LMFDB - Embedded newform 432.9.o.a.127.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,9,Mod(127,432)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("432.127"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	432.o (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,-147,0,-2769] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$175.987559546$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			127.16
Character	$\chi$	$=$	432.127
Dual form			432.9.o.a.415.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(607.471 + 1052.17i) q^{5} +(-4030.55 - 2327.04i) q^{7} +(-3674.67 - 2121.57i) q^{11} +(-13902.8 - 24080.3i) q^{13} +1913.35 q^{17} -96672.2i q^{19} +(-283599. + 163736. i) q^{23} +(-542730. + 940036. i) q^{25} +(399355. - 691703. i) q^{29} +(-561470. + 324165. i) q^{31} -5.65444e6i q^{35} +2.23855e6 q^{37} +(491649. + 851562. i) q^{41} +(-3.68522e6 - 2.12767e6i) q^{43} +(-2.98466e6 - 1.72320e6i) q^{47} +(7.94784e6 + 1.37661e7i) q^{49} +5.99110e6 q^{53} -5.15517e6i q^{55} +(1.21091e7 - 6.99118e6i) q^{59} +(-650503. + 1.12670e6i) q^{61} +(1.68911e7 - 2.92562e7i) q^{65} +(9.82409e6 - 5.67194e6i) q^{67} +6.82352e6i q^{71} +3.84732e7 q^{73} +(9.87397e6 + 1.71022e7i) q^{77} +(-1.73833e6 - 1.00363e6i) q^{79} +(3.73630e7 + 2.15715e7i) q^{83} +(1.16230e6 + 2.01317e6i) q^{85} -7.34418e7 q^{89} +1.29409e8i q^{91} +(1.01716e8 - 5.87255e7i) q^{95} +(-1.67437e7 + 2.90010e7i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 147 q^{5} - 2769 q^{7} - 17082 q^{11} + 1685 q^{13} + 6402 q^{17} + 4941 q^{23} - 1080803 q^{25} + 639219 q^{29} + 920745 q^{31} - 2765636 q^{37} + 2229390 q^{41} + 1788381 q^{47} + 13397285 q^{49}+ \cdots + 74963282 q^{97}+O(q^{100}) $ 32 * q - 147 * q^5 - 2769 * q^7 - 17082 * q^11 + 1685 * q^13 + 6402 * q^17 + 4941 * q^23 - 1080803 * q^25 + 639219 * q^29 + 920745 * q^31 - 2765636 * q^37 + 2229390 * q^41 + 1788381 * q^47 + 13397285 * q^49 + 16055532 * q^53 + 42176466 * q^59 - 6097553 * q^61 - 14325141 * q^65 + 1476300 * q^67 + 44269442 * q^73 + 22487769 * q^77 + 65644341 * q^79 - 67089393 * q^83 + 28601250 * q^85 - 133904424 * q^89 - 7225380 * q^95 + 74963282 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	607.471	+	1052.17i	0.971954	+	1.68347i	0.689642	+	0.724151i	$0.257768\pi$
$5$	607.471	+	1052.17i	0.971954	+	1.68347i	0.282312	+	0.959323i	$0.408899\pi$
$6$		0			0
$7$	−4030.55	−	2327.04i	−1.67870	−	0.969197i	−0.962493	−	0.271307i	$-0.912544\pi$
$7$	−4030.55	−	2327.04i	−1.67870	−	0.969197i	−0.716205	−	0.697890i	$-0.754123\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3674.67	−	2121.57i	−0.250985	−	0.144906i	0.369230	−	0.929338i	$-0.379621\pi$
$11$	−3674.67	−	2121.57i	−0.250985	−	0.144906i	−0.620215	+	0.784432i	$0.712955\pi$
$12$		0			0
$13$	−13902.8	−	24080.3i	−0.486775	−	0.843119i	0.513109	−	0.858323i	$-0.328494\pi$
$13$	−13902.8	−	24080.3i	−0.486775	−	0.843119i	−0.999884	+	0.0152040i	$0.995160\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1913.35			0.0229086			0.0114543	−	0.999934i	$-0.496354\pi$
$17$	1913.35			0.0229086			0.0114543	+	0.999934i	$0.496354\pi$
$18$		0			0
$19$		−	96672.2i		−	0.741800i	−0.928673	−	0.370900i	$-0.879049\pi$
$19$		−	96672.2i		−	0.741800i	0.928673	−	0.370900i	$-0.120951\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−283599.	+	163736.i	−1.01343	+	0.585104i	−0.912194	−	0.409759i	$-0.865613\pi$
$23$	−283599.	+	163736.i	−1.01343	+	0.585104i	−0.101235	+	0.994862i	$0.532280\pi$
$24$		0			0
$25$	−542730.	+	940036.i	−1.38939	+	2.40649i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	399355.	−	691703.i	0.564634	−	0.977974i	−0.432450	−	0.901658i	$-0.642351\pi$
$29$	399355.	−	691703.i	0.564634	−	0.977974i	0.997084	−	0.0763163i	$-0.0243159\pi$
$30$		0			0
$31$	−561470.	+	324165.i	−0.607966	+	0.351010i	−0.772169	−	0.635417i	$-0.780828\pi$
$31$	−561470.	+	324165.i	−0.607966	+	0.351010i	0.164203	+	0.986427i	$0.447495\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	5.65444e6i		−	3.76806i
$36$		0			0
$37$	2.23855e6			1.19443			0.597214	−	0.802082i	$-0.296274\pi$
$37$	2.23855e6			1.19443			0.597214	+	0.802082i	$0.296274\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	491649.	+	851562.i	0.173988	+	0.301357i	0.939811	−	0.341696i	$-0.111001\pi$
$41$	491649.	+	851562.i	0.173988	+	0.301357i	−0.765822	+	0.643052i	$0.777668\pi$
$42$		0			0
$43$	−3.68522e6	−	2.12767e6i	−1.07793	−	0.622343i	−0.147592	−	0.989048i	$-0.547152\pi$
$43$	−3.68522e6	−	2.12767e6i	−1.07793	−	0.622343i	−0.930337	+	0.366706i	$0.880486\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.98466e6	−	1.72320e6i	−0.611651	−	0.353137i	0.161960	−	0.986797i	$-0.448218\pi$
$47$	−2.98466e6	−	1.72320e6i	−0.611651	−	0.353137i	−0.773611	+	0.633660i	$0.781552\pi$
$48$		0			0
$49$	7.94784e6	+	1.37661e7i	1.37868	+	2.38795i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.99110e6			0.759282			0.379641	−	0.925134i	$-0.376048\pi$
$53$	5.99110e6			0.759282			0.379641	+	0.925134i	$0.376048\pi$
$54$		0			0
$55$		−	5.15517e6i		−	0.563368i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.21091e7	−	6.99118e6i	0.999317	−	0.576956i	0.0912709	−	0.995826i	$-0.470907\pi$
$59$	1.21091e7	−	6.99118e6i	0.999317	−	0.576956i	0.908046	+	0.418870i	$0.137574\pi$
$60$		0			0
$61$	−650503.	+	1.12670e6i	−0.0469819	+	0.0813750i	−0.888560	−	0.458760i	$-0.848294\pi$
$61$	−650503.	+	1.12670e6i	−0.0469819	+	0.0813750i	0.841578	+	0.540135i	$0.181627\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.68911e7	−	2.92562e7i	0.946246	−	1.63895i
$66$		0			0
$67$	9.82409e6	−	5.67194e6i	0.487521	−	0.281470i	−0.236025	−	0.971747i	$-0.575845\pi$
$67$	9.82409e6	−	5.67194e6i	0.487521	−	0.281470i	0.723545	+	0.690277i	$0.242511\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6.82352e6i			0.268519i	0.990946	+	0.134260i	$0.0428656\pi$
$71$			6.82352e6i			0.268519i	−0.990946	+	0.134260i	$0.957134\pi$
$72$		0			0
$73$	3.84732e7			1.35477			0.677387	−	0.735627i	$-0.263112\pi$
$73$	3.84732e7			1.35477			0.677387	+	0.735627i	$0.263112\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.87397e6	+	1.71022e7i	0.280885	+	0.486507i
$78$		0			0
$79$	−1.73833e6	−	1.00363e6i	−0.0446297	−	0.0257670i	0.477519	−	0.878621i	$-0.341536\pi$
$79$	−1.73833e6	−	1.00363e6i	−0.0446297	−	0.0257670i	−0.522149	+	0.852854i	$0.674869\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.73630e7	+	2.15715e7i	0.787280	+	0.454536i	0.839004	−	0.544125i	$-0.183138\pi$
$83$	3.73630e7	+	2.15715e7i	0.787280	+	0.454536i	−0.0517243	+	0.998661i	$0.516472\pi$
$84$		0			0
$85$	1.16230e6	+	2.01317e6i	0.0222661	+	0.0385660i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.34418e7			−1.17053			−0.585266	−	0.810841i	$-0.699010\pi$
$89$	−7.34418e7			−1.17053			−0.585266	+	0.810841i	$0.699010\pi$
$90$		0			0
$91$			1.29409e8i			1.88712i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.01716e8	−	5.87255e7i	1.24880	−	0.720996i
$96$		0			0
$97$	−1.67437e7	+	2.90010e7i	−0.189132	+	0.327586i	−0.944961	−	0.327183i	$-0.893901\pi$
$97$	−1.67437e7	+	2.90010e7i	−0.189132	+	0.327586i	0.755829	+	0.654769i	$0.227234\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.73164e7	−	9.92750e7i	0.550800	−	0.954013i	−0.447417	−	0.894325i	$-0.647656\pi$
$101$	5.73164e7	−	9.92750e7i	0.550800	−	0.954013i	0.998217	−	0.0596877i	$-0.0190105\pi$
$102$		0			0
$103$	−4.02438e7	+	2.32347e7i	−0.357561	+	0.206438i	−0.668010	−	0.744152i	$-0.732854\pi$
$103$	−4.02438e7	+	2.32347e7i	−0.357561	+	0.206438i	0.310450	+	0.950590i	$0.399520\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			9.06367e7i			0.691463i	0.938333	+	0.345732i	$0.112369\pi$
$107$			9.06367e7i			0.691463i	−0.938333	+	0.345732i	$0.887631\pi$
$108$		0			0
$109$	−1.17994e8			−0.835899			−0.417949	−	0.908470i	$-0.637251\pi$
$109$	−1.17994e8			−0.835899			−0.417949	+	0.908470i	$0.637251\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.04763e7	+	1.22069e8i	0.432244	+	0.748669i	0.997066	−	0.0765437i	$-0.0243885\pi$
$113$	7.04763e7	+	1.22069e8i	0.432244	+	0.748669i	−0.564822	+	0.825213i	$0.691055\pi$
$114$		0			0
$115$	−3.44556e8	−	1.98930e8i	−1.97001	−	1.13739i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.71186e6	−	4.45244e6i	−0.0384566	−	0.0222029i
$120$		0			0
$121$	−9.81773e7	−	1.70048e8i	−0.458004	−	0.793287i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.44184e8			−3.45778
$126$		0			0
$127$			2.31625e8i			0.890372i	0.895438	+	0.445186i	$0.146862\pi$
$127$			2.31625e8i			0.890372i	−0.895438	+	0.445186i	$0.853138\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.06791e8	−	1.77126e8i	1.04173	−	0.601445i	0.121410	−	0.992602i	$-0.461258\pi$
$131$	3.06791e8	−	1.77126e8i	1.04173	−	0.601445i	0.920324	+	0.391157i	$0.127925\pi$
$132$		0			0
$133$	−2.24960e8	+	3.89642e8i	−0.718950	+	1.24526i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.33444e8	+	4.04337e8i	−0.662675	+	1.14779i	0.317235	+	0.948347i	$0.397246\pi$
$137$	−2.33444e8	+	4.04337e8i	−0.662675	+	1.14779i	−0.979910	+	0.199440i	$0.936088\pi$
$138$		0			0
$139$	−1.04248e8	+	6.01878e7i	−0.279261	+	0.161231i	−0.633089	−	0.774079i	$-0.718213\pi$
$139$	−1.04248e8	+	6.01878e7i	−0.279261	+	0.161231i	0.353828	+	0.935311i	$0.384880\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.17983e8i			0.282147i
$144$		0			0
$145$	9.70386e8			2.19519
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.44973e7	+	1.46354e8i	0.171434	+	0.296933i	0.938922	−	0.344131i	$-0.111827\pi$
$149$	8.44973e7	+	1.46354e8i	0.171434	+	0.296933i	−0.767487	+	0.641064i	$0.778493\pi$
$150$		0			0
$151$	−2.16515e8	−	1.25005e8i	−0.416467	−	0.240447i	0.277098	−	0.960842i	$-0.410627\pi$
$151$	−2.16515e8	−	1.25005e8i	−0.416467	−	0.240447i	−0.693565	+	0.720394i	$0.743961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.82153e8	−	3.93841e8i	−1.18183	−	0.682330i
$156$		0			0
$157$	8.04433e7	+	1.39332e8i	0.132401	+	0.229325i	0.924602	−	0.380935i	$-0.124398\pi$
$157$	8.04433e7	+	1.39332e8i	0.132401	+	0.229325i	−0.792201	+	0.610261i	$0.791065\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.52408e9			2.26832
$162$		0			0
$163$		−	3.45045e7i		−	0.0488793i	−0.999701	−	0.0244397i	$-0.992220\pi$
$163$		−	3.45045e7i		−	0.0488793i	0.999701	−	0.0244397i	$-0.00778016\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.60697e8	−	1.50513e8i	0.335173	−	0.193512i	−0.322962	−	0.946412i	$-0.604679\pi$
$167$	2.60697e8	−	1.50513e8i	0.335173	−	0.193512i	0.658136	+	0.752899i	$0.271345\pi$
$168$		0			0
$169$	2.12905e7	−	3.68762e7i	0.0260999	−	0.0452064i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.12232e8	−	5.40802e8i	0.348573	−	0.603746i	−0.637423	−	0.770514i	$-0.720000\pi$
$173$	3.12232e8	−	5.40802e8i	0.348573	−	0.603746i	0.985996	+	0.166768i	$0.0533331\pi$
$174$		0			0
$175$	4.37500e9	−	2.52591e9i	4.66473	−	2.69318i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			1.31519e9i			1.28108i	0.767924	+	0.640541i	$0.221290\pi$
$179$			1.31519e9i			1.28108i	−0.767924	+	0.640541i	$0.778710\pi$
$180$		0			0
$181$	9.93254e8			0.925435			0.462718	−	0.886506i	$-0.346874\pi$
$181$	9.93254e8			0.925435			0.462718	+	0.886506i	$0.346874\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.35986e9	+	2.35534e9i	1.16093	+	2.01079i
$186$		0			0
$187$	−7.03093e6	−	4.05931e6i	−0.00574971	−	0.00331960i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.79972e9	+	1.03907e9i	1.35229	+	0.780747i	0.988570	−	0.150760i	$-0.0481720\pi$
$191$	1.79972e9	+	1.03907e9i	1.35229	+	0.780747i	0.363723	+	0.931507i	$0.381505\pi$
$192$		0			0
$193$	6.87423e8	+	1.19065e9i	0.495444	+	0.858135i	0.999986	−	0.00525240i	$-0.00167190\pi$
$193$	6.87423e8	+	1.19065e9i	0.495444	+	0.858135i	−0.504542	+	0.863387i	$0.668339\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.37288e8			0.356732			0.178366	−	0.983964i	$-0.442919\pi$
$197$	5.37288e8			0.356732			0.178366	+	0.983964i	$0.442919\pi$
$198$		0			0
$199$			1.21091e9i			0.772148i	0.922468	+	0.386074i	$0.126169\pi$
$199$			1.21091e9i			0.772148i	−0.922468	+	0.386074i	$0.873831\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.21924e9	+	1.85863e9i	−1.89570	+	1.09448i
$204$		0			0
$205$	−5.97326e8	+	1.03460e9i	−0.338217	+	0.585809i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.05097e8	+	3.55238e8i	−0.107491	+	0.186181i
$210$		0			0
$211$	3.88737e8	−	2.24437e8i	0.196122	−	0.113231i	−0.398723	−	0.917071i	$-0.630547\pi$
$211$	3.88737e8	−	2.24437e8i	0.196122	−	0.113231i	0.594845	+	0.803840i	$0.297213\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	5.16998e9i		−	2.41955i
$216$		0			0
$217$	3.01738e9			1.36079
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.66009e7	−	4.60741e7i	−0.0111513	−	0.0193147i
$222$		0			0
$223$	9.29382e8	+	5.36579e8i	0.375816	+	0.216977i	0.675996	−	0.736905i	$-0.263714\pi$
$223$	9.29382e8	+	5.36579e8i	0.375816	+	0.216977i	−0.300181	+	0.953882i	$0.597047\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.45562e9	+	8.40401e8i	0.548206	+	0.316507i	0.748398	−	0.663250i	$-0.230823\pi$
$227$	1.45562e9	+	8.40401e8i	0.548206	+	0.316507i	−0.200192	+	0.979757i	$0.564157\pi$
$228$		0			0
$229$	−6.58317e8	−	1.14024e9i	−0.239383	−	0.414624i	0.721154	−	0.692774i	$-0.243612\pi$
$229$	−6.58317e8	−	1.14024e9i	−0.239383	−	0.414624i	−0.960537	+	0.278151i	$0.910279\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.04257e9			−1.37162			−0.685811	−	0.727780i	$-0.740552\pi$
$233$	−4.04257e9			−1.37162			−0.685811	+	0.727780i	$0.740552\pi$
$234$		0			0
$235$		−	4.18717e9i		−	1.37293i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.29471e9	−	1.90220e9i	1.00978	−	0.582995i	0.0986502	−	0.995122i	$-0.468547\pi$
$239$	3.29471e9	−	1.90220e9i	1.00978	−	0.582995i	0.911126	+	0.412127i	$0.135214\pi$
$240$		0			0
$241$	−1.63069e9	+	2.82444e9i	−0.483396	+	0.837267i	−0.999818	−	0.0190675i	$-0.993930\pi$
$241$	−1.63069e9	+	2.82444e9i	−0.483396	+	0.837267i	0.516422	+	0.856334i	$0.327264\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.65617e9	+	1.67250e10i	−2.68004	+	4.64196i
$246$		0			0
$247$	−2.32790e9	+	1.34401e9i	−0.625426	+	0.361090i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			6.31615e9i			1.59132i	0.605744	+	0.795660i	$0.292876\pi$
$251$			6.31615e9i			1.59132i	−0.605744	+	0.795660i	$0.707124\pi$
$252$		0			0
$253$	1.38951e9			0.339141
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.47301e9	−	2.55133e9i	−0.337655	−	0.584836i	0.646336	−	0.763053i	$-0.276300\pi$
$257$	−1.47301e9	−	2.55133e9i	−0.337655	−	0.584836i	−0.983991	+	0.178217i	$0.942967\pi$
$258$		0			0
$259$	−9.02261e9	−	5.20920e9i	−2.00509	−	1.15764i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.93310e9	+	2.84813e9i	1.03109	+	0.595300i	0.917297	−	0.398204i	$-0.130366\pi$
$263$	4.93310e9	+	2.84813e9i	1.03109	+	0.595300i	0.113794	+	0.993504i	$0.463700\pi$
$264$		0			0
$265$	3.63942e9	+	6.30366e9i	0.737987	+	1.27823i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.91442e9			0.747582			0.373791	−	0.927513i	$-0.378058\pi$
$269$	3.91442e9			0.747582			0.373791	+	0.927513i	$0.378058\pi$
$270$		0			0
$271$			2.30909e9i			0.428119i	0.976821	+	0.214060i	$0.0686686\pi$
$271$			2.30909e9i			0.428119i	−0.976821	+	0.214060i	$0.931331\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.98871e9	−	2.30288e9i	0.697431	−	0.402662i
$276$		0			0
$277$	4.47038e9	−	7.74293e9i	0.759321	−	1.31518i	−0.183876	−	0.982949i	$-0.558865\pi$
$277$	4.47038e9	−	7.74293e9i	0.759321	−	1.31518i	0.943197	−	0.332233i	$-0.107802\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.02784e8	+	3.51232e8i	−0.0325243	+	0.0563338i	−0.881829	−	0.471569i	$-0.843688\pi$
$281$	−2.02784e8	+	3.51232e8i	−0.0325243	+	0.0563338i	0.849305	+	0.527902i	$0.177021\pi$
$282$		0			0
$283$	2.00591e9	−	1.15811e9i	0.312727	−	0.180553i	−0.335419	−	0.942069i	$-0.608878\pi$
$283$	2.00591e9	−	1.15811e9i	0.312727	−	0.180553i	0.648146	+	0.761516i	$0.275545\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	4.57635e9i		−	0.674516i
$288$		0			0
$289$	−6.97210e9			−0.999475
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.52957e8	+	7.84544e8i	0.0614591	+	0.106450i	0.895118	−	0.445830i	$-0.147091\pi$
$293$	4.52957e8	+	7.84544e8i	0.0614591	+	0.106450i	−0.833659	+	0.552280i	$0.813758\pi$
$294$		0			0
$295$	1.47118e10	+	8.49388e9i	1.94258	+	1.12155i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.88563e9	+	4.55277e9i	0.986624	+	0.569628i
$300$		0			0
$301$	9.90233e9	+	1.71513e10i	1.20634	+	2.08945i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.58065e9			−0.182657
$306$		0			0
$307$		−	2.08049e8i		−	0.0234214i	−0.999931	−	0.0117107i	$-0.996272\pi$
$307$		−	2.08049e8i		−	0.0234214i	0.999931	−	0.0117107i	$-0.00372771\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.31637e9	−	3.06941e9i	0.568295	−	0.328105i	−0.188173	−	0.982136i	$-0.560257\pi$
$311$	5.31637e9	−	3.06941e9i	0.568295	−	0.328105i	0.756468	+	0.654030i	$0.226923\pi$
$312$		0			0
$313$	−6.79299e9	+	1.17658e10i	−0.707756	+	1.22587i	0.257931	+	0.966163i	$0.416959\pi$
$313$	−6.79299e9	+	1.17658e10i	−0.707756	+	1.22587i	−0.965687	+	0.259707i	$0.916374\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.40217e9	−	4.16068e9i	0.237885	−	0.412029i	−0.722222	−	0.691661i	$-0.756879\pi$
$317$	2.40217e9	−	4.16068e9i	0.237885	−	0.412029i	0.960107	+	0.279632i	$0.0902125\pi$
$318$		0			0
$319$	−2.93499e9	+	1.69452e9i	−0.283429	+	0.163638i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.84968e8i		−	0.0169936i
$324$		0			0
$325$	3.01818e10			2.70528
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.01990e9	+	1.38909e10i	0.684519	+	1.18562i
$330$		0			0
$331$	1.86683e10	+	1.07782e10i	1.55522	+	0.897909i	0.997703	+	0.0677462i	$0.0215808\pi$
$331$	1.86683e10	+	1.07782e10i	1.55522	+	0.897909i	0.557521	+	0.830163i	$0.311753\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.19357e10	+	6.89108e9i	0.947695	+	0.547152i
$336$		0			0
$337$	5.99212e9	+	1.03787e10i	0.464580	+	0.804677i	0.999182	−	0.0404270i	$-0.0128718\pi$
$337$	5.99212e9	+	1.03787e10i	0.464580	+	0.804677i	−0.534602	+	0.845104i	$0.679538\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.75095e9			0.203454
$342$		0			0
$343$		−	4.71500e10i		−	3.40647i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.07137e10	−	6.18557e9i	0.738963	−	0.426640i	−0.0827294	−	0.996572i	$-0.526364\pi$
$347$	1.07137e10	−	6.18557e9i	0.738963	−	0.426640i	0.821692	+	0.569932i	$0.193030\pi$
$348$		0			0
$349$	8.76411e9	−	1.51799e10i	0.590753	−	1.02321i	−0.403378	−	0.915033i	$-0.632164\pi$
$349$	8.76411e9	−	1.51799e10i	0.590753	−	1.02321i	0.994131	−	0.108181i	$-0.0345025\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.08654e9	−	1.57383e10i	0.585194	−	1.01359i	−0.409657	−	0.912239i	$-0.634352\pi$
$353$	9.08654e9	−	1.57383e10i	0.585194	−	1.01359i	0.994851	−	0.101346i	$-0.0323149\pi$
$354$		0			0
$355$	−7.17951e9	+	4.14509e9i	−0.452045	+	0.260988i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	4.59629e9i		−	0.276713i	−0.990382	−	0.138357i	$-0.955818\pi$
$359$		−	4.59629e9i		−	0.276713i	0.990382	−	0.138357i	$-0.0441820\pi$
$360$		0			0
$361$	7.63806e9			0.449732
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.33714e10	+	4.04804e10i	1.31678	+	2.28073i
$366$		0			0
$367$	−1.33414e10	−	7.70264e9i	−0.735421	−	0.424596i	0.0849810	−	0.996383i	$-0.472917\pi$
$367$	−1.33414e10	−	7.70264e9i	−0.735421	−	0.424596i	−0.820402	+	0.571787i	$0.806250\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.41475e10	−	1.39415e10i	−1.27461	−	0.735894i
$372$		0			0
$373$	1.58316e10	+	2.74212e10i	0.817881	+	1.41661i	0.907241	+	0.420612i	$0.138184\pi$
$373$	1.58316e10	+	2.74212e10i	0.817881	+	1.41661i	−0.0893601	+	0.995999i	$0.528482\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.22086e10			−1.09940
$378$		0			0
$379$		−	1.69138e10i		−	0.819758i	−0.912140	−	0.409879i	$-0.865571\pi$
$379$		−	1.69138e10i		−	0.819758i	0.912140	−	0.409879i	$-0.134429\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.60642e10	+	9.27469e9i	−0.746560	+	0.431027i	−0.824450	−	0.565935i	$-0.808515\pi$
$383$	−1.60642e10	+	9.27469e9i	−0.746560	+	0.431027i	0.0778896	+	0.996962i	$0.475182\pi$
$384$		0			0
$385$	−1.19963e10	+	2.07782e10i	−0.546015	+	0.945725i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.41141e10	+	2.44463e10i	−0.616387	+	1.06761i	0.373753	+	0.927528i	$0.378071\pi$
$389$	−1.41141e10	+	2.44463e10i	−0.616387	+	1.06761i	−0.990140	+	0.140085i	$0.955262\pi$
$390$		0			0
$391$	−5.42624e8	+	3.13284e8i	−0.0232162	+	0.0134039i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	2.43870e9i		−	0.100177i
$396$		0			0
$397$	−3.74343e10			−1.50698			−0.753491	−	0.657458i	$-0.771632\pi$
$397$	−3.74343e10			−1.50698			−0.753491	+	0.657458i	$0.771632\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.27517e10	−	2.20866e10i	−0.493164	−	0.854185i	0.506805	−	0.862061i	$-0.330826\pi$
$401$	−1.27517e10	−	2.20866e10i	−0.493164	−	0.854185i	−0.999969	+	0.00787548i	$0.997493\pi$
$402$		0			0
$403$	1.56120e10	+	9.01358e9i	0.591886	+	0.341725i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.22594e9	−	4.74925e9i	−0.299784	−	0.173080i
$408$		0			0
$409$	−7.90200e9	−	1.36867e10i	−0.282386	−	0.489107i	0.689586	−	0.724204i	$-0.257793\pi$
$409$	−7.90200e9	−	1.36867e10i	−0.282386	−	0.489107i	−0.971972	+	0.235097i	$0.924459\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.50751e10			−2.23674
$414$		0			0
$415$			5.24163e10i			1.76715i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.71376e10	−	1.56679e10i	0.880473	−	0.508341i	0.00965836	−	0.999953i	$-0.496926\pi$
$419$	2.71376e10	−	1.56679e10i	0.880473	−	0.508341i	0.870814	+	0.491612i	$0.163592\pi$
$420$		0			0
$421$	2.89117e10	−	5.00765e10i	0.920334	−	1.59406i	0.121435	−	0.992599i	$-0.461250\pi$
$421$	2.89117e10	−	5.00765e10i	0.920334	−	1.59406i	0.798899	−	0.601466i	$-0.205416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.03843e9	+	1.79862e9i	−0.0318289	+	0.0551293i
$426$		0			0
$427$	5.24378e9	−	3.02750e9i	0.157737	−	0.0910693i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			2.33112e10i			0.675547i	0.941227	+	0.337774i	$0.109674\pi$
$431$			2.33112e10i			0.675547i	−0.941227	+	0.337774i	$0.890326\pi$
$432$		0			0
$433$	−2.72421e10			−0.774978			−0.387489	−	0.921874i	$-0.626658\pi$
$433$	−2.72421e10			−0.774978			−0.387489	+	0.921874i	$0.626658\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.58287e10	+	2.74161e10i	0.434030	+	0.751762i
$438$		0			0
$439$	−2.05703e10	−	1.18763e10i	−0.553838	−	0.319758i	0.196831	−	0.980437i	$-0.436935\pi$
$439$	−2.05703e10	−	1.18763e10i	−0.553838	−	0.319758i	−0.750668	+	0.660679i	$0.770268\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.45520e10	+	1.41751e10i	0.637489	+	0.368054i	0.783647	−	0.621207i	$-0.213357\pi$
$443$	2.45520e10	+	1.41751e10i	0.637489	+	0.368054i	−0.146158	+	0.989261i	$0.546691\pi$
$444$		0			0
$445$	−4.46138e10	−	7.72733e10i	−1.13770	−	1.97056i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.71106e10			1.40518			0.702589	−	0.711596i	$-0.252027\pi$
$449$	5.71106e10			1.40518			0.702589	+	0.711596i	$0.252027\pi$
$450$		0			0
$451$		−	4.17228e9i		−	0.100848i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.36161e11	+	7.86125e10i	−3.17692	+	1.83420i
$456$		0			0
$457$	−1.51344e10	+	2.62136e10i	−0.346977	+	0.600981i	−0.985711	−	0.168446i	$-0.946125\pi$
$457$	−1.51344e10	+	2.62136e10i	−0.346977	+	0.600981i	0.638734	+	0.769428i	$0.279458\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.81118e10	−	4.86911e10i	0.622422	−	1.07807i	−0.366611	−	0.930374i	$-0.619482\pi$
$461$	2.81118e10	−	4.86911e10i	0.622422	−	1.07807i	0.989033	−	0.147693i	$-0.0471847\pi$
$462$		0			0
$463$	1.51399e10	−	8.74103e9i	0.329457	−	0.190212i	−0.326143	−	0.945321i	$-0.605749\pi$
$463$	1.51399e10	−	8.74103e9i	0.329457	−	0.190212i	0.655600	+	0.755108i	$0.272416\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			2.50581e10i			0.526842i	0.964681	+	0.263421i	$0.0848509\pi$
$467$			2.50581e10i			0.526842i	−0.964681	+	0.263421i	$0.915149\pi$
$468$		0			0
$469$	−5.27953e10			−1.09120
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.02799e9	+	1.56369e10i	0.180363	+	0.312397i
$474$		0			0
$475$	9.08753e10	+	5.24669e10i	1.78514	+	1.03065i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.59329e10	+	3.80664e10i	1.25245	+	0.723101i	0.971595	−	0.236649i	$-0.0760492\pi$
$479$	6.59329e10	+	3.80664e10i	1.25245	+	0.723101i	0.280853	+	0.959751i	$0.409383\pi$
$480$		0			0
$481$	−3.11221e10	−	5.39051e10i	−0.581418	−	1.00705i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.06853e10			−0.735311
$486$		0			0
$487$		−	7.10799e9i		−	0.126366i	−0.998002	−	0.0631831i	$-0.979875\pi$
$487$		−	7.10799e9i		−	0.126366i	0.998002	−	0.0631831i	$-0.0201252\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.34310e10	−	1.35279e10i	0.403148	−	0.232758i	−0.284693	−	0.958619i	$-0.591892\pi$
$491$	2.34310e10	−	1.35279e10i	0.403148	−	0.232758i	0.687841	+	0.725861i	$0.258558\pi$
$492$		0			0
$493$	7.64105e8	−	1.32347e9i	0.0129350	−	0.0224040i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.58786e10	−	2.75026e10i	0.260248	−	0.450763i
$498$		0			0
$499$	6.06207e10	−	3.49994e10i	0.977730	−	0.564493i	0.0761458	−	0.997097i	$-0.475739\pi$
$499$	6.06207e10	−	3.49994e10i	0.977730	−	0.564493i	0.901584	+	0.432604i	$0.142405\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8.22545e10i		−	1.28496i	−0.766304	−	0.642478i	$-0.777907\pi$
$503$		−	8.22545e10i		−	1.28496i	0.766304	−	0.642478i	$-0.222093\pi$
$504$		0			0
$505$	1.39272e11			2.14141
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.93217e10	−	8.54277e10i	−0.734797	−	1.27271i	−0.954813	−	0.297209i	$-0.903944\pi$
$509$	−4.93217e10	−	8.54277e10i	−0.734797	−	1.27271i	0.220016	−	0.975496i	$-0.429389\pi$
$510$		0			0
$511$	−1.55068e11	−	8.95288e10i	−2.27426	−	1.31304i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.88938e10	−	2.82289e10i	−0.695065	−	0.401296i
$516$		0			0
$517$	7.31177e9	+	1.26644e10i	0.102343	+	0.177264i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.19634e11			1.62370			0.811849	−	0.583867i	$-0.198461\pi$
$521$	1.19634e11			1.62370			0.811849	+	0.583867i	$0.198461\pi$
$522$		0			0
$523$			1.42288e11i			1.90179i	0.309511	+	0.950896i	$0.399835\pi$
$523$			1.42288e11i			1.90179i	−0.309511	+	0.950896i	$0.600165\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.07429e9	+	6.20240e8i	−0.0139277	+	0.00804114i
$528$		0			0
$529$	1.44635e10	−	2.50515e10i	0.184693	−	0.319897i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.36706e10	−	2.36782e10i	0.169386	−	0.293386i
$534$		0			0
$535$	−9.53653e10	+	5.50592e10i	−1.16406	+	0.672070i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	6.74477e10i		−	0.799120i
$540$		0			0
$541$	−5.29348e10			−0.617949			−0.308974	−	0.951070i	$-0.599986\pi$
$541$	−5.29348e10			−0.617949			−0.308974	+	0.951070i	$0.599986\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.16779e10	−	1.24150e11i	−0.812455	−	1.40721i
$546$		0			0
$547$	−1.24424e11	−	7.18359e10i	−1.38980	−	0.802403i	−0.396510	−	0.918030i	$-0.629779\pi$
$547$	−1.24424e11	−	7.18359e10i	−1.38980	−	0.802403i	−0.993293	+	0.115627i	$0.963112\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.68684e10	−	3.86065e10i	−0.725461	−	0.418845i
$552$		0			0
$553$	4.67096e9	+	8.09034e9i	0.0499466	+	0.0865100i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.08728e10			0.840198			0.420099	−	0.907478i	$-0.361995\pi$
$557$	8.08728e10			0.840198			0.420099	+	0.907478i	$0.361995\pi$
$558$		0			0
$559$			1.18322e11i			1.21176i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.74793e10	−	1.58652e10i	0.273510	−	0.157911i	−0.356972	−	0.934115i	$-0.616191\pi$
$563$	2.74793e10	−	1.58652e10i	0.273510	−	0.157911i	0.630482	+	0.776204i	$0.282857\pi$
$564$		0			0
$565$	−8.56246e10	+	1.48306e11i	−0.840243	+	1.45534i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.43237e10	−	7.67708e10i	0.422850	−	0.732398i	−0.573367	−	0.819299i	$-0.694363\pi$
$569$	4.43237e10	−	7.67708e10i	0.422850	−	0.732398i	0.996217	+	0.0869008i	$0.0276963\pi$
$570$		0			0
$571$	6.40257e10	−	3.69653e10i	0.602296	−	0.347736i	−0.167648	−	0.985847i	$-0.553617\pi$
$571$	6.40257e10	−	3.69653e10i	0.602296	−	0.347736i	0.769944	+	0.638111i	$0.220284\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	3.55458e11i		−	3.25175i
$576$		0			0
$577$	6.29684e9			0.0568093			0.0284046	−	0.999597i	$-0.490957\pi$
$577$	6.29684e9			0.0568093			0.0284046	+	0.999597i	$0.490957\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.00396e11	−	1.73890e11i	−0.881070	−	1.52606i
$582$		0			0
$583$	−2.20153e10	−	1.27105e10i	−0.190568	−	0.110025i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.39287e9	+	8.04174e8i	0.0117316	+	0.00677326i	0.505854	−	0.862619i	$-0.331177\pi$
$587$	1.39287e9	+	8.04174e8i	0.0117316	+	0.00677326i	−0.494123	+	0.869392i	$0.664511\pi$
$588$		0			0
$589$	3.13377e10	+	5.42785e10i	0.260379	+	0.450990i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.51442e10			−0.365076			−0.182538	−	0.983199i	$-0.558431\pi$
$593$	−4.51442e10			−0.365076			−0.182538	+	0.983199i	$0.558431\pi$
$594$		0			0
$595$		−	1.08189e10i		−	0.0863209i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.81446e11	+	1.04758e11i	−1.40942	+	0.813727i	−0.995332	−	0.0965118i	$-0.969231\pi$
$599$	−1.81446e11	+	1.04758e11i	−1.40942	+	0.813727i	−0.414084	+	0.910239i	$0.635898\pi$
$600$		0			0
$601$	−1.05737e11	+	1.83142e11i	−0.810456	+	1.40375i	0.102089	+	0.994775i	$0.467447\pi$
$601$	−1.05737e11	+	1.83142e11i	−0.810456	+	1.40375i	−0.912545	+	0.408976i	$0.865886\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.19280e11	−	2.06599e11i	0.890318	−	1.54208i
$606$		0			0
$607$	−1.86455e11	+	1.07650e11i	−1.37347	+	0.792972i	−0.991363	−	0.131146i	$-0.958134\pi$
$607$	−1.86455e11	+	1.07650e11i	−1.37347	+	0.792972i	−0.382106	+	0.924119i	$0.624801\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			9.58289e10i			0.687593i
$612$		0			0
$613$	−8.72845e10			−0.618152			−0.309076	−	0.951037i	$-0.600020\pi$
$613$	−8.72845e10			−0.618152			−0.309076	+	0.951037i	$0.600020\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.30451e11	+	2.25948e11i	0.900136	+	1.55908i	0.827317	+	0.561736i	$0.189866\pi$
$617$	1.30451e11	+	2.25948e11i	0.900136	+	1.55908i	0.0728191	+	0.997345i	$0.476800\pi$
$618$		0			0
$619$	−1.47623e11	−	8.52304e10i	−1.00552	−	0.580540i	−0.0956464	−	0.995415i	$-0.530492\pi$
$619$	−1.47623e11	−	8.52304e10i	−1.00552	−	0.580540i	−0.909878	+	0.414875i	$0.863825\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.96011e11	+	1.70902e11i	1.96497	+	1.13448i
$624$		0			0
$625$	−3.00814e11	−	5.21024e11i	−1.97141	−	3.41459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.28313e9			0.0273627
$630$		0			0
$631$			2.63602e10i			0.166277i	0.996538	+	0.0831384i	$0.0264944\pi$
$631$			2.63602e10i			0.166277i	−0.996538	+	0.0831384i	$0.973506\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.43709e11	+	1.40706e11i	−1.49892	+	0.865400i
$636$		0			0
$637$	2.20994e11	−	3.82773e11i	1.34222	−	2.32479i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.35455e10	−	7.54231e10i	0.257936	−	0.446758i	−0.707753	−	0.706460i	$-0.750291\pi$
$641$	4.35455e10	−	7.54231e10i	0.257936	−	0.446758i	0.965689	+	0.259702i	$0.0836244\pi$
$642$		0			0
$643$	−6.39709e10	+	3.69336e10i	−0.374230	+	0.216062i	−0.675305	−	0.737539i	$-0.735988\pi$
$643$	−6.39709e10	+	3.69336e10i	−0.374230	+	0.216062i	0.301075	+	0.953601i	$0.402655\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.94284e11i		−	1.10871i	−0.832279	−	0.554357i	$-0.812964\pi$
$647$		−	1.94284e11i		−	1.10871i	0.832279	−	0.554357i	$-0.187036\pi$
$648$		0			0
$649$	−5.93292e10			−0.334418
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.18524e9	−	1.59093e10i	−0.0505170	−	0.0874980i	0.839661	−	0.543111i	$-0.182754\pi$
$653$	−9.18524e9	−	1.59093e10i	−0.0505170	−	0.0874980i	−0.890178	+	0.455613i	$0.849420\pi$
$654$		0			0
$655$	3.72733e11	+	2.15197e11i	2.02503	+	1.16915i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.85945e11	+	1.07355e11i	0.985922	+	0.569222i	0.904053	−	0.427421i	$-0.140578\pi$
$659$	1.85945e11	+	1.07355e11i	0.985922	+	0.569222i	0.0818689	+	0.996643i	$0.473911\pi$
$660$		0			0
$661$	5.00632e10	+	8.67120e10i	0.262248	+	0.454228i	0.966839	−	0.255386i	$-0.0822027\pi$
$661$	5.00632e10	+	8.67120e10i	0.262248	+	0.454228i	−0.704591	+	0.709614i	$0.748869\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.46627e11			−2.79515
$666$		0			0
$667$			2.61555e11i			1.32148i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.78077e9	−	2.76018e9i	0.0235835	−	0.0136159i
$672$		0			0
$673$	6.67551e10	−	1.15623e11i	0.325405	−	0.563618i	−0.656189	−	0.754596i	$-0.727833\pi$
$673$	6.67551e10	−	1.15623e11i	0.325405	−	0.563618i	0.981594	+	0.190978i	$0.0611660\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.25836e10	+	7.37569e10i	−0.202716	+	0.351114i	−0.949403	−	0.314062i	$-0.898310\pi$
$677$	−4.25836e10	+	7.37569e10i	−0.202716	+	0.351114i	0.746687	+	0.665176i	$0.231643\pi$
$678$		0			0
$679$	1.34973e11	−	7.79267e10i	0.634992	−	0.366613i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			3.43809e11i			1.57992i	0.613161	+	0.789958i	$0.289898\pi$
$683$			3.43809e11i			1.57992i	−0.613161	+	0.789958i	$0.710102\pi$
$684$		0			0
$685$	−5.67242e11			−2.57636
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−8.32930e10	−	1.44268e11i	−0.369600	−	0.640165i
$690$		0			0
$691$	1.21006e11	+	6.98626e10i	0.530754	+	0.306431i	0.741323	−	0.671148i	$-0.234198\pi$
$691$	1.21006e11	+	6.98626e10i	0.530754	+	0.306431i	−0.210570	+	0.977579i	$0.567532\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.26656e11	−	7.31248e10i	−0.542857	−	0.313419i
$696$		0			0
$697$	9.40697e8	+	1.62933e9i	0.00398583	+	0.00690366i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.67800e10			−0.276550			−0.138275	−	0.990394i	$-0.544156\pi$
$701$	−6.67800e10			−0.276550			−0.138275	+	0.990394i	$0.544156\pi$
$702$		0			0
$703$		−	2.16406e11i		−	0.886028i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.62034e11	+	2.66755e11i	−1.84925	+	1.06767i
$708$		0			0
$709$	6.21617e10	−	1.07667e11i	0.246002	−	0.426087i	−0.716411	−	0.697678i	$-0.754216\pi$
$709$	6.21617e10	−	1.07667e11i	0.246002	−	0.426087i	0.962413	+	0.271591i	$0.0875498\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.06155e11	−	1.83866e11i	0.410754	−	0.711447i
$714$		0			0
$715$	−1.24138e11	+	7.16713e10i	−0.474987	+	0.274234i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1.22418e11i			0.458068i	0.973418	+	0.229034i	$0.0735567\pi$
$719$			1.22418e11i			0.458068i	−0.973418	+	0.229034i	$0.926443\pi$
$720$		0			0
$721$	2.16273e11			0.800315
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.33483e11	+	7.50815e11i	1.56899	+	2.71757i
$726$		0			0
$727$	−1.80496e11	−	1.04209e11i	−0.646144	−	0.373051i	0.140834	−	0.990033i	$-0.455022\pi$
$727$	−1.80496e11	−	1.04209e11i	−0.646144	−	0.373051i	−0.786977	+	0.616982i	$0.788355\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.05112e9	−	4.07097e9i	−0.0246938	−	0.0142570i
$732$		0			0
$733$	−7.40916e10	−	1.28330e11i	−0.256657	−	0.444543i	0.708687	−	0.705523i	$-0.249288\pi$
$733$	−7.40916e10	−	1.28330e11i	−0.256657	−	0.444543i	−0.965344	+	0.260980i	$0.915954\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.81337e10			−0.163147
$738$		0			0
$739$		−	5.18644e11i		−	1.73897i	−0.493962	−	0.869484i	$-0.664452\pi$
$739$		−	5.18644e11i		−	1.73897i	0.493962	−	0.869484i	$-0.335548\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.12933e11	−	1.80672e11i	1.02682	−	0.592837i	0.110751	−	0.993848i	$-0.464674\pi$
$743$	3.12933e11	−	1.80672e11i	1.02682	−	0.592837i	0.916073	+	0.401011i	$0.131341\pi$
$744$		0			0
$745$	−1.02659e11	+	1.77811e11i	−0.333253	+	0.577211i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.10915e11	−	3.65316e11i	0.670164	−	1.16076i
$750$		0			0
$751$	1.61051e11	−	9.29826e10i	0.506294	−	0.292309i	−0.225015	−	0.974355i	$-0.572243\pi$
$751$	1.61051e11	−	9.29826e10i	0.506294	−	0.292309i	0.731309	+	0.682046i	$0.238910\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	3.03748e11i		−	0.934815i
$756$		0			0
$757$	3.70073e11			1.12695			0.563475	−	0.826133i	$-0.309464\pi$
$757$	3.70073e11			1.12695			0.563475	+	0.826133i	$0.309464\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.57796e11	+	2.73311e11i	0.470497	+	0.814926i	0.999431	−	0.0337379i	$-0.0107411\pi$
$761$	1.57796e11	+	2.73311e11i	0.470497	+	0.814926i	−0.528933	+	0.848663i	$0.677408\pi$
$762$		0			0
$763$	4.75581e11	+	2.74577e11i	1.40322	+	0.810151i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.36700e11	−	1.94394e11i	−0.972885	−	0.561696i
$768$		0			0
$769$	1.67469e11	+	2.90065e11i	0.478883	+	0.829450i	0.999707	−	0.0242141i	$-0.00770832\pi$
$769$	1.67469e11	+	2.90065e11i	0.478883	+	0.829450i	−0.520823	+	0.853664i	$0.674375\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.84555e10			0.219738			0.109869	−	0.993946i	$-0.464957\pi$
$773$	7.84555e10			0.219738			0.109869	+	0.993946i	$0.464957\pi$
$774$		0			0
$775$		−	7.03735e11i		−	1.95075i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.23223e10	−	4.75288e10i	0.223546	−	0.129065i
$780$		0			0
$781$	1.44766e10	−	2.50742e10i	0.0389101	−	0.0673943i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.77340e10	+	1.69280e11i	−0.257375	+	0.445787i
$786$		0			0
$787$	−4.65348e11	+	2.68669e11i	−1.21305	+	0.700354i	−0.963422	−	0.267988i	$-0.913641\pi$
$787$	−4.65348e11	+	2.68669e11i	−1.21305	+	0.700354i	−0.249627	+	0.968342i	$0.580308\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	6.56005e11i		−	1.67572i
$792$		0			0
$793$	3.61752e10			0.0914784
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.41509e11	+	2.45100e11i	0.350711	+	0.607449i	0.986374	−	0.164517i	$-0.0526066\pi$
$797$	1.41509e11	+	2.45100e11i	0.350711	+	0.607449i	−0.635663	+	0.771967i	$0.719273\pi$
$798$		0			0
$799$	−5.71070e9	−	3.29708e9i	−0.0140121	−	0.00808988i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.41376e11	−	8.16237e10i	−0.340028	−	0.196315i
$804$		0			0
$805$	9.25836e11	+	1.60359e12i	2.20470	+	3.81866i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.37475e11			−0.554401			−0.277200	−	0.960812i	$-0.589407\pi$
$809$	−2.37475e11			−0.554401			−0.277200	+	0.960812i	$0.589407\pi$
$810$		0			0
$811$		−	1.65197e11i		−	0.381874i	−0.981602	−	0.190937i	$-0.938847\pi$
$811$		−	1.65197e11i		−	0.381874i	0.981602	−	0.190937i	$-0.0611526\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.63046e10	−	2.09605e10i	0.0822871	−	0.0475085i
$816$		0			0
$817$	−2.05686e11	+	3.56259e11i	−0.461654	+	0.799608i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.24341e11	−	3.88571e11i	0.493784	−	0.855258i	−0.506191	−	0.862422i	$-0.668947\pi$
$821$	2.24341e11	−	3.88571e11i	0.493784	−	0.855258i	0.999974	+	0.00716314i	$0.00228012\pi$
$822$		0			0
$823$	2.61680e11	−	1.51081e11i	0.570389	−	0.329315i	−0.186915	−	0.982376i	$-0.559849\pi$
$823$	2.61680e11	−	1.51081e11i	0.570389	−	0.329315i	0.757305	+	0.653062i	$0.226516\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	3.50332e11i		−	0.748959i	−0.927235	−	0.374479i	$-0.877821\pi$
$827$		−	3.50332e11i		−	0.748959i	0.927235	−	0.374479i	$-0.122179\pi$
$828$		0			0
$829$	−3.42927e11			−0.726078			−0.363039	−	0.931774i	$-0.618261\pi$
$829$	−3.42927e11			−0.726078			−0.363039	+	0.931774i	$0.618261\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.52070e10	+	2.63393e10i	0.0315837	+	0.0547046i
$834$		0			0
$835$	3.16731e11	+	1.82865e11i	0.651546	+	0.376170i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.97330e11	+	1.71664e11i	0.600055	+	0.346442i	0.769063	−	0.639173i	$-0.220723\pi$
$839$	2.97330e11	+	1.71664e11i	0.600055	+	0.346442i	−0.169008	+	0.985615i	$0.554057\pi$
$840$		0			0
$841$	−6.88451e10	−	1.19243e11i	−0.137622	−	0.238369i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.17335e10			0.101472
$846$		0			0
$847$			9.13851e11i			1.77559i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.34851e11	+	3.66532e11i	−1.21047	+	0.698865i
$852$		0			0
$853$	3.38710e11	−	5.86662e11i	0.639781	−	1.10813i	−0.345699	−	0.938345i	$-0.612358\pi$
$853$	3.38710e11	−	5.86662e11i	0.639781	−	1.10813i	0.985481	−	0.169788i	$-0.0543084\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.20591e11	+	7.28485e11i	−0.779717	+	1.35051i	0.152388	+	0.988321i	$0.451304\pi$
$857$	−4.20591e11	+	7.28485e11i	−0.779717	+	1.35051i	−0.932105	+	0.362188i	$0.882030\pi$
$858$		0			0
$859$	8.51944e11	−	4.91870e11i	1.56473	−	0.903396i	0.567960	−	0.823056i	$-0.307733\pi$
$859$	8.51944e11	−	4.91870e11i	1.56473	−	0.903396i	0.996768	−	0.0803391i	$-0.0256003\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	6.71168e11i		−	1.21001i	−0.796223	−	0.605004i	$-0.793172\pi$
$863$		−	6.71168e11i		−	1.21001i	0.796223	−	0.605004i	$-0.206828\pi$
$864$		0			0
$865$	7.58688e11			1.35519
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.25853e9	+	7.37599e9i	0.00746759	+	0.0129342i
$870$		0			0
$871$	−2.73164e11	−	1.57711e11i	−0.474626	−	0.274025i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.40253e12	+	1.96445e12i	5.80456	+	3.35127i
$876$		0			0
$877$	−5.17243e10	−	8.95892e10i	−0.0874372	−	0.151446i	0.818990	−	0.573808i	$-0.194534\pi$
$877$	−5.17243e10	−	8.95892e10i	−0.0874372	−	0.151446i	−0.906427	+	0.422362i	$0.861201\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.24789e11			0.207144			0.103572	−	0.994622i	$-0.466973\pi$
$881$	1.24789e11			0.207144			0.103572	+	0.994622i	$0.466973\pi$
$882$		0			0
$883$		−	9.40863e11i		−	1.54769i	−0.633376	−	0.773844i	$-0.718331\pi$
$883$		−	9.40863e11i		−	1.54769i	0.633376	−	0.773844i	$-0.281669\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.79707e11	+	5.07899e11i	−1.42116	+	0.820508i	−0.996398	−	0.0847958i	$-0.972976\pi$
$887$	−8.79707e11	+	5.07899e11i	−1.42116	+	0.820508i	−0.424764	+	0.905304i	$0.639643\pi$
$888$		0			0
$889$	5.39002e11	−	9.33579e11i	0.862945	−	1.49466i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.66585e11	+	2.88534e11i	−0.261957	+	0.453723i
$894$		0			0
$895$	−1.38381e12	+	7.98941e11i	−2.15667	+	1.24515i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			5.17827e11i			0.792767i
$900$		0			0
$901$	1.14631e10			0.0173941
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.03373e11	+	1.04507e12i	0.899480	+	1.55795i
$906$		0			0
$907$	7.32476e10	+	4.22895e10i	0.108234	+	0.0624890i	0.553140	−	0.833088i	$-0.313430\pi$
$907$	7.32476e10	+	4.22895e10i	0.108234	+	0.0624890i	−0.444906	+	0.895577i	$0.646763\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7.32487e11	+	4.22902e11i	1.06347	+	0.613997i	0.926391	−	0.376563i	$-0.122894\pi$
$911$	7.32487e11	+	4.22902e11i	1.06347	+	0.613997i	0.137082	+	0.990560i	$0.456228\pi$
$912$		0			0
$913$	−9.15310e10	−	1.58536e11i	−0.131730	−	0.228163i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.64871e12			−2.33168
$918$		0			0
$919$		−	4.67279e11i		−	0.655110i	−0.944832	−	0.327555i	$-0.893775\pi$
$919$		−	4.67279e11i		−	0.655110i	0.944832	−	0.327555i	$-0.106225\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.64313e11	−	9.48660e10i	0.226394	−	0.130708i
$924$		0			0
$925$	−1.21493e12	+	2.10432e12i	−1.65953	+	2.87438i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.74314e11	+	4.75127e11i	−0.368287	+	0.637891i	−0.989298	−	0.145911i	$-0.953389\pi$
$929$	−2.74314e11	+	4.75127e11i	−0.368287	+	0.637891i	0.621011	+	0.783802i	$0.286722\pi$
$930$		0			0
$931$	1.33080e12	−	7.68335e11i	1.77138	−	1.02271i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	9.86365e9i		−	0.0129060i
$936$		0			0
$937$	1.40712e12			1.82546			0.912729	−	0.408565i	$-0.133971\pi$
$937$	1.40712e12			1.82546			0.912729	+	0.408565i	$0.133971\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.57797e11	−	1.31254e12i	−0.966483	−	1.67400i	−0.705577	−	0.708633i	$-0.749312\pi$
$941$	−7.57797e11	−	1.31254e12i	−0.966483	−	1.67400i	−0.260906	−	0.965364i	$-0.584021\pi$
$942$		0			0
$943$	−2.78863e11	−	1.61001e11i	−0.352650	−	0.203602i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.84092e11	−	1.64021e11i	−0.353231	−	0.203938i	0.312876	−	0.949794i	$-0.398708\pi$
$947$	−2.84092e11	−	1.64021e11i	−0.353231	−	0.203938i	−0.666108	+	0.745856i	$0.732041\pi$
$948$		0			0
$949$	−5.34885e11	−	9.26448e11i	−0.659471	−	1.14224i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.41370e11			−0.292625			−0.146312	−	0.989238i	$-0.546741\pi$
$953$	−2.41370e11			−0.292625			−0.146312	+	0.989238i	$0.546741\pi$
$954$		0			0
$955$			2.52481e12i			3.03540i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.88182e12	−	1.08647e12i	2.22486	−	1.28452i
$960$		0			0
$961$	−2.16280e11	+	3.74608e11i	−0.253585	+	0.439221i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.35179e11	+	1.44657e12i	−0.963098	+	1.66813i
$966$		0			0
$967$	−7.52074e11	+	4.34210e11i	−0.860111	+	0.496585i	−0.864049	−	0.503407i	$-0.832080\pi$
$967$	−7.52074e11	+	4.34210e11i	−0.860111	+	0.496585i	0.00393849	+	0.999992i	$0.498746\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1.50474e12i		−	1.69272i	−0.532611	−	0.846360i	$-0.678789\pi$
$971$		−	1.50474e12i		−	1.69272i	0.532611	−	0.846360i	$-0.321211\pi$
$972$		0			0
$973$	5.60238e11			0.625060
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.72174e11	+	8.17830e11i	0.518232	+	0.897603i	0.999776	+	0.0211816i	$0.00674281\pi$
$977$	4.72174e11	+	8.17830e11i	0.518232	+	0.897603i	−0.481544	+	0.876422i	$0.659924\pi$
$978$		0			0
$979$	2.69874e11	+	1.55812e11i	0.293786	+	0.169617i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.01487e12	−	5.85933e11i	−1.08691	−	0.627529i	−0.154159	−	0.988046i	$-0.549267\pi$
$983$	−1.01487e12	−	5.85933e11i	−1.08691	−	0.627529i	−0.932753	+	0.360517i	$0.882600\pi$
$984$		0			0
$985$	3.26387e11	+	5.65319e11i	0.346727	+	0.600549i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.39350e12			1.45654
$990$		0			0
$991$		−	3.52592e11i		−	0.365576i	−0.983152	−	0.182788i	$-0.941488\pi$
$991$		−	3.52592e11i		−	0.365576i	0.983152	−	0.182788i	$-0.0585122\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.27409e12	+	7.35595e11i	−1.29989	+	0.750493i
$996$		0			0
$997$	1.11469e11	−	1.93071e11i	0.112817	−	0.195405i	−0.804088	−	0.594510i	$-0.797346\pi$
$997$	1.11469e11	−	1.93071e11i	0.112817	−	0.195405i	0.916905	+	0.399105i	$0.130679\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.9.o.a.127.16		32
3.2	odd	2		144.9.o.c.79.16	yes	32
4.3	odd	2		432.9.o.b.127.16		32
9.4	even	3		432.9.o.b.415.16		32
9.5	odd	6		144.9.o.a.31.1	✓	32
12.11	even	2		144.9.o.a.79.1	yes	32
36.23	even	6		144.9.o.c.31.16	yes	32
36.31	odd	6	inner	432.9.o.a.415.16		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.9.o.a.31.1	✓	32	9.5	odd	6
144.9.o.a.79.1	yes	32	12.11	even	2
144.9.o.c.31.16	yes	32	36.23	even	6
144.9.o.c.79.16	yes	32	3.2	odd	2
432.9.o.a.127.16		32	1.1	even	1	trivial
432.9.o.a.415.16		32	36.31	odd	6	inner
432.9.o.b.127.16		32	4.3	odd	2
432.9.o.b.415.16		32	9.4	even	3

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	432.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(607.471 + 1052.17i) q^{5} +(-4030.55 - 2327.04i) q^{7} +(-3674.67 - 2121.57i) q^{11} +(-13902.8 - 24080.3i) q^{13} +1913.35 q^{17} -96672.2i q^{19} +(-283599. + 163736. i) q^{23} +(-542730. + 940036. i) q^{25} +(399355. - 691703. i) q^{29} +(-561470. + 324165. i) q^{31} -5.65444e6i q^{35} +2.23855e6 q^{37} +(491649. + 851562. i) q^{41} +(-3.68522e6 - 2.12767e6i) q^{43} +(-2.98466e6 - 1.72320e6i) q^{47} +(7.94784e6 + 1.37661e7i) q^{49} +5.99110e6 q^{53} -5.15517e6i q^{55} +(1.21091e7 - 6.99118e6i) q^{59} +(-650503. + 1.12670e6i) q^{61} +(1.68911e7 - 2.92562e7i) q^{65} +(9.82409e6 - 5.67194e6i) q^{67} +6.82352e6i q^{71} +3.84732e7 q^{73} +(9.87397e6 + 1.71022e7i) q^{77} +(-1.73833e6 - 1.00363e6i) q^{79} +(3.73630e7 + 2.15715e7i) q^{83} +(1.16230e6 + 2.01317e6i) q^{85} -7.34418e7 q^{89} +1.29409e8i q^{91} +(1.01716e8 - 5.87255e7i) q^{95} +(-1.67437e7 + 2.90010e7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 147 q^{5} - 2769 q^{7} - 17082 q^{11} + 1685 q^{13} + 6402 q^{17} + 4941 q^{23} - 1080803 q^{25} + 639219 q^{29} + 920745 q^{31} - 2765636 q^{37} + 2229390 q^{41} + 1788381 q^{47} + 13397285 q^{49}+ \cdots + 74963282 q^{97}+O(q^{100}) \) 32 * q - 147 * q^5 - 2769 * q^7 - 17082 * q^11 + 1685 * q^13 + 6402 * q^17 + 4941 * q^23 - 1080803 * q^25 + 639219 * q^29 + 920745 * q^31 - 2765636 * q^37 + 2229390 * q^41 + 1788381 * q^47 + 13397285 * q^49 + 16055532 * q^53 + 42176466 * q^59 - 6097553 * q^61 - 14325141 * q^65 + 1476300 * q^67 + 44269442 * q^73 + 22487769 * q^77 + 65644341 * q^79 - 67089393 * q^83 + 28601250 * q^85 - 133904424 * q^89 - 7225380 * q^95 + 74963282 * q^97

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