36$		0			0
$37$	−3.50000	−	6.06218i	−0.575396	−	0.996616i	−0.995998	−	0.0893706i	$-0.971514\pi$
$37$	−3.50000	−	6.06218i	−0.575396	−	0.996616i	0.420602	−	0.907245i	$-0.361819\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	−0.530831	−	0.847477i	$-0.678120\pi$
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	0.999353	+	0.0359748i	$0.0114536\pi$
$42$		0			0
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	0.672264	−	0.740312i	$-0.265322\pi$
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	−0.977261	+	0.212041i	$0.931989\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.50000	−	2.59808i	0.218797	−	0.378968i	−0.735643	−	0.677369i	$-0.763120\pi$
$47$	1.50000	−	2.59808i	0.218797	−	0.378968i	0.954441	+	0.298401i	$0.0964533\pi$
$48$		0			0
$49$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.50000	+	2.59808i	0.618123	+	0.356873i	0.776138	−	0.630563i	$-0.217176\pi$
$53$	4.50000	+	2.59808i	0.618123	+	0.356873i	−0.158015	+	0.987437i	$0.550509\pi$
$54$		0			0
$55$		−	15.5885i		−	2.10195i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	0.751710	−	0.659494i	$-0.229229\pi$
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	−0.946993	+	0.321253i	$0.895896\pi$
$60$		0			0
$61$	10.5000	+	6.06218i	1.34439	+	0.776182i	0.987448	−	0.157945i	$-0.0504869\pi$
$61$	10.5000	+	6.06218i	1.34439	+	0.776182i	0.356939	+	0.934128i	$0.383820\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$	−10.5000	−	6.06218i	−1.22893	−	0.709524i	−0.262126	−	0.965034i	$-0.584423\pi$
$73$	−10.5000	−	6.06218i	−1.22893	−	0.709524i	−0.966807	+	0.255510i	$0.917757\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13.5000	+	2.59808i	1.53847	+	0.296078i
$78$		0			0
$79$	0.500000	−	0.866025i	0.0562544	−	0.0974355i	−0.836527	−	0.547926i	$-0.815418\pi$
$79$	0.500000	−	0.866025i	0.0562544	−	0.0974355i	0.892781	+	0.450490i	$0.148751\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	−0.980982	−	0.194099i	$-0.937822\pi$
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	0.322396	−	0.946605i	$-0.395512\pi$
$84$		0			0
$85$	−4.50000	+	7.79423i	−0.488094	+	0.845403i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.50000	−	7.79423i	−0.476999	−	0.826187i	0.522654	−	0.852545i	$-0.324942\pi$
$89$	−4.50000	−	7.79423i	−0.476999	−	0.826187i	−0.999653	+	0.0263586i	$0.991609\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.50000	+	2.59808i	−0.461690	+	0.266557i
$96$		0			0
$97$	6.00000	−	3.46410i	0.609208	−	0.351726i	−0.163448	−	0.986552i	$-0.552261\pi$
$97$	6.00000	−	3.46410i	0.609208	−	0.351726i	0.772655	+	0.634826i	$0.218928\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.00000			0.895533			0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000			0.895533			0.447767	+	0.894150i	$0.352219\pi$
$102$		0			0
$103$		−	5.19615i		−	0.511992i	−0.966678	−	0.255996i	$-0.917597\pi$
$103$		−	5.19615i		−	0.511992i	0.966678	−	0.255996i	$-0.0824034\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.5000	+	7.79423i	−1.30509	+	0.753497i	−0.981273	−	0.192622i	$-0.938301\pi$
$107$	−13.5000	+	7.79423i	−1.30509	+	0.753497i	−0.323821	+	0.946118i	$0.604968\pi$
$108$		0			0
$109$	8.50000	−	14.7224i	0.814152	−	1.41015i	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	8.50000	−	14.7224i	0.814152	−	1.41015i	0.909935	−	0.414751i	$-0.136131\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.0000	−	10.3923i	−1.69330	−	0.977626i	−0.951825	−	0.306643i	$-0.900794\pi$
$113$	−18.0000	−	10.3923i	−1.69330	−	0.977626i	−0.741473	−	0.670983i	$-0.765872\pi$
$114$		0			0
$115$		−	15.5885i		−	1.45363i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.00000	−	5.19615i	−0.550019	−	0.476331i
$120$		0			0
$121$	−16.0000			−1.45455
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.00000			0.268328
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.00000			0.786334			0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000			0.786334			0.393167	+	0.919467i	$0.371379\pi$
$132$		0			0
$133$	−1.50000	−	4.33013i	−0.130066	−	0.375470i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	5.19615i		−	0.443937i	−0.975054	−	0.221969i	$-0.928752\pi$
$137$		−	5.19615i		−	0.443937i	0.975054	−	0.221969i	$-0.0712483\pi$
$138$		0			0
$139$	9.00000	+	5.19615i	0.763370	+	0.440732i	0.830504	−	0.557012i	$-0.188052\pi$
$139$	9.00000	+	5.19615i	0.763370	+	0.440732i	−0.0671344	+	0.997744i	$0.521386\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			5.19615i			0.425685i	0.977086	+	0.212843i	$0.0682722\pi$
$149$			5.19615i			0.425685i	−0.977086	+	0.212843i	$0.931728\pi$
$150$		0			0
$151$	13.0000			1.05792			0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000			1.05792			0.528962	+	0.848645i	$0.322581\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.50000	+	2.59808i	−0.361449	+	0.208683i
$156$		0			0
$157$	4.50000	−	2.59808i	0.359139	−	0.207349i	−0.309564	−	0.950879i	$-0.600183\pi$
$157$	4.50000	−	2.59808i	0.359139	−	0.207349i	0.668703	+	0.743530i	$0.266850\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	13.5000	+	2.59808i	1.06395	+	0.204757i
$162$		0			0
$163$	0.500000	+	0.866025i	0.0391630	+	0.0678323i	0.884943	−	0.465700i	$-0.154198\pi$
$163$	0.500000	+	0.866025i	0.0391630	+	0.0678323i	−0.845780	+	0.533533i	$0.820864\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	−6.50000	−	11.2583i	−0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.50000	+	7.79423i	−0.342129	+	0.592584i	−0.984828	−	0.173534i	$-0.944481\pi$
$173$	−4.50000	+	7.79423i	−0.342129	+	0.592584i	0.642699	+	0.766119i	$0.277815\pi$
$174$		0			0
$175$	2.00000	−	10.3923i	0.151186	−	0.785584i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.50000	−	2.59808i	−0.336346	−	0.194189i	0.322309	−	0.946634i	$-0.395541\pi$
$179$	−4.50000	−	2.59808i	−0.336346	−	0.194189i	−0.658655	+	0.752445i	$0.728874\pi$
$180$		0			0
$181$		−	6.92820i		−	0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.5000	+	18.1865i	0.771975	+	1.33710i
$186$		0			0
$187$	13.5000	+	7.79423i	0.987218	+	0.569970i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.5000	+	7.79423i	0.976826	+	0.563971i	0.901310	−	0.433174i	$-0.142606\pi$
$191$	13.5000	+	7.79423i	0.976826	+	0.563971i	0.0755154	+	0.997145i	$0.475940\pi$
$192$		0			0
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	0.941865	−	0.335993i	$-0.109072\pi$
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	−0.761911	+	0.647682i	$0.775738\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−1.50000	−	0.866025i	−0.106332	−	0.0613909i	0.445891	−	0.895087i	$-0.352887\pi$
$199$	−1.50000	−	0.866025i	−0.106332	−	0.0613909i	−0.552223	+	0.833696i	$0.686220\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−9.00000	+	15.5885i	−0.628587	+	1.08875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.50000	+	7.79423i	0.311272	+	0.539138i
$210$		0			0
$211$	10.0000	−	17.3205i	0.688428	−	1.19239i	−0.283918	−	0.958849i	$-0.591634\pi$
$211$	10.0000	−	17.3205i	0.688428	−	1.19239i	0.972346	−	0.233544i	$-0.0750324\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.00000	+	10.3923i	0.409197	+	0.708749i
$216$		0			0
$217$	−1.50000	−	4.33013i	−0.101827	−	0.293948i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	9.00000	−	5.19615i	0.602685	−	0.347960i	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	9.00000	−	5.19615i	0.602685	−	0.347960i	0.770097	+	0.637927i	$0.220208\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.00000			−0.597351			−0.298675	−	0.954355i	$-0.596545\pi$
$227$	−9.00000			−0.597351			−0.298675	+	0.954355i	$0.596545\pi$
$228$		0			0
$229$			22.5167i			1.48794i	0.668211	+	0.743971i	$0.267060\pi$
$229$			22.5167i			1.48794i	−0.668211	+	0.743971i	$0.732940\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.50000	−	2.59808i	0.294805	−	0.170206i	−0.345302	−	0.938492i	$-0.612223\pi$
$233$	4.50000	−	2.59808i	0.294805	−	0.170206i	0.640107	+	0.768286i	$0.278890\pi$
$234$		0			0
$235$	−4.50000	+	7.79423i	−0.293548	+	0.508439i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.00000	−	5.19615i	−0.582162	−	0.336111i	0.179830	−	0.983698i	$-0.442445\pi$
$239$	−9.00000	−	5.19615i	−0.582162	−	0.336111i	−0.761992	+	0.647586i	$0.775778\pi$
$240$		0			0
$241$		−	1.73205i		−	0.111571i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$		−	1.73205i		−	0.111571i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	19.5000	+	7.79423i	1.24581	+	0.497955i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$	−27.0000			−1.69748
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−27.0000			−1.68421			−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000			−1.68421			−0.842107	+	0.539311i	$0.818685\pi$
$258$		0			0
$259$	−17.5000	+	6.06218i	−1.08740	+	0.376685i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5.19615i			0.320408i	0.987084	+	0.160204i	$0.0512153\pi$
$263$			5.19615i			0.320408i	−0.987084	+	0.160204i	$0.948785\pi$
$264$		0			0
$265$	−13.5000	−	7.79423i	−0.829298	−	0.478796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.5000	+	18.1865i	−0.640196	+	1.10885i	0.345192	+	0.938532i	$0.387814\pi$
$269$	−10.5000	+	18.1865i	−0.640196	+	1.10885i	−0.985389	+	0.170321i	$0.945520\pi$
$270$		0			0
$271$	19.5000	−	11.2583i	1.18454	−	0.683895i	0.227480	−	0.973783i	$-0.426951\pi$
$271$	19.5000	−	11.2583i	1.18454	−	0.683895i	0.957061	+	0.289888i	$0.0936180\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			20.7846i			1.25336i
$276$		0			0
$277$	−13.0000			−0.781094			−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000			−0.781094			−0.390547	+	0.920583i	$0.627714\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.0000	+	10.3923i	−1.07379	+	0.619953i	−0.929214	−	0.369541i	$-0.879515\pi$
$281$	−18.0000	+	10.3923i	−1.07379	+	0.619953i	−0.144575	+	0.989494i	$0.546182\pi$
$282$		0			0
$283$	7.50000	−	4.33013i	0.445829	−	0.257399i	−0.260238	−	0.965544i	$-0.583801\pi$
$283$	7.50000	−	4.33013i	0.445829	−	0.257399i	0.706067	+	0.708145i	$0.250468\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.0000	−	10.3923i	−0.708338	−	0.613438i
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.00000	−	5.19615i	0.175262	−	0.303562i	−0.764990	−	0.644042i	$-0.777256\pi$
$293$	3.00000	−	5.19615i	0.175262	−	0.303562i	0.940252	+	0.340480i	$0.110589\pi$
$294$		0			0
$295$	4.50000	+	7.79423i	0.262000	+	0.453798i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−10.0000	+	3.46410i	−0.576390	+	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−31.5000	−	18.1865i	−1.80368	−	1.04136i
$306$		0			0
$307$			17.3205i			0.988534i	0.869310	+	0.494267i	$0.164563\pi$
$307$			17.3205i			0.988534i	−0.869310	+	0.494267i	$0.835437\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.50000	−	12.9904i	−0.425286	−	0.736617i	0.571161	−	0.820838i	$-0.306493\pi$
$311$	−7.50000	−	12.9904i	−0.425286	−	0.736617i	−0.996447	+	0.0842210i	$0.973160\pi$
$312$		0			0
$313$	10.5000	+	6.06218i	0.593495	+	0.342655i	0.766478	−	0.642270i	$-0.222007\pi$
$313$	10.5000	+	6.06218i	0.593495	+	0.342655i	−0.172983	+	0.984925i	$0.555341\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−22.5000	−	12.9904i	−1.26373	−	0.729612i	−0.289933	−	0.957047i	$-0.593633\pi$
$317$	−22.5000	−	12.9904i	−1.26373	−	0.729612i	−0.973793	+	0.227435i	$0.926966\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5.19615i		−	0.289122i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.00000	−	5.19615i	−0.330791	−	0.286473i
$330$		0			0
$331$	−15.5000	+	26.8468i	−0.851957	+	1.47563i	0.0274825	+	0.999622i	$0.491251\pi$
$331$	−15.5000	+	26.8468i	−0.851957	+	1.47563i	−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.50000	+	12.9904i	0.409769	+	0.709740i
$336$		0			0
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	−0.837503	−	0.546433i	$-0.815985\pi$
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	0.891976	+	0.452082i	$0.149319\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.50000	+	7.79423i	0.243689	+	0.422081i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.5000	+	7.79423i	−0.724718	+	0.418416i	−0.816487	−	0.577364i	$-0.804081\pi$
$347$	−13.5000	+	7.79423i	−0.724718	+	0.418416i	0.0917687	+	0.995780i	$0.470748\pi$
$348$		0			0
$349$	12.0000	−	6.92820i	0.642345	−	0.370858i	−0.143172	−	0.989698i	$-0.545730\pi$
$349$	12.0000	−	6.92820i	0.642345	−	0.370858i	0.785517	+	0.618840i	$0.212397\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.00000			−0.159674			−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000			−0.159674			−0.0798369	+	0.996808i	$0.525440\pi$
$354$		0			0
$355$			31.1769i			1.65470i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	22.5000	−	12.9904i	1.18750	−	0.685606i	0.229766	−	0.973246i	$-0.426204\pi$
$359$	22.5000	−	12.9904i	1.18750	−	0.685606i	0.957739	+	0.287640i	$0.0928706\pi$
$360$		0			0
$361$	−8.00000	+	13.8564i	−0.421053	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	31.5000	+	18.1865i	1.64879	+	0.951927i
$366$		0			0
$367$		−	32.9090i		−	1.71783i	−0.512115	−	0.858917i	$-0.671138\pi$
$367$		−	32.9090i		−	1.71783i	0.512115	−	0.858917i	$-0.328862\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.00000	−	10.3923i	0.467257	−	0.539542i
$372$		0			0
$373$	−25.0000			−1.29445			−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000			−1.29445			−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	21.0000			1.07305			0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000			1.07305			0.536525	+	0.843884i	$0.319737\pi$
$384$		0			0
$385$	−40.5000	−	7.79423i	−2.06407	−	0.397231i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	25.9808i		−	1.31728i	−0.752460	−	0.658638i	$-0.771133\pi$
$389$		−	25.9808i		−	1.31728i	0.752460	−	0.658638i	$-0.228867\pi$
$390$		0			0
$391$	13.5000	+	7.79423i	0.682724	+	0.394171i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.50000	+	2.59808i	−0.0754732	+	0.130723i
$396$		0			0
$397$	7.50000	−	4.33013i	0.376414	−	0.217323i	−0.299843	−	0.953989i	$-0.596934\pi$
$397$	7.50000	−	4.33013i	0.376414	−	0.217323i	0.676257	+	0.736666i	$0.263601\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	36.3731i		−	1.81638i	−0.418554	−	0.908192i	$-0.637463\pi$
$401$		−	36.3731i		−	1.81638i	0.418554	−	0.908192i	$-0.362537\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	31.5000	−	18.1865i	1.56140	−	0.901473i
$408$		0			0
$409$	4.50000	−	2.59808i	0.222511	−	0.128467i	−0.384602	−	0.923083i	$-0.625661\pi$
$409$	4.50000	−	2.59808i	0.222511	−	0.128467i	0.607112	+	0.794616i	$0.292328\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.50000	+	2.59808i	−0.369051	+	0.127843i
$414$		0			0
$415$	18.0000	+	31.1769i	0.883585	+	1.53041i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	−0.681426	−	0.731887i	$-0.738640\pi$
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	0.974546	+	0.224189i	$0.0719734\pi$
$420$		0			0
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.00000	−	10.3923i	0.291043	−	0.504101i
$426$		0			0
$427$	21.0000	−	24.2487i	1.01626	−	1.17348i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13.5000	+	7.79423i	0.650272	+	0.375435i	0.788560	−	0.614957i	$-0.210827\pi$
$431$	13.5000	+	7.79423i	0.650272	+	0.375435i	−0.138288	+	0.990392i	$0.544160\pi$
$432$		0			0
$433$			34.6410i			1.66474i	0.554220	+	0.832370i	$0.313017\pi$
$433$			34.6410i			1.66474i	−0.554220	+	0.832370i	$0.686983\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.50000	+	7.79423i	0.215264	+	0.372849i
$438$		0			0
$439$	−13.5000	−	7.79423i	−0.644320	−	0.371998i	0.141957	−	0.989873i	$-0.454661\pi$
$439$	−13.5000	−	7.79423i	−0.644320	−	0.371998i	−0.786277	+	0.617875i	$0.787994\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.5000	+	7.79423i	0.641404	+	0.370315i	0.785155	−	0.619299i	$-0.212583\pi$
$443$	13.5000	+	7.79423i	0.641404	+	0.370315i	−0.143751	+	0.989614i	$0.545916\pi$
$444$		0			0
$445$	13.5000	+	23.3827i	0.639961	+	1.10845i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			20.7846i			0.980886i	0.871473	+	0.490443i	$0.163165\pi$
$449$			20.7846i			0.980886i	−0.871473	+	0.490443i	$0.836835\pi$
$450$		0			0
$451$	27.0000	+	15.5885i	1.27138	+	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	0.233890	+	0.972263i	$0.424854\pi$
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	−0.958950	+	0.283577i	$0.908479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	−0.968945	−	0.247276i	$-0.920465\pi$
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	0.270326	−	0.962769i	$-0.412869\pi$
$462$		0			0
$463$	8.00000	−	13.8564i	0.371792	−	0.643962i	−0.618050	−	0.786139i	$-0.712077\pi$
$463$	8.00000	−	13.8564i	0.371792	−	0.643962i	0.989841	+	0.142177i	$0.0454103\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.50000	−	7.79423i	−0.208235	−	0.360674i	0.742923	−	0.669376i	$-0.233439\pi$
$467$	−4.50000	−	7.79423i	−0.208235	−	0.360674i	−0.951159	+	0.308702i	$0.900105\pi$
$468$		0			0
$469$	−12.5000	+	4.33013i	−0.577196	+	0.199947i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18.0000	−	10.3923i	0.827641	−	0.477839i
$474$		0			0
$475$	6.00000	−	3.46410i	0.275299	−	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−33.0000			−1.50781			−0.753904	−	0.656984i	$-0.771832\pi$
$479$	−33.0000			−1.50781			−0.753904	+	0.656984i	$0.771832\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18.0000	+	10.3923i	−0.817338	+	0.471890i
$486$		0			0
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	−0.877132	−	0.480250i	$-0.840546\pi$
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	0.854475	+	0.519493i	$0.173879\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.00000	+	5.19615i	0.406164	+	0.234499i	0.689140	−	0.724628i	$-0.257988\pi$
$491$	9.00000	+	5.19615i	0.406164	+	0.234499i	−0.282976	+	0.959127i	$0.591322\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−27.0000	−	5.19615i	−1.21112	−	0.233079i
$498$		0			0
$499$	7.00000			0.313363			0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000			0.313363			0.156682	+	0.987649i	$0.449920\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	−27.0000			−1.20148
$506$		0			0
$507$		0			0
$508$		0			0
$509$	21.0000			0.930809			0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000			0.930809			0.465404	+	0.885098i	$0.345909\pi$
$510$		0			0
$511$	−21.0000	+	24.2487i	−0.928985	+	1.07270i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			15.5885i			0.686909i
$516$		0			0
$517$	13.5000	+	7.79423i	0.593729	+	0.342790i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7.50000	−	12.9904i	0.328581	−	0.569119i	−0.653650	−	0.756797i	$-0.726763\pi$
$521$	7.50000	−	12.9904i	0.328581	−	0.569119i	0.982231	+	0.187678i	$0.0600963\pi$
$522$		0			0
$523$	−22.5000	+	12.9904i	−0.983856	+	0.568030i	−0.903432	−	0.428731i	$-0.858961\pi$
$523$	−22.5000	+	12.9904i	−0.983856	+	0.568030i	−0.0804241	+	0.996761i	$0.525627\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	5.19615i		−	0.226348i
$528$		0			0
$529$	−4.00000			−0.173913
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	40.5000	−	23.3827i	1.75097	−	1.01092i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13.5000	−	33.7750i	0.581486	−	1.45479i
$540$		0			0
$541$	18.5000	+	32.0429i	0.795377	+	1.37763i	0.922599	+	0.385759i	$0.126061\pi$
$541$	18.5000	+	32.0429i	0.795377	+	1.37763i	−0.127222	+	0.991874i	$0.540606\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−25.5000	+	44.1673i	−1.09230	+	1.89192i
$546$		0			0
$547$	−16.0000	−	27.7128i	−0.684111	−	1.18491i	−0.973715	−	0.227768i	$-0.926857\pi$
$547$	−16.0000	−	27.7128i	−0.684111	−	1.18491i	0.289605	−	0.957146i	$-0.406476\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.00000	−	1.73205i	−0.0850487	−	0.0736543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22.5000	+	12.9904i	0.953356	+	0.550420i	0.894122	−	0.447824i	$-0.147801\pi$
$557$	22.5000	+	12.9904i	0.953356	+	0.550420i	0.0592339	+	0.998244i	$0.481134\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−13.5000	−	23.3827i	−0.568957	−	0.985463i	−0.996669	−	0.0815478i	$-0.974014\pi$
$563$	−13.5000	−	23.3827i	−0.568957	−	0.985463i	0.427712	−	0.903915i	$-0.359320\pi$
$564$		0			0
$565$	54.0000	+	31.1769i	2.27180	+	1.31162i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.5000	+	7.79423i	0.565949	+	0.326751i	0.755530	−	0.655114i	$-0.227379\pi$
$569$	13.5000	+	7.79423i	0.565949	+	0.326751i	−0.189580	+	0.981865i	$0.560713\pi$
$570$		0			0
$571$	3.50000	+	6.06218i	0.146470	+	0.253694i	0.929921	−	0.367760i	$-0.119875\pi$
$571$	3.50000	+	6.06218i	0.146470	+	0.253694i	−0.783450	+	0.621455i	$0.786542\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			20.7846i			0.866778i
$576$		0			0
$577$	−22.5000	−	12.9904i	−0.936687	−	0.540797i	−0.0477669	−	0.998859i	$-0.515210\pi$
$577$	−22.5000	−	12.9904i	−0.936687	−	0.540797i	−0.888920	+	0.458062i	$0.848544\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−30.0000	+	10.3923i	−1.24461	+	0.431145i
$582$		0			0
$583$	−13.5000	+	23.3827i	−0.559113	+	0.968412i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.0000	+	31.1769i	0.742940	+	1.28681i	0.951151	+	0.308725i	$0.0999023\pi$
$587$	18.0000	+	31.1769i	0.742940	+	1.28681i	−0.208212	+	0.978084i	$0.566764\pi$
$588$		0			0
$589$	1.50000	−	2.59808i	0.0618064	−	0.107052i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.5000	−	18.1865i	−0.431183	−	0.746831i	0.565792	−	0.824548i	$-0.308570\pi$
$593$	−10.5000	−	18.1865i	−0.431183	−	0.746831i	−0.996976	+	0.0777165i	$0.975237\pi$
$594$		0			0
$595$	18.0000	+	15.5885i	0.737928	+	0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.50000	−	2.59808i	0.183865	−	0.106155i	−0.405242	−	0.914209i	$-0.632813\pi$
$599$	4.50000	−	2.59808i	0.183865	−	0.106155i	0.589107	+	0.808055i	$0.299480\pi$
$600$		0			0
$601$	−30.0000	+	17.3205i	−1.22373	+	0.706518i	−0.965710	−	0.259623i	$-0.916402\pi$
$601$	−30.0000	+	17.3205i	−1.22373	+	0.706518i	−0.258015	+	0.966141i	$0.583069\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	48.0000			1.95148
$606$		0			0
$607$			29.4449i			1.19513i	0.801820	+	0.597565i	$0.203865\pi$
$607$			29.4449i			1.19513i	−0.801820	+	0.597565i	$0.796135\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.500000	−	0.866025i	0.0201948	−	0.0349784i	−0.855751	−	0.517387i	$-0.826905\pi$
$613$	0.500000	−	0.866025i	0.0201948	−	0.0349784i	0.875946	+	0.482409i	$0.160238\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.0000	−	10.3923i	−0.724653	−	0.418378i	0.0918100	−	0.995777i	$-0.470735\pi$
$617$	−18.0000	−	10.3923i	−0.724653	−	0.418378i	−0.816463	+	0.577398i	$0.804068\pi$
$618$		0			0
$619$		−	25.9808i		−	1.04425i	−0.852867	−	0.522127i	$-0.825139\pi$
$619$		−	25.9808i		−	1.04425i	0.852867	−	0.522127i	$-0.174861\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−22.5000	+	7.79423i	−0.901443	+	0.312269i
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−21.0000			−0.837325
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.0000			−0.952411
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			15.5885i			0.615707i	0.951434	+	0.307854i	$0.0996107\pi$
$641$			15.5885i			0.615707i	−0.951434	+	0.307854i	$0.900389\pi$
$642$		0			0
$643$	−3.00000	−	1.73205i	−0.118308	−	0.0683054i	0.439678	−	0.898155i	$-0.355093\pi$
$643$	−3.00000	−	1.73205i	−0.118308	−	0.0683054i	−0.557986	+	0.829850i	$0.688426\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.50000	−	7.79423i	0.176913	−	0.306423i	−0.763908	−	0.645325i	$-0.776722\pi$
$647$	4.50000	−	7.79423i	0.176913	−	0.306423i	0.940822	+	0.338902i	$0.110055\pi$
$648$		0			0
$649$	13.5000	−	7.79423i	0.529921	−	0.305950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	15.5885i		−	0.610023i	−0.952349	−	0.305012i	$-0.901340\pi$
$653$		−	15.5885i		−	0.610023i	0.952349	−	0.305012i	$-0.0986604\pi$
$654$		0			0
$655$	−27.0000			−1.05498
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.00000	−	5.19615i	0.350590	−	0.202413i	−0.314355	−	0.949306i	$-0.601788\pi$
$659$	9.00000	−	5.19615i	0.350590	−	0.202413i	0.664945	+	0.746892i	$0.268455\pi$
$660$		0			0
$661$	−19.5000	+	11.2583i	−0.758462	+	0.437898i	−0.828743	−	0.559629i	$-0.810944\pi$
$661$	−19.5000	+	11.2583i	−0.758462	+	0.437898i	0.0702812	+	0.997527i	$0.477610\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.50000	+	12.9904i	0.174503	+	0.503745i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−31.5000	+	54.5596i	−1.21604	+	2.10625i
$672$		0			0
$673$	−11.0000	−	19.0526i	−0.424019	−	0.734422i	0.572309	−	0.820038i	$-0.306048\pi$
$673$	−11.0000	−	19.0526i	−0.424019	−	0.734422i	−0.996328	+	0.0856156i	$0.972714\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.5000	+	38.9711i	−0.864745	+	1.49778i	0.00255466	+	0.999997i	$0.499187\pi$
$677$	−22.5000	+	38.9711i	−0.864745	+	1.49778i	−0.867300	+	0.497786i	$0.834147\pi$
$678$		0			0
$679$	−6.00000	−	17.3205i	−0.230259	−	0.664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	31.5000	+	18.1865i	1.20531	+	0.695888i	0.961732	−	0.273992i	$-0.0883442\pi$
$683$	31.5000	+	18.1865i	1.20531	+	0.695888i	0.243582	+	0.969880i	$0.421677\pi$
$684$		0			0
$685$			15.5885i			0.595604i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	28.5000	+	16.4545i	1.08419	+	0.625958i	0.932024	−	0.362397i	$-0.118041\pi$
$691$	28.5000	+	16.4545i	1.08419	+	0.625958i	0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−27.0000	−	15.5885i	−1.02417	−	0.591304i
$696$		0			0
$697$	−9.00000	−	15.5885i	−0.340899	−	0.590455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	20.7846i		−	0.785024i	−0.919747	−	0.392512i	$-0.871606\pi$
$701$		−	20.7846i		−	0.785024i	0.919747	−	0.392512i	$-0.128394\pi$
$702$		0			0
$703$	−10.5000	−	6.06218i	−0.396015	−	0.228639i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.50000	−	23.3827i	0.169240	−	0.879396i
$708$		0			0
$709$	−15.5000	+	26.8468i	−0.582115	+	1.00825i	0.413114	+	0.910679i	$0.364441\pi$
$709$	−15.5000	+	26.8468i	−0.582115	+	1.00825i	−0.995228	+	0.0975728i	$0.968892\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.50000	+	7.79423i	0.168526	+	0.291896i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.50000	+	12.9904i	0.279703	+	0.484459i	0.971311	−	0.237814i	$-0.0764307\pi$
$719$	7.50000	+	12.9904i	0.279703	+	0.484459i	−0.691608	+	0.722273i	$0.743097\pi$
$720$		0			0
$721$	−13.5000	−	2.59808i	−0.502766	−	0.0967574i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	21.0000	−	12.1244i	0.778847	−	0.449667i	−0.0571746	−	0.998364i	$-0.518209\pi$
$727$	21.0000	−	12.1244i	0.778847	−	0.449667i	0.836021	+	0.548697i	$0.184876\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.0000			−0.443836
$732$		0			0
$733$			29.4449i			1.08757i	0.839224	+	0.543785i	$0.183009\pi$
$733$			29.4449i			1.08757i	−0.839224	+	0.543785i	$0.816991\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	22.5000	−	12.9904i	0.828798	−	0.478507i
$738$		0			0
$739$	−8.50000	+	14.7224i	−0.312678	+	0.541573i	−0.978941	−	0.204143i	$-0.934559\pi$
$739$	−8.50000	+	14.7224i	−0.312678	+	0.541573i	0.666264	+	0.745716i	$0.267893\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	45.0000	+	25.9808i	1.65089	+	0.953142i	0.976707	+	0.214577i	$0.0688374\pi$
$743$	45.0000	+	25.9808i	1.65089	+	0.953142i	0.674183	+	0.738564i	$0.264496\pi$
$744$		0			0
$745$		−	15.5885i		−	0.571117i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13.5000	+	38.9711i	0.493279	+	1.42397i
$750$		0			0
$751$	−29.0000			−1.05823			−0.529113	−	0.848552i	$-0.677475\pi$
$751$	−29.0000			−1.05823			−0.529113	+	0.848552i	$0.677475\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−39.0000			−1.41936
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	21.0000			0.761249			0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000			0.761249			0.380625	+	0.924730i	$0.375709\pi$
$762$		0			0
$763$	−34.0000	−	29.4449i	−1.23088	−	1.06598i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	30.0000	+	17.3205i	1.08183	+	0.624593i	0.931389	−	0.364026i	$-0.118598\pi$
$769$	30.0000	+	17.3205i	1.08183	+	0.624593i	0.150439	+	0.988619i	$0.451931\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.5000	−	33.7750i	0.701366	−	1.21480i	−0.266621	−	0.963802i	$-0.585907\pi$
$773$	19.5000	−	33.7750i	0.701366	−	1.21480i	0.967987	−	0.251000i	$-0.0807596\pi$
$774$		0			0
$775$	6.00000	−	3.46410i	0.215526	−	0.124434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	10.3923i		−	0.372343i
$780$		0			0
$781$	54.0000			1.93227
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13.5000	+	7.79423i	−0.481836	+	0.278188i
$786$		0			0
$787$	1.50000	−	0.866025i	0.0534692	−	0.0308705i	−0.473027	−	0.881048i	$-0.656839\pi$
$787$	1.50000	−	0.866025i	0.0534692	−	0.0308705i	0.526496	+	0.850177i	$0.323505\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−36.0000	+	41.5692i	−1.28001	+	1.47803i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.00000	−	15.5885i	0.318796	−	0.552171i	−0.661441	−	0.749997i	$-0.730055\pi$
$797$	9.00000	−	15.5885i	0.318796	−	0.552171i	0.980237	+	0.197826i	$0.0633881\pi$
$798$		0			0
$799$	−4.50000	−	7.79423i	−0.159199	−	0.275740i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	31.5000	−	54.5596i	1.11161	−	1.92537i
$804$		0			0
$805$	−40.5000	−	7.79423i	−1.42744	−	0.274710i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22.5000	+	12.9904i	0.791058	+	0.456717i	0.840335	−	0.542068i	$-0.182358\pi$
$809$	22.5000	+	12.9904i	0.791058	+	0.456717i	−0.0492770	+	0.998785i	$0.515692\pi$
$810$		0			0
$811$		−	51.9615i		−	1.82462i	−0.409505	−	0.912308i	$-0.634299\pi$
$811$		−	51.9615i		−	1.82462i	0.409505	−	0.912308i	$-0.365701\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.50000	−	2.59808i	−0.0525427	−	0.0910066i
$816$		0			0
$817$	−6.00000	−	3.46410i	−0.209913	−	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.50000	−	2.59808i	−0.157051	−	0.0906735i	0.419415	−	0.907795i	$-0.362235\pi$
$821$	−4.50000	−	2.59808i	−0.157051	−	0.0906735i	−0.576466	+	0.817121i	$0.695569\pi$
$822$		0			0
$823$	3.50000	+	6.06218i	0.122002	+	0.211314i	0.920557	−	0.390608i	$-0.127735\pi$
$823$	3.50000	+	6.06218i	0.122002	+	0.211314i	−0.798555	+	0.601922i	$0.794402\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			10.3923i			0.361376i	0.983540	+	0.180688i	$0.0578324\pi$
$827$			10.3923i			0.361376i	−0.983540	+	0.180688i	$0.942168\pi$
$828$		0			0
$829$	1.50000	+	0.866025i	0.0520972	+	0.0300783i	0.525822	−	0.850594i	$-0.323758\pi$
$829$	1.50000	+	0.866025i	0.0520972	+	0.0300783i	−0.473725	+	0.880673i	$0.657091\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.5000	+	12.9904i	−0.571691	+	0.450090i
$834$		0			0
$835$	−18.0000	+	31.1769i	−0.622916	+	1.07892i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.0000	−	20.7846i	−0.414286	−	0.717564i	0.581067	−	0.813856i	$-0.302635\pi$
$839$	−12.0000	−	20.7846i	−0.414286	−	0.717564i	−0.995353	+	0.0962912i	$0.969302\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	19.5000	+	33.7750i	0.670820	+	1.16190i
$846$		0			0
$847$	−8.00000	+	41.5692i	−0.274883	+	1.42834i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	31.5000	−	18.1865i	1.07981	−	0.623426i
$852$		0			0
$853$	12.0000	−	6.92820i	0.410872	−	0.237217i	−0.280292	−	0.959915i	$-0.590431\pi$
$853$	12.0000	−	6.92820i	0.410872	−	0.237217i	0.691164	+	0.722698i	$0.257098\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	33.0000			1.12726			0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000			1.12726			0.563629	+	0.826028i	$0.309405\pi$
$858$		0			0
$859$		−	39.8372i		−	1.35923i	−0.733571	−	0.679613i	$-0.762148\pi$
$859$		−	39.8372i		−	1.35923i	0.733571	−	0.679613i	$-0.237852\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.50000	−	2.59808i	0.153182	−	0.0884395i	−0.421450	−	0.906852i	$-0.638479\pi$
$863$	4.50000	−	2.59808i	0.153182	−	0.0884395i	0.574632	+	0.818412i	$0.305145\pi$
$864$		0			0
$865$	13.5000	−	23.3827i	0.459014	−	0.795035i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.50000	+	2.59808i	0.152652	+	0.0881337i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.50000	−	7.79423i	0.0507093	−	0.263493i
$876$		0			0
$877$	11.0000			0.371444			0.185722	−	0.982602i	$-0.440538\pi$
$877$	11.0000			0.371444			0.185722	+	0.982602i	$0.440538\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.00000			−0.100730			−0.0503651	−	0.998731i	$-0.516038\pi$
$887$	−3.00000			−0.100730			−0.0503651	+	0.998731i	$0.516038\pi$
$888$		0			0
$889$	4.00000	−	20.7846i	0.134156	−	0.697093i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	5.19615i		−	0.173883i
$894$		0			0
$895$	13.5000	+	7.79423i	0.451255	+	0.260532i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	13.5000	−	7.79423i	0.449750	−	0.259663i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			20.7846i			0.690904i
$906$		0			0
$907$	37.0000			1.22856			0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000			1.22856			0.614282	+	0.789086i	$0.289446\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.00000	+	5.19615i	−0.298183	+	0.172156i	−0.641626	−	0.767017i	$-0.721740\pi$
$911$	−9.00000	+	5.19615i	−0.298183	+	0.172156i	0.343443	+	0.939173i	$0.388407\pi$
$912$		0			0
$913$	54.0000	−	31.1769i	1.78714	−	1.03181i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.50000	−	23.3827i	0.148603	−	0.772164i
$918$		0			0
$919$	−9.50000	−	16.4545i	−0.313376	−	0.542783i	0.665715	−	0.746206i	$-0.268127\pi$
$919$	−9.50000	−	16.4545i	−0.313376	−	0.542783i	−0.979091	+	0.203423i	$0.934793\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−14.0000	−	24.2487i	−0.460317	−	0.797293i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19.5000	−	33.7750i	0.639774	−	1.10812i	−0.345708	−	0.938342i	$-0.612361\pi$
$929$	19.5000	−	33.7750i	0.639774	−	1.10812i	0.985482	−	0.169779i	$-0.0543055\pi$
$930$		0			0
$931$	−12.0000	+	1.73205i	−0.393284	+	0.0567657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−40.5000	−	23.3827i	−1.32449	−	0.764696i
$936$		0			0
$937$		−	41.5692i		−	1.35801i	−0.734135	−	0.679004i	$-0.762412\pi$
$937$		−	41.5692i		−	1.35801i	0.734135	−	0.679004i	$-0.237588\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.5000	−	18.1865i	−0.342290	−	0.592864i	0.642567	−	0.766229i	$-0.277869\pi$
$941$	−10.5000	−	18.1865i	−0.342290	−	0.592864i	−0.984858	+	0.173365i	$0.944536\pi$
$942$		0			0
$943$	27.0000	+	15.5885i	0.879241	+	0.507630i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−40.5000	−	23.3827i	−1.31607	−	0.759835i	−0.332979	−	0.942934i	$-0.608054\pi$
$947$	−40.5000	−	23.3827i	−1.31607	−	0.759835i	−0.983094	+	0.183099i	$0.941387\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			20.7846i			0.673280i	0.941634	+	0.336640i	$0.109290\pi$
$953$			20.7846i			0.673280i	−0.941634	+	0.336640i	$0.890710\pi$
$954$		0			0
$955$	−40.5000	−	23.3827i	−1.31055	−	0.756646i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−13.5000	−	2.59808i	−0.435938	−	0.0838963i
$960$		0			0
$961$	−14.0000	+	24.2487i	−0.451613	+	0.782216i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.50000	−	12.9904i	−0.241434	−	0.418175i
$966$		0			0
$967$	20.0000	−	34.6410i	0.643157	−	1.11398i	−0.341567	−	0.939857i	$-0.610958\pi$
$967$	20.0000	−	34.6410i	0.643157	−	1.11398i	0.984724	−	0.174123i	$-0.0557089\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.5000	−	49.3634i	−0.914609	−	1.58415i	−0.807473	−	0.589904i	$-0.799166\pi$
$971$	−28.5000	−	49.3634i	−0.914609	−	1.58415i	−0.107135	−	0.994244i	$-0.534168\pi$
$972$		0			0
$973$	18.0000	−	20.7846i	0.577054	−	0.666324i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	31.5000	−	18.1865i	1.00777	−	0.581839i	0.0972351	−	0.995261i	$-0.469000\pi$
$977$	31.5000	−	18.1865i	1.00777	−	0.581839i	0.910539	+	0.413423i	$0.135667\pi$
$978$		0			0
$979$	40.5000	−	23.3827i	1.29439	−	0.747314i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.00000			0.0956851			0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000			0.0956851			0.0478426	+	0.998855i	$0.484765\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	18.0000	−	10.3923i	0.572367	−	0.330456i
$990$		0			0
$991$	−12.5000	+	21.6506i	−0.397076	+	0.687755i	−0.993364	−	0.115015i	$-0.963308\pi$
$991$	−12.5000	+	21.6506i	−0.397076	+	0.687755i	0.596288	+	0.802771i	$0.296642\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.50000	+	2.59808i	0.142660	+	0.0823646i
$996$		0			0
$997$		−	22.5167i		−	0.713110i	−0.934274	−	0.356555i	$-0.883951\pi$
$997$		−	22.5167i		−	0.713110i	0.934274	−	0.356555i	$-0.116049\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.bm.a.1025.1		2
3.2	odd	2		2268.2.bm.f.1025.1		2
7.5	odd	6		2268.2.w.a.1349.1		2
9.2	odd	6		2268.2.w.a.269.1		2
9.4	even	3		84.2.k.a.17.1	yes	2
9.5	odd	6		84.2.k.b.17.1	yes	2
9.7	even	3		2268.2.w.f.269.1		2
21.5	even	6		2268.2.w.f.1349.1		2
36.23	even	6		336.2.bc.b.17.1		2
36.31	odd	6		336.2.bc.d.17.1		2
45.4	even	6		2100.2.bi.f.101.1		2
45.13	odd	12		2100.2.bo.a.1949.1		4
45.14	odd	6		2100.2.bi.e.101.1		2
45.22	odd	12		2100.2.bo.a.1949.2		4
45.23	even	12		2100.2.bo.f.1949.2		4
45.32	even	12		2100.2.bo.f.1949.1		4
63.4	even	3		588.2.f.c.293.2		2
63.5	even	6		84.2.k.a.5.1	✓	2
63.13	odd	6		588.2.k.d.521.1		2
63.23	odd	6		588.2.k.d.509.1		2
63.31	odd	6		588.2.f.a.293.1		2
63.32	odd	6		588.2.f.a.293.2		2
63.40	odd	6		84.2.k.b.5.1	yes	2
63.41	even	6		588.2.k.c.521.1		2
63.47	even	6	inner	2268.2.bm.a.593.1		2
63.58	even	3		588.2.k.c.509.1		2
63.59	even	6		588.2.f.c.293.1		2
63.61	odd	6		2268.2.bm.f.593.1		2
252.31	even	6		2352.2.k.d.881.2		2
252.59	odd	6		2352.2.k.a.881.2		2
252.67	odd	6		2352.2.k.a.881.1		2
252.95	even	6		2352.2.k.d.881.1		2
252.103	even	6		336.2.bc.b.257.1		2
252.131	odd	6		336.2.bc.d.257.1		2
315.68	odd	12		2100.2.bo.a.1349.2		4
315.103	even	12		2100.2.bo.f.1349.1		4
315.194	even	6		2100.2.bi.f.1601.1		2
315.229	odd	6		2100.2.bi.e.1601.1		2
315.257	odd	12		2100.2.bo.a.1349.1		4
315.292	even	12		2100.2.bo.f.1349.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.k.a.5.1	✓	2	63.5	even	6
84.2.k.a.17.1	yes	2	9.4	even	3
84.2.k.b.5.1	yes	2	63.40	odd	6
84.2.k.b.17.1	yes	2	9.5	odd	6
336.2.bc.b.17.1		2	36.23	even	6
336.2.bc.b.257.1		2	252.103	even	6
336.2.bc.d.17.1		2	36.31	odd	6
336.2.bc.d.257.1		2	252.131	odd	6
588.2.f.a.293.1		2	63.31	odd	6
588.2.f.a.293.2		2	63.32	odd	6
588.2.f.c.293.1		2	63.59	even	6
588.2.f.c.293.2		2	63.4	even	3
588.2.k.c.509.1		2	63.58	even	3
588.2.k.c.521.1		2	63.41	even	6
588.2.k.d.509.1		2	63.23	odd	6
588.2.k.d.521.1		2	63.13	odd	6
2100.2.bi.e.101.1		2	45.14	odd	6
2100.2.bi.e.1601.1		2	315.229	odd	6
2100.2.bi.f.101.1		2	45.4	even	6
2100.2.bi.f.1601.1		2	315.194	even	6
2100.2.bo.a.1349.1		4	315.257	odd	12
2100.2.bo.a.1349.2		4	315.68	odd	12
2100.2.bo.a.1949.1		4	45.13	odd	12
2100.2.bo.a.1949.2		4	45.22	odd	12
2100.2.bo.f.1349.1		4	315.103	even	12
2100.2.bo.f.1349.2		4	315.292	even	12
2100.2.bo.f.1949.1		4	45.32	even	12
2100.2.bo.f.1949.2		4	45.23	even	12
2268.2.w.a.269.1		2	9.2	odd	6
2268.2.w.a.1349.1		2	7.5	odd	6
2268.2.w.f.269.1		2	9.7	even	3
2268.2.w.f.1349.1		2	21.5	even	6
2268.2.bm.a.593.1		2	63.47	even	6	inner
2268.2.bm.a.1025.1		2	1.1	even	1	trivial
2268.2.bm.f.593.1		2	63.61	odd	6
2268.2.bm.f.1025.1		2	3.2	odd	2
2352.2.k.a.881.1		2	252.67	odd	6
2352.2.k.a.881.2		2	252.59	odd	6
2352.2.k.d.881.1		2	252.95	even	6
2352.2.k.d.881.2		2	252.31	even	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.bm (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{5} +(0.500000 - 2.59808i) q^{7} +5.19615i q^{11} +(1.50000 - 2.59808i) q^{17} +(1.50000 - 0.866025i) q^{19} +5.19615i q^{23} +4.00000 q^{25} +(1.50000 - 0.866025i) q^{31} +(-1.50000 + 7.79423i) q^{35} +(-3.50000 - 6.06218i) q^{37} +(3.00000 - 5.19615i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(1.50000 - 2.59808i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(4.50000 + 2.59808i) q^{53} -15.5885i q^{55} +(-1.50000 - 2.59808i) q^{59} +(10.5000 + 6.06218i) q^{61} +(-2.50000 - 4.33013i) q^{67} -10.3923i q^{71} +(-10.5000 - 6.06218i) q^{73} +(13.5000 + 2.59808i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(-4.50000 - 7.79423i) q^{89} +(-4.50000 + 2.59808i) q^{95} +(6.00000 - 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + q^{7} + 3 q^{17} + 3 q^{19} + 8 q^{25} + 3 q^{31} - 3 q^{35} - 7 q^{37} + 6 q^{41} - 4 q^{43} + 3 q^{47} - 13 q^{49} + 9 q^{53} - 3 q^{59} + 21 q^{61} - 5 q^{67} - 21 q^{73} + 27 q^{77}+ \cdots + 12 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + q^7 + 3 * q^17 + 3 * q^19 + 8 * q^25 + 3 * q^31 - 3 * q^35 - 7 * q^37 + 6 * q^41 - 4 * q^43 + 3 * q^47 - 13 * q^49 + 9 * q^53 - 3 * q^59 + 21 * q^61 - 5 * q^67 - 21 * q^73 + 27 * q^77 + q^79 - 12 * q^83 - 9 * q^85 - 9 * q^89 - 9 * q^95 + 12 * q^97

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