LMFDB - Embedded newform 432.9.e.l.161.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,9,Mod(161,432)] 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("432.161"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	432.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-456] 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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 6x^{6} + 264x^{5} - 1872x^{4} + 360x^{3} + 12062x^{2} - 106892x + 822753 $ x^8 - 4x^7 + 6x^6 + 264x^5 - 1872x^4 + 360x^3 + 12062x^2 - 106892*x + 822753
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{47}\cdot 3^{12}\cdot 19^{2} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.5
Root			$-5.69537 - 0.664009i$ of defining polynomial
Character	$\chi$	$=$	432.161
Dual form			432.9.e.l.161.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+383.685i q^{5} +1056.61 q^{7} +21019.0i q^{11} -44547.1 q^{13} -49959.5i q^{17} -195658. q^{19} -443798. i q^{23} +243411. q^{25} -876037. i q^{29} -131242. q^{31} +405405. i q^{35} +1.91241e6 q^{37} +1.42112e6i q^{41} +5.05657e6 q^{43} -5.53056e6i q^{47} -4.64838e6 q^{49} +4.70100e6i q^{53} -8.06468e6 q^{55} +1.25330e7i q^{59} +1.18239e7 q^{61} -1.70921e7i q^{65} +1.42031e7 q^{67} -3.83376e7i q^{71} -4.03426e7 q^{73} +2.22088e7i q^{77} -2.04301e7 q^{79} -4.39231e7i q^{83} +1.91687e7 q^{85} +2.15810e7i q^{89} -4.70688e7 q^{91} -7.50712e7i q^{95} +6.36854e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 456 q^{7} - 75016 q^{13} + 132536 q^{19} - 1187768 q^{25} + 1301072 q^{31} - 2559432 q^{37} + 2958544 q^{43} + 18410368 q^{49} - 10936256 q^{55} + 20663672 q^{61} - 59168392 q^{67} - 98757640 q^{73}+ \cdots - 103615304 q^{97}+O(q^{100}) $ 8 * q - 456 * q^7 - 75016 * q^13 + 132536 * q^19 - 1187768 * q^25 + 1301072 * q^31 - 2559432 * q^37 + 2958544 * q^43 + 18410368 * q^49 - 10936256 * q^55 + 20663672 * q^61 - 59168392 * q^67 - 98757640 * q^73 + 107248888 * q^79 + 151752256 * q^85 + 263862984 * q^91 - 103615304 * q^97

Character values

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

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	383.685i	0.613896i	0.951726	+	0.306948i	$0.0993078\pi$
$5$	383.685i	0.613896i	−0.951726	+	0.306948i	$0.900692\pi$
$6$	0	0
$7$	1056.61	0.440070	0.220035	−	0.975492i	$-0.429383\pi$
$7$	1056.61	0.440070	0.220035	+	0.975492i	$0.429383\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	21019.0i	1.43563i	0.696236	+	0.717813i	$0.254857\pi$
$11$	21019.0i	1.43563i	−0.696236	+	0.717813i	$0.745143\pi$
$12$	0	0
$13$	−44547.1	−1.55972	−0.779859	−	0.625955i	$-0.784709\pi$
$13$	−44547.1	−1.55972	−0.779859	+	0.625955i	$0.784709\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 49959.5i	− 0.598167i	−0.954227	−	0.299083i	$-0.903319\pi$
$17$	− 49959.5i	− 0.598167i	0.954227	−	0.299083i	$-0.0966809\pi$
$18$	0	0
$19$	−195658.	−1.50136	−0.750679	−	0.660667i	$-0.770274\pi$
$19$	−195658.	−1.50136	−0.750679	+	0.660667i	$0.770274\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 443798.i	− 1.58589i	−0.609290	−	0.792947i	$-0.708545\pi$
$23$	− 443798.i	− 1.58589i	0.609290	−	0.792947i	$-0.291455\pi$
$24$	0	0
$25$	243411.	0.623132
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 876037.i	− 1.23860i	−0.785155	−	0.619299i	$-0.787417\pi$
$29$	− 876037.i	− 1.23860i	0.785155	−	0.619299i	$-0.212583\pi$
$30$	0	0
$31$	−131242.	−0.142110	−0.0710552	−	0.997472i	$-0.522637\pi$
$31$	−131242.	−0.142110	−0.0710552	+	0.997472i	$0.522637\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	405405.i	0.270157i
$36$	0	0
$37$	1.91241e6	1.02041	0.510203	−	0.860054i	$-0.329570\pi$
$37$	1.91241e6	1.02041	0.510203	+	0.860054i	$0.329570\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.42112e6i	0.502916i	0.967868	+	0.251458i	$0.0809100\pi$
$41$	1.42112e6i	0.502916i	−0.967868	+	0.251458i	$0.919090\pi$
$42$	0	0
$43$	5.05657e6	1.47905	0.739524	−	0.673130i	$-0.235050\pi$
$43$	5.05657e6	1.47905	0.739524	+	0.673130i	$0.235050\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.53056e6i	− 1.13339i	−0.823929	−	0.566693i	$-0.808223\pi$
$47$	− 5.53056e6i	− 1.13339i	0.823929	−	0.566693i	$-0.191777\pi$
$48$	0	0
$49$	−4.64838e6	−0.806338
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.70100e6i	0.595781i	0.954600	+	0.297890i	$0.0962830\pi$
$53$	4.70100e6i	0.595781i	−0.954600	+	0.297890i	$0.903717\pi$
$54$	0	0
$55$	−8.06468e6	−0.881325
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.25330e7i	1.03430i	0.855894	+	0.517151i	$0.173007\pi$
$59$	1.25330e7i	1.03430i	−0.855894	+	0.517151i	$0.826993\pi$
$60$	0	0
$61$	1.18239e7	0.853964	0.426982	−	0.904260i	$-0.359577\pi$
$61$	1.18239e7	0.853964	0.426982	+	0.904260i	$0.359577\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.70921e7i	− 0.957505i
$66$	0	0
$67$	1.42031e7	0.704828	0.352414	−	0.935844i	$-0.385361\pi$
$67$	1.42031e7	0.704828	0.352414	+	0.935844i	$0.385361\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 3.83376e7i	− 1.50866i	−0.656496	−	0.754329i	$-0.727962\pi$
$71$	− 3.83376e7i	− 1.50866i	0.656496	−	0.754329i	$-0.272038\pi$
$72$	0	0
$73$	−4.03426e7	−1.42060	−0.710301	−	0.703898i	$-0.751441\pi$
$73$	−4.03426e7	−1.42060	−0.710301	+	0.703898i	$0.751441\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.22088e7i	0.631776i
$78$	0	0
$79$	−2.04301e7	−0.524519	−0.262259	−	0.964997i	$-0.584468\pi$
$79$	−2.04301e7	−0.524519	−0.262259	+	0.964997i	$0.584468\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.39231e7i	− 0.925509i	−0.886487	−	0.462754i	$-0.846861\pi$
$83$	− 4.39231e7i	− 0.925509i	0.886487	−	0.462754i	$-0.153139\pi$
$84$	0	0
$85$	1.91687e7	0.367212
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.15810e7i	0.343963i	0.985100	+	0.171982i	$0.0550169\pi$
$89$	2.15810e7i	0.343963i	−0.985100	+	0.171982i	$0.944983\pi$
$90$	0	0
$91$	−4.70688e7	−0.686385
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 7.50712e7i	− 0.921678i
$96$	0	0
$97$	6.36854e7	0.719371	0.359685	−	0.933074i	$-0.382884\pi$
$97$	6.36854e7	0.719371	0.359685	+	0.933074i	$0.382884\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.10700e8i	1.06380i	0.846806	+	0.531901i	$0.178522\pi$
$101$	1.10700e8i	1.06380i	−0.846806	+	0.531901i	$0.821478\pi$
$102$	0	0
$103$	5.92788e7	0.526685	0.263342	−	0.964702i	$-0.415175\pi$
$103$	5.92788e7	0.526685	0.263342	+	0.964702i	$0.415175\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.74525e7i	0.590882i	0.955361	+	0.295441i	$0.0954665\pi$
$107$	7.74525e7i	0.590882i	−0.955361	+	0.295441i	$0.904534\pi$
$108$	0	0
$109$	1.64242e8	1.16353	0.581766	−	0.813356i	$-0.302362\pi$
$109$	1.64242e8	1.16353	0.581766	+	0.813356i	$0.302362\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.57156e8i	1.57719i	0.614916	+	0.788593i	$0.289190\pi$
$113$	2.57156e8i	1.57719i	−0.614916	+	0.788593i	$0.710810\pi$
$114$	0	0
$115$	1.70279e8	0.973575
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 5.27876e7i	− 0.263235i
$120$	0	0
$121$	−2.27440e8	−1.06102
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.43270e8i	0.996434i
$126$	0	0
$127$	1.54595e8	0.594267	0.297133	−	0.954836i	$-0.403969\pi$
$127$	1.54595e8	0.594267	0.297133	+	0.954836i	$0.403969\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.85589e8i	− 0.630184i	−0.949061	−	0.315092i	$-0.897965\pi$
$131$	− 1.85589e8i	− 0.630184i	0.949061	−	0.315092i	$-0.102035\pi$
$132$	0	0
$133$	−2.06734e8	−0.660703
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.58751e8i	1.01838i	0.860654	+	0.509191i	$0.170055\pi$
$137$	3.58751e8i	1.01838i	−0.860654	+	0.509191i	$0.829945\pi$
$138$	0	0
$139$	−2.76495e8	−0.740674	−0.370337	−	0.928897i	$-0.620758\pi$
$139$	−2.76495e8	−0.740674	−0.370337	+	0.928897i	$0.620758\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 9.36336e8i	− 2.23917i
$144$	0	0
$145$	3.36122e8	0.760371
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.98480e7i	0.0605578i	0.999541	+	0.0302789i	$0.00963955\pi$
$149$	2.98480e7i	0.0605578i	−0.999541	+	0.0302789i	$0.990360\pi$
$150$	0	0
$151$	−1.28363e8	−0.246906	−0.123453	−	0.992350i	$-0.539397\pi$
$151$	−1.28363e8	−0.246906	−0.123453	+	0.992350i	$0.539397\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 5.03556e7i	− 0.0872410i
$156$	0	0
$157$	5.83413e8	0.960236	0.480118	−	0.877204i	$-0.340594\pi$
$157$	5.83413e8	0.960236	0.480118	+	0.877204i	$0.340594\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.68921e8i	− 0.697905i
$162$	0	0
$163$	1.06572e9	1.50971	0.754854	−	0.655892i	$-0.227707\pi$
$163$	1.06572e9	1.50971	0.754854	+	0.655892i	$0.227707\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.65775e8i	− 0.855976i	−0.903784	−	0.427988i	$-0.859223\pi$
$167$	− 6.65775e8i	− 0.855976i	0.903784	−	0.427988i	$-0.140777\pi$
$168$	0	0
$169$	1.16871e9	1.43272
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.56305e9i	1.74497i	0.488640	+	0.872485i	$0.337493\pi$
$173$	1.56305e9i	1.74497i	−0.488640	+	0.872485i	$0.662507\pi$
$174$	0	0
$175$	2.57190e8	0.274222
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.55656e9i	− 1.51619i	−0.652143	−	0.758096i	$-0.726130\pi$
$179$	− 1.55656e9i	− 1.51619i	0.652143	−	0.758096i	$-0.273870\pi$
$180$	0	0
$181$	−9.14872e8	−0.852405	−0.426202	−	0.904628i	$-0.640149\pi$
$181$	−9.14872e8	−0.852405	−0.426202	+	0.904628i	$0.640149\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.33762e8i	0.626424i
$186$	0	0
$187$	1.05010e9	0.858744
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.62389e9i	1.22018i	0.792332	+	0.610090i	$0.208867\pi$
$191$	1.62389e9i	1.22018i	−0.792332	+	0.610090i	$0.791133\pi$
$192$	0	0
$193$	1.41342e9	1.01869	0.509347	−	0.860561i	$-0.329887\pi$
$193$	1.41342e9	1.01869	0.509347	+	0.860561i	$0.329887\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.77963e7i	− 0.0250948i	−0.999921	−	0.0125474i	$-0.996006\pi$
$197$	− 3.77963e7i	− 0.0250948i	0.999921	−	0.0125474i	$-0.00399407\pi$
$198$	0	0
$199$	−1.66396e9	−1.06104	−0.530518	−	0.847673i	$-0.678003\pi$
$199$	−1.66396e9	−1.06104	−0.530518	+	0.847673i	$0.678003\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 9.25628e8i	− 0.545070i
$204$	0	0
$205$	−5.45262e8	−0.308738
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.11255e9i	− 2.15539i
$210$	0	0
$211$	1.40054e9	0.706586	0.353293	−	0.935513i	$-0.385062\pi$
$211$	1.40054e9	0.706586	0.353293	+	0.935513i	$0.385062\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.94013e9i	0.907982i
$216$	0	0
$217$	−1.38671e8	−0.0625385
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.22555e9i	0.932972i
$222$	0	0
$223$	1.61380e9	0.652574	0.326287	−	0.945271i	$-0.394202\pi$
$223$	1.61380e9	0.652574	0.326287	+	0.945271i	$0.394202\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 9.59192e8i	− 0.361245i	−0.983552	−	0.180623i	$-0.942189\pi$
$227$	− 9.59192e8i	− 0.361245i	0.983552	−	0.180623i	$-0.0578112\pi$
$228$	0	0
$229$	2.43698e9	0.886156	0.443078	−	0.896483i	$-0.353887\pi$
$229$	2.43698e9	0.886156	0.443078	+	0.896483i	$0.353887\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.84126e9i	1.30332i	0.758512	+	0.651659i	$0.225926\pi$
$233$	3.84126e9i	1.30332i	−0.758512	+	0.651659i	$0.774074\pi$
$234$	0	0
$235$	2.12199e9	0.695781
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.11611e9i	0.648555i	0.945962	+	0.324278i	$0.105121\pi$
$239$	2.11611e9i	0.648555i	−0.945962	+	0.324278i	$0.894879\pi$
$240$	0	0
$241$	2.38434e9	0.706804	0.353402	−	0.935471i	$-0.385025\pi$
$241$	2.38434e9	0.706804	0.353402	+	0.935471i	$0.385025\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.78351e9i	− 0.495008i
$246$	0	0
$247$	8.71602e9	2.34170
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.77080e8i	0.220976i	0.993877	+	0.110488i	$0.0352413\pi$
$251$	8.77080e8i	0.220976i	−0.993877	+	0.110488i	$0.964759\pi$
$252$	0	0
$253$	9.32820e9	2.27675
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.09444e9i	− 1.39702i	−0.715603	−	0.698508i	$-0.753848\pi$
$257$	− 6.09444e9i	− 1.39702i	0.715603	−	0.698508i	$-0.246152\pi$
$258$	0	0
$259$	2.02066e9	0.449050
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.15154e9i	0.658719i	0.944205	+	0.329360i	$0.106833\pi$
$263$	3.15154e9i	0.658719i	−0.944205	+	0.329360i	$0.893167\pi$
$264$	0	0
$265$	−1.80370e9	−0.365747
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 8.90106e9i	− 1.69994i	−0.526835	−	0.849968i	$-0.676621\pi$
$269$	− 8.90106e9i	− 1.69994i	0.526835	−	0.849968i	$-0.323379\pi$
$270$	0	0
$271$	4.11284e9	0.762543	0.381271	−	0.924463i	$-0.375486\pi$
$271$	4.11284e9	0.762543	0.381271	+	0.924463i	$0.375486\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.11625e9i	0.894584i
$276$	0	0
$277$	5.38125e9	0.914038	0.457019	−	0.889457i	$-0.348917\pi$
$277$	5.38125e9	0.914038	0.457019	+	0.889457i	$0.348917\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 8.09671e9i	− 1.29862i	−0.760522	−	0.649312i	$-0.775057\pi$
$281$	− 8.09671e9i	− 1.29862i	0.760522	−	0.649312i	$-0.224943\pi$
$282$	0	0
$283$	−9.74985e9	−1.52003	−0.760015	−	0.649905i	$-0.774809\pi$
$283$	−9.74985e9	−1.52003	−0.760015	+	0.649905i	$0.774809\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.50157e9i	0.221318i
$288$	0	0
$289$	4.47981e9	0.642196
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.31046e10i	1.77809i	0.457815	+	0.889047i	$0.348632\pi$
$293$	1.31046e10i	1.77809i	−0.457815	+	0.889047i	$0.651368\pi$
$294$	0	0
$295$	−4.80873e9	−0.634954
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.97699e10i	2.47355i
$300$	0	0
$301$	5.34282e9	0.650885
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.53664e9i	0.524245i
$306$	0	0
$307$	1.40665e10	1.58355	0.791775	−	0.610812i	$-0.209157\pi$
$307$	1.40665e10	1.58355	0.791775	+	0.610812i	$0.209157\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 1.05439e10i	− 1.12709i	−0.826086	−	0.563544i	$-0.809438\pi$
$311$	− 1.05439e10i	− 1.12709i	0.826086	−	0.563544i	$-0.190562\pi$
$312$	0	0
$313$	−7.59473e9	−0.791288	−0.395644	−	0.918404i	$-0.629479\pi$
$313$	−7.59473e9	−0.791288	−0.395644	+	0.918404i	$0.629479\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.29322e10i	− 1.28067i	−0.768098	−	0.640333i	$-0.778797\pi$
$317$	− 1.29322e10i	− 1.28067i	0.768098	−	0.640333i	$-0.221203\pi$
$318$	0	0
$319$	1.84134e10	1.77816
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.77500e9i	0.898063i
$324$	0	0
$325$	−1.08432e10	−0.971910
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 5.84363e9i	− 0.498769i
$330$	0	0
$331$	6.48342e9	0.540123	0.270061	−	0.962843i	$-0.412956\pi$
$331$	6.48342e9	0.540123	0.270061	+	0.962843i	$0.412956\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.44950e9i	0.432691i
$336$	0	0
$337$	1.40148e10	1.08660	0.543298	−	0.839540i	$-0.317175\pi$
$337$	1.40148e10	1.08660	0.543298	+	0.839540i	$0.317175\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.75857e9i	− 0.204017i
$342$	0	0
$343$	−1.10027e10	−0.794915
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.16873e10i	− 1.49585i	−0.663785	−	0.747924i	$-0.731051\pi$
$347$	− 2.16873e10i	− 1.49585i	0.663785	−	0.747924i	$-0.268949\pi$
$348$	0	0
$349$	2.13901e10	1.44182	0.720911	−	0.693027i	$-0.243724\pi$
$349$	2.13901e10	1.44182	0.720911	+	0.693027i	$0.243724\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.47496e10i	− 1.59393i	−0.604026	−	0.796964i	$-0.706438\pi$
$353$	− 2.47496e10i	− 1.59393i	0.604026	−	0.796964i	$-0.293562\pi$
$354$	0	0
$355$	1.47095e10	0.926160
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 2.03107e10i	− 1.22278i	−0.791331	−	0.611388i	$-0.790612\pi$
$359$	− 2.03107e10i	− 1.22278i	0.791331	−	0.611388i	$-0.209388\pi$
$360$	0	0
$361$	2.12987e10	1.25408
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1.54788e10i	− 0.872102i
$366$	0	0
$367$	2.43749e10	1.34363	0.671813	−	0.740721i	$-0.265516\pi$
$367$	2.43749e10	1.34363	0.671813	+	0.740721i	$0.265516\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.96711e9i	0.262185i
$372$	0	0
$373$	3.24806e9	0.167799	0.0838994	−	0.996474i	$-0.473263\pi$
$373$	3.24806e9	0.167799	0.0838994	+	0.996474i	$0.473263\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.90249e10i	1.93186i
$378$	0	0
$379$	−1.53147e10	−0.742251	−0.371125	−	0.928583i	$-0.621028\pi$
$379$	−1.53147e10	−0.742251	−0.371125	+	0.928583i	$0.621028\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.17685e9i	0.194113i	0.995279	+	0.0970563i	$0.0309427\pi$
$383$	4.17685e9i	0.194113i	−0.995279	+	0.0970563i	$0.969057\pi$
$384$	0	0
$385$	−8.52120e9	−0.387845
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.11838e9i	0.223529i	0.993735	+	0.111765i	$0.0356503\pi$
$389$	5.11838e9i	0.223529i	−0.993735	+	0.111765i	$0.964350\pi$
$390$	0	0
$391$	−2.21719e10	−0.948630
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 7.83871e9i	− 0.322000i
$396$	0	0
$397$	−2.88076e10	−1.15970	−0.579849	−	0.814724i	$-0.696889\pi$
$397$	−2.88076e10	−1.15970	−0.579849	+	0.814724i	$0.696889\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.20349e10i	1.23893i	0.785025	+	0.619464i	$0.212650\pi$
$401$	3.20349e10i	1.23893i	−0.785025	+	0.619464i	$0.787350\pi$
$402$	0	0
$403$	5.84645e9	0.221652
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.01969e10i	1.46492i
$408$	0	0
$409$	3.16750e10	1.13194	0.565969	−	0.824426i	$-0.308502\pi$
$409$	3.16750e10	1.13194	0.565969	+	0.824426i	$0.308502\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.32425e10i	0.455165i
$414$	0	0
$415$	1.68526e10	0.568166
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 5.42311e10i	− 1.75951i	−0.475424	−	0.879757i	$-0.657705\pi$
$419$	− 5.42311e10i	− 1.75951i	0.475424	−	0.879757i	$-0.342295\pi$
$420$	0	0
$421$	9.30916e9	0.296334	0.148167	−	0.988962i	$-0.452663\pi$
$421$	9.30916e9	0.296334	0.148167	+	0.988962i	$0.452663\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.21607e10i	− 0.372737i
$426$	0	0
$427$	1.24932e10	0.375804
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 1.90914e10i	− 0.553260i	−0.960976	−	0.276630i	$-0.910782\pi$
$431$	− 1.90914e10i	− 0.553260i	0.960976	−	0.276630i	$-0.0892177\pi$
$432$	0	0
$433$	−2.94043e10	−0.836486	−0.418243	−	0.908335i	$-0.637354\pi$
$433$	−2.94043e10	−0.836486	−0.418243	+	0.908335i	$0.637354\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.68329e10i	2.38100i
$438$	0	0
$439$	−2.10343e10	−0.566332	−0.283166	−	0.959071i	$-0.591385\pi$
$439$	−2.10343e10	−0.566332	−0.283166	+	0.959071i	$0.591385\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.52740e10i	0.656234i	0.944637	+	0.328117i	$0.106414\pi$
$443$	2.52740e10i	0.656234i	−0.944637	+	0.328117i	$0.893586\pi$
$444$	0	0
$445$	−8.28031e9	−0.211158
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.33057e10i	0.327380i	0.986512	+	0.163690i	$0.0523397\pi$
$449$	1.33057e10i	0.327380i	−0.986512	+	0.163690i	$0.947660\pi$
$450$	0	0
$451$	−2.98705e10	−0.721999
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1.80596e10i	− 0.421369i
$456$	0	0
$457$	−2.06957e9	−0.0474477	−0.0237238	−	0.999719i	$-0.507552\pi$
$457$	−2.06957e9	−0.0474477	−0.0237238	+	0.999719i	$0.507552\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 4.44364e10i	− 0.983864i	−0.870634	−	0.491932i	$-0.836291\pi$
$461$	− 4.44364e10i	− 0.983864i	0.870634	−	0.491932i	$-0.163709\pi$
$462$	0	0
$463$	−5.02008e9	−0.109241	−0.0546207	−	0.998507i	$-0.517395\pi$
$463$	−5.02008e9	−0.109241	−0.0546207	+	0.998507i	$0.517395\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.91015e10i	− 0.822102i	−0.911612	−	0.411051i	$-0.865162\pi$
$467$	− 3.91015e10i	− 0.822102i	0.911612	−	0.411051i	$-0.134838\pi$
$468$	0	0
$469$	1.50071e10	0.310174
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.06284e11i	2.12336i
$474$	0	0
$475$	−4.76254e10	−0.935544
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 7.40399e10i	− 1.40645i	−0.710969	−	0.703224i	$-0.751743\pi$
$479$	− 7.40399e10i	− 1.40645i	0.710969	−	0.703224i	$-0.248257\pi$
$480$	0	0
$481$	−8.51922e10	−1.59155
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.44351e10i	0.441619i
$486$	0	0
$487$	2.90300e10	0.516097	0.258049	−	0.966132i	$-0.416921\pi$
$487$	2.90300e10	0.516097	0.258049	+	0.966132i	$0.416921\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 2.93492e10i	− 0.504976i	−0.967600	−	0.252488i	$-0.918751\pi$
$491$	− 2.93492e10i	− 0.504976i	0.967600	−	0.252488i	$-0.0812488\pi$
$492$	0	0
$493$	−4.37664e10	−0.740888
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.05078e10i	− 0.663916i
$498$	0	0
$499$	5.62186e10	0.906729	0.453365	−	0.891325i	$-0.350224\pi$
$499$	5.62186e10	0.906729	0.453365	+	0.891325i	$0.350224\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.85472e10i	0.289739i	0.989451	+	0.144870i	$0.0462763\pi$
$503$	1.85472e10i	0.289739i	−0.989451	+	0.144870i	$0.953724\pi$
$504$	0	0
$505$	−4.24738e10	−0.653064
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 9.62814e10i	− 1.43440i	−0.696866	−	0.717202i	$-0.745423\pi$
$509$	− 9.62814e10i	− 1.43440i	0.696866	−	0.717202i	$-0.254577\pi$
$510$	0	0
$511$	−4.26263e10	−0.625164
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.27444e10i	0.323330i
$516$	0	0
$517$	1.16247e11	1.62712
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.76457e10i	− 0.239490i	−0.992805	−	0.119745i	$-0.961792\pi$
$521$	− 1.76457e10i	− 0.239490i	0.992805	−	0.119745i	$-0.0382078\pi$
$522$	0	0
$523$	−9.05572e10	−1.21036	−0.605182	−	0.796087i	$-0.706900\pi$
$523$	−9.05572e10	−1.21036	−0.605182	+	0.796087i	$0.706900\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.55678e9i	0.0850057i
$528$	0	0
$529$	−1.18646e11	−1.51506
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 6.33068e10i	− 0.784407i
$534$	0	0
$535$	−2.97174e10	−0.362740
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 9.77043e10i	− 1.15760i
$540$	0	0
$541$	2.88132e10	0.336359	0.168180	−	0.985756i	$-0.446211\pi$
$541$	2.88132e10	0.336359	0.168180	+	0.985756i	$0.446211\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.30172e10i	0.714288i
$546$	0	0
$547$	−9.13910e10	−1.02083	−0.510416	−	0.859928i	$-0.670508\pi$
$547$	−9.13910e10	−1.02083	−0.510416	+	0.859928i	$0.670508\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.71404e11i	1.85958i
$552$	0	0
$553$	−2.15866e10	−0.230825
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.05061e10i	0.940280i	0.882592	+	0.470140i	$0.155797\pi$
$557$	9.05061e10i	0.940280i	−0.882592	+	0.470140i	$0.844203\pi$
$558$	0	0
$559$	−2.25256e11	−2.30690
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4.53503e10i	− 0.451385i	−0.974199	−	0.225692i	$-0.927536\pi$
$563$	− 4.53503e10i	− 0.451385i	0.974199	−	0.225692i	$-0.0724644\pi$
$564$	0	0
$565$	−9.86669e10	−0.968228
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.12610e11i	1.07430i	0.843485	+	0.537152i	$0.180500\pi$
$569$	1.12610e11i	1.07430i	−0.843485	+	0.537152i	$0.819500\pi$
$570$	0	0
$571$	1.59157e11	1.49721	0.748605	−	0.663017i	$-0.230724\pi$
$571$	1.59157e11	1.49721	0.748605	+	0.663017i	$0.230724\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1.08025e11i	− 0.988221i
$576$	0	0
$577$	−7.23141e10	−0.652409	−0.326205	−	0.945299i	$-0.605770\pi$
$577$	−7.23141e10	−0.652409	−0.326205	+	0.945299i	$0.605770\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.64095e10i	− 0.407289i
$582$	0	0
$583$	−9.88103e10	−0.855318
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.14664e11i	1.80804i	0.427495	+	0.904018i	$0.359396\pi$
$587$	2.14664e11i	1.80804i	−0.427495	+	0.904018i	$0.640604\pi$
$588$	0	0
$589$	2.56786e10	0.213359
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 2.24928e11i	− 1.81896i	−0.415743	−	0.909482i	$-0.636478\pi$
$593$	− 2.24928e11i	− 1.81896i	0.415743	−	0.909482i	$-0.363522\pi$
$594$	0	0
$595$	2.02538e10	0.161599
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1.40038e10i	− 0.108778i	−0.998520	−	0.0543889i	$-0.982679\pi$
$599$	− 1.40038e10i	− 0.108778i	0.998520	−	0.0543889i	$-0.0173211\pi$
$600$	0	0
$601$	−2.01335e10	−0.154320	−0.0771600	−	0.997019i	$-0.524585\pi$
$601$	−2.01335e10	−0.154320	−0.0771600	+	0.997019i	$0.524585\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 8.72652e10i	− 0.651357i
$606$	0	0
$607$	−3.32218e10	−0.244720	−0.122360	−	0.992486i	$-0.539046\pi$
$607$	−3.32218e10	−0.244720	−0.122360	+	0.992486i	$0.539046\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.46370e11i	1.76776i
$612$	0	0
$613$	−2.64787e11	−1.87523	−0.937614	−	0.347678i	$-0.886970\pi$
$613$	−2.64787e11	−1.87523	−0.937614	+	0.347678i	$0.886970\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.32856e11i	− 1.60675i	−0.595475	−	0.803374i	$-0.703036\pi$
$617$	− 2.32856e11i	− 1.60675i	0.595475	−	0.803374i	$-0.296964\pi$
$618$	0	0
$619$	7.41619e10	0.505147	0.252574	−	0.967578i	$-0.418723\pi$
$619$	7.41619e10	0.505147	0.252574	+	0.967578i	$0.418723\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.28027e10i	0.151368i
$624$	0	0
$625$	1.74323e9	0.0114245
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 9.55429e10i	− 0.610373i
$630$	0	0
$631$	1.79703e11	1.13354	0.566772	−	0.823875i	$-0.308192\pi$
$631$	1.79703e11	1.13354	0.566772	+	0.823875i	$0.308192\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.93159e10i	0.364818i
$636$	0	0
$637$	2.07072e11	1.25766
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 5.38915e10i	− 0.319219i	−0.987180	−	0.159609i	$-0.948977\pi$
$641$	− 5.38915e10i	− 0.319219i	0.987180	−	0.159609i	$-0.0510234\pi$
$642$	0	0
$643$	−1.86975e11	−1.09380	−0.546901	−	0.837197i	$-0.684193\pi$
$643$	−1.86975e11	−1.09380	−0.546901	+	0.837197i	$0.684193\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.36662e10i	− 0.534522i	−0.963624	−	0.267261i	$-0.913881\pi$
$647$	− 9.36662e10i	− 0.534522i	0.963624	−	0.267261i	$-0.0861186\pi$
$648$	0	0
$649$	−2.63431e11	−1.48487
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.39391e11i	1.31660i	0.752754	+	0.658302i	$0.228725\pi$
$653$	2.39391e11i	1.31660i	−0.752754	+	0.658302i	$0.771275\pi$
$654$	0	0
$655$	7.12078e10	0.386868
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 8.85271e10i	− 0.469391i	−0.972069	−	0.234696i	$-0.924591\pi$
$659$	− 8.85271e10i	− 0.469391i	0.972069	−	0.234696i	$-0.0754093\pi$
$660$	0	0
$661$	2.44646e11	1.28154	0.640771	−	0.767732i	$-0.278615\pi$
$661$	2.44646e11	1.28154	0.640771	+	0.767732i	$0.278615\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 7.93209e10i	− 0.405603i
$666$	0	0
$667$	−3.88784e11	−1.96429
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.48526e11i	1.22597i
$672$	0	0
$673$	−1.47419e11	−0.718612	−0.359306	−	0.933220i	$-0.616987\pi$
$673$	−1.47419e11	−0.718612	−0.359306	+	0.933220i	$0.616987\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 3.76509e10i	− 0.179234i	−0.995976	−	0.0896170i	$-0.971436\pi$
$677$	− 3.76509e10i	− 0.179234i	0.995976	−	0.0896170i	$-0.0285643\pi$
$678$	0	0
$679$	6.72905e10	0.316573
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.50809e11i	1.61208i	0.591859	+	0.806041i	$0.298394\pi$
$683$	3.50809e11i	1.61208i	−0.591859	+	0.806041i	$0.701606\pi$
$684$	0	0
$685$	−1.37647e11	−0.625180
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 2.09416e11i	− 0.929250i
$690$	0	0
$691$	−3.96740e11	−1.74018	−0.870090	−	0.492893i	$-0.835939\pi$
$691$	−3.96740e11	−1.74018	−0.870090	+	0.492893i	$0.835939\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.06087e11i	− 0.454697i
$696$	0	0
$697$	7.09984e10	0.300828
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.36905e11i	0.566952i	0.958979	+	0.283476i	$0.0914877\pi$
$701$	1.36905e11i	0.566952i	−0.958979	+	0.283476i	$0.908512\pi$
$702$	0	0
$703$	−3.74179e11	−1.53200
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.16966e11i	0.468148i
$708$	0	0
$709$	−4.79530e11	−1.89771	−0.948857	−	0.315706i	$-0.897759\pi$
$709$	−4.79530e11	−1.89771	−0.948857	+	0.315706i	$0.897759\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.82450e10i	0.225372i
$714$	0	0
$715$	3.59258e11	1.37462
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.09505e11i	0.409749i	0.978788	+	0.204874i	$0.0656786\pi$
$719$	1.09505e11i	0.409749i	−0.978788	+	0.204874i	$0.934321\pi$
$720$	0	0
$721$	6.26345e10	0.231778
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.13237e11i	− 0.771810i
$726$	0	0
$727$	1.16526e11	0.417143	0.208571	−	0.978007i	$-0.433119\pi$
$727$	1.16526e11	0.417143	0.208571	+	0.978007i	$0.433119\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.52624e11i	− 0.884718i
$732$	0	0
$733$	2.19970e11	0.761986	0.380993	−	0.924578i	$-0.375582\pi$
$733$	2.19970e11	0.761986	0.380993	+	0.924578i	$0.375582\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.98534e11i	1.01187i
$738$	0	0
$739$	1.88576e11	0.632277	0.316139	−	0.948713i	$-0.397614\pi$
$739$	1.88576e11	0.632277	0.316139	+	0.948713i	$0.397614\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.97746e11i	0.648863i	0.945909	+	0.324431i	$0.105173\pi$
$743$	1.97746e11i	0.648863i	−0.945909	+	0.324431i	$0.894827\pi$
$744$	0	0
$745$	−1.14522e10	−0.0371762
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.18370e10i	0.260029i
$750$	0	0
$751$	6.21441e11	1.95362	0.976811	−	0.214105i	$-0.0686835\pi$
$751$	6.21441e11	1.95362	0.976811	+	0.214105i	$0.0686835\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 4.92508e10i	− 0.151574i
$756$	0	0
$757$	4.99965e11	1.52250	0.761248	−	0.648461i	$-0.224587\pi$
$757$	4.99965e11	1.52250	0.761248	+	0.648461i	$0.224587\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.81400e10i	0.232989i	0.993191	+	0.116494i	$0.0371657\pi$
$761$	7.81400e10i	0.232989i	−0.993191	+	0.116494i	$0.962834\pi$
$762$	0	0
$763$	1.73539e11	0.512036
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.58309e11i	− 1.61322i
$768$	0	0
$769$	4.12936e11	1.18080	0.590402	−	0.807110i	$-0.298969\pi$
$769$	4.12936e11	1.18080	0.590402	+	0.807110i	$0.298969\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 4.00086e10i	− 0.112056i	−0.998429	−	0.0560280i	$-0.982156\pi$
$773$	− 4.00086e10i	− 0.112056i	0.998429	−	0.0560280i	$-0.0178436\pi$
$774$	0	0
$775$	−3.19457e10	−0.0885535
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.78054e11i	− 0.755057i
$780$	0	0
$781$	8.05817e11	2.16587
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.23847e11i	0.589485i
$786$	0	0
$787$	−6.52745e10	−0.170155	−0.0850775	−	0.996374i	$-0.527114\pi$
$787$	−6.52745e10	−0.170155	−0.0850775	+	0.996374i	$0.527114\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.71713e11i	0.694072i
$792$	0	0
$793$	−5.26719e11	−1.33194
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.29050e11i	− 0.319835i	−0.987130	−	0.159917i	$-0.948877\pi$
$797$	− 1.29050e11i	− 0.319835i	0.987130	−	0.159917i	$-0.0511228\pi$
$798$	0	0
$799$	−2.76304e11	−0.677953
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 8.47961e11i	− 2.03945i
$804$	0	0
$805$	1.79918e11	0.428441
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 7.15122e11i	− 1.66950i	−0.550630	−	0.834750i	$-0.685612\pi$
$809$	− 7.15122e11i	− 1.66950i	0.550630	−	0.834750i	$-0.314388\pi$
$810$	0	0
$811$	2.61429e11	0.604325	0.302162	−	0.953256i	$-0.402292\pi$
$811$	2.61429e11	0.604325	0.302162	+	0.953256i	$0.402292\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.08901e11i	0.926804i
$816$	0	0
$817$	−9.89362e11	−2.22058
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.39193e11i	1.40689i	0.710750	+	0.703444i	$0.248356\pi$
$821$	6.39193e11i	1.40689i	−0.710750	+	0.703444i	$0.751644\pi$
$822$	0	0
$823$	6.82246e11	1.48710	0.743552	−	0.668678i	$-0.233140\pi$
$823$	6.82246e11	1.48710	0.743552	+	0.668678i	$0.233140\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.96972e11i	− 0.848668i	−0.905506	−	0.424334i	$-0.860508\pi$
$827$	− 3.96972e11i	− 0.848668i	0.905506	−	0.424334i	$-0.139492\pi$
$828$	0	0
$829$	2.44430e11	0.517532	0.258766	−	0.965940i	$-0.416684\pi$
$829$	2.44430e11	0.517532	0.258766	+	0.965940i	$0.416684\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.32231e11i	0.482325i
$834$	0	0
$835$	2.55448e11	0.525480
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.74819e10i	0.0958253i	0.998852	+	0.0479126i	$0.0152569\pi$
$839$	4.74819e10i	0.0958253i	−0.998852	+	0.0479126i	$0.984743\pi$
$840$	0	0
$841$	−2.67194e11	−0.534125
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.48418e11i	0.879542i
$846$	0	0
$847$	−2.40314e11	−0.466924
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 8.48723e11i	− 1.61826i
$852$	0	0
$853$	8.31571e11	1.57074	0.785368	−	0.619029i	$-0.212474\pi$
$853$	8.31571e11	1.57074	0.785368	+	0.619029i	$0.212474\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.02068e11i	0.189220i	0.995514	+	0.0946101i	$0.0301604\pi$
$857$	1.02068e11i	0.189220i	−0.995514	+	0.0946101i	$0.969840\pi$
$858$	0	0
$859$	2.38390e11	0.437839	0.218920	−	0.975743i	$-0.429747\pi$
$859$	2.38390e11	0.437839	0.218920	+	0.975743i	$0.429747\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 3.90151e11i	− 0.703380i	−0.936117	−	0.351690i	$-0.885607\pi$
$863$	− 3.90151e11i	− 0.703380i	0.936117	−	0.351690i	$-0.114393\pi$
$864$	0	0
$865$	−5.99719e11	−1.07123
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 4.29419e11i	− 0.753013i
$870$	0	0
$871$	−6.32706e11	−1.09933
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.57041e11i	0.438501i
$876$	0	0
$877$	1.08094e12	1.82727	0.913635	−	0.406536i	$-0.133264\pi$
$877$	1.08094e12	1.82727	0.913635	+	0.406536i	$0.133264\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.52799e11i	1.58160i	0.612072	+	0.790802i	$0.290336\pi$
$881$	9.52799e11i	1.58160i	−0.612072	+	0.790802i	$0.709664\pi$
$882$	0	0
$883$	−1.62997e11	−0.268125	−0.134063	−	0.990973i	$-0.542802\pi$
$883$	−1.62997e11	−0.268125	−0.134063	+	0.990973i	$0.542802\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.33342e11i	− 0.376963i	−0.982077	−	0.188481i	$-0.939643\pi$
$887$	− 2.33342e11i	− 0.376963i	0.982077	−	0.188481i	$-0.0603565\pi$
$888$	0	0
$889$	1.63347e11	0.261519
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.08210e12i	1.70162i
$894$	0	0
$895$	5.97230e11	0.930785
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.14973e11i	0.176018i
$900$	0	0
$901$	2.34859e11	0.356376
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 3.51023e11i	− 0.523288i
$906$	0	0
$907$	7.28967e11	1.07716	0.538578	−	0.842575i	$-0.318962\pi$
$907$	7.28967e11	1.07716	0.538578	+	0.842575i	$0.318962\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1.28380e12i	− 1.86391i	−0.362571	−	0.931956i	$-0.618101\pi$
$911$	− 1.28380e12i	− 1.86391i	0.362571	−	0.931956i	$-0.381899\pi$
$912$	0	0
$913$	9.23219e11	1.32868
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.96095e11i	− 0.277325i
$918$	0	0
$919$	−3.67453e11	−0.515158	−0.257579	−	0.966257i	$-0.582925\pi$
$919$	−3.67453e11	−0.515158	−0.257579	+	0.966257i	$0.582925\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.70783e12i	2.35308i
$924$	0	0
$925$	4.65500e11	0.635848
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.45683e12i	1.95589i	0.208857	+	0.977946i	$0.433026\pi$
$929$	1.45683e12i	1.95589i	−0.208857	+	0.977946i	$0.566974\pi$
$930$	0	0
$931$	9.09495e11	1.21060
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.02907e11i	0.527180i
$936$	0	0
$937$	−1.97945e11	−0.256795	−0.128397	−	0.991723i	$-0.540983\pi$
$937$	−1.97945e11	−0.256795	−0.128397	+	0.991723i	$0.540983\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 7.34556e10i	− 0.0936842i	−0.998902	−	0.0468421i	$-0.985084\pi$
$941$	− 7.34556e10i	− 0.0936842i	0.998902	−	0.0468421i	$-0.0149158\pi$
$942$	0	0
$943$	6.30691e11	0.797572
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.72305e11i	0.711586i	0.934565	+	0.355793i	$0.115789\pi$
$947$	5.72305e11i	0.711586i	−0.934565	+	0.355793i	$0.884211\pi$
$948$	0	0
$949$	1.79715e12	2.21574
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 2.21067e11i	− 0.268010i	−0.990981	−	0.134005i	$-0.957216\pi$
$953$	− 2.21067e11i	− 0.268010i	0.990981	−	0.134005i	$-0.0427839\pi$
$954$	0	0
$955$	−6.23064e11	−0.749064
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.79059e11i	0.448159i
$960$	0	0
$961$	−8.35667e11	−0.979805
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.42310e11i	0.625372i
$966$	0	0
$967$	1.41164e12	1.61442	0.807211	−	0.590263i	$-0.200976\pi$
$967$	1.41164e12	1.61442	0.807211	+	0.590263i	$0.200976\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.48773e11i	0.504835i	0.967618	+	0.252418i	$0.0812256\pi$
$971$	4.48773e11i	0.504835i	−0.967618	+	0.252418i	$0.918774\pi$
$972$	0	0
$973$	−2.92146e11	−0.325949
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 7.04915e11i	− 0.773675i	−0.922148	−	0.386837i	$-0.873568\pi$
$977$	− 7.04915e11i	− 0.773675i	0.922148	−	0.386837i	$-0.126432\pi$
$978$	0	0
$979$	−4.53611e11	−0.493802
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 5.08211e11i	− 0.544290i	−0.962256	−	0.272145i	$-0.912267\pi$
$983$	− 5.08211e11i	− 0.544290i	0.962256	−	0.272145i	$-0.0877330\pi$
$984$	0	0
$985$	1.45019e10	0.0154056
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2.24410e12i	− 2.34562i
$990$	0	0
$991$	2.22910e11	0.231118	0.115559	−	0.993301i	$-0.463134\pi$
$991$	2.22910e11	0.231118	0.115559	+	0.993301i	$0.463134\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 6.38436e11i	− 0.651366i
$996$	0	0
$997$	−1.60476e12	−1.62416	−0.812080	−	0.583546i	$-0.801665\pi$
$997$	−1.60476e12	−1.62416	−0.812080	+	0.583546i	$0.801665\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.9.e.l.161.5		8
3.2	odd	2	inner	432.9.e.l.161.4		8
4.3	odd	2		216.9.e.b.161.5	yes	8
12.11	even	2		216.9.e.b.161.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.9.e.b.161.4	✓	8	12.11	even	2
216.9.e.b.161.5	yes	8	4.3	odd	2
432.9.e.l.161.4		8	3.2	odd	2	inner
432.9.e.l.161.5		8	1.1	even	1	trivial

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.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+383.685i q^{5} +1056.61 q^{7} +21019.0i q^{11} -44547.1 q^{13} -49959.5i q^{17} -195658. q^{19} -443798. i q^{23} +243411. q^{25} -876037. i q^{29} -131242. q^{31} +405405. i q^{35} +1.91241e6 q^{37} +1.42112e6i q^{41} +5.05657e6 q^{43} -5.53056e6i q^{47} -4.64838e6 q^{49} +4.70100e6i q^{53} -8.06468e6 q^{55} +1.25330e7i q^{59} +1.18239e7 q^{61} -1.70921e7i q^{65} +1.42031e7 q^{67} -3.83376e7i q^{71} -4.03426e7 q^{73} +2.22088e7i q^{77} -2.04301e7 q^{79} -4.39231e7i q^{83} +1.91687e7 q^{85} +2.15810e7i q^{89} -4.70688e7 q^{91} -7.50712e7i q^{95} +6.36854e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 456 q^{7} - 75016 q^{13} + 132536 q^{19} - 1187768 q^{25} + 1301072 q^{31} - 2559432 q^{37} + 2958544 q^{43} + 18410368 q^{49} - 10936256 q^{55} + 20663672 q^{61} - 59168392 q^{67} - 98757640 q^{73}+ \cdots - 103615304 q^{97}+O(q^{100}) \) 8 * q - 456 * q^7 - 75016 * q^13 + 132536 * q^19 - 1187768 * q^25 + 1301072 * q^31 - 2559432 * q^37 + 2958544 * q^43 + 18410368 * q^49 - 10936256 * q^55 + 20663672 * q^61 - 59168392 * q^67 - 98757640 * q^73 + 107248888 * q^79 + 151752256 * q^85 + 263862984 * q^91 - 103615304 * q^97

Introduction

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