LMFDB - Embedded newform 144.9.g.d.127.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,9,Mod(127,144)] 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("144.127"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	144.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-516] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$58.6625198488$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	144.127
Dual form			144.9.g.d.127.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-258.000 q^{5} -3297.82i q^{7} +23154.1i q^{11} +19138.0 q^{13} +58686.0 q^{17} +152573. i q^{19} -270117. i q^{23} -324061. q^{25} -842178. q^{29} -1.05120e6i q^{31} +850839. i q^{35} +2.54861e6 q^{37} +4.32416e6 q^{41} +2.03702e6i q^{43} -7.23321e6i q^{47} -5.11085e6 q^{49} -1.19219e6 q^{53} -5.97375e6i q^{55} +338249. i q^{59} +8.41479e6 q^{61} -4.93760e6 q^{65} -1.74247e7i q^{67} -3.08765e7i q^{71} +1.27359e7 q^{73} +7.63580e7 q^{77} +6.28665e6i q^{79} -8.30740e7i q^{83} -1.51410e7 q^{85} +1.68028e7 q^{89} -6.31138e7i q^{91} -3.93638e7i q^{95} +1.20995e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 516 q^{5} + 38276 q^{13} + 117372 q^{17} - 648122 q^{25} - 1684356 q^{29} + 5097220 q^{37} + 8648316 q^{41} - 10221694 q^{49} - 2384388 q^{53} + 16829572 q^{61} - 9875208 q^{65} + 25471748 q^{73}+ \cdots + 241989764 q^{97}+O(q^{100}) $ 2 * q - 516 * q^5 + 38276 * q^13 + 117372 * q^17 - 648122 * q^25 - 1684356 * q^29 + 5097220 * q^37 + 8648316 * q^41 - 10221694 * q^49 - 2384388 * q^53 + 16829572 * q^61 - 9875208 * q^65 + 25471748 * q^73 + 152716032 * q^77 - 30281976 * q^85 + 33605628 * q^89 + 241989764 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	−258.000	−0.412800	−0.206400	−	0.978468i	$-0.566175\pi$
$5$	−258.000	−0.412800	−0.206400	+	0.978468i	$0.566175\pi$
$6$	0	0
$7$	− 3297.82i	− 1.37352i	−0.726884	−	0.686761i	$-0.759032\pi$
$7$	− 3297.82i	− 1.37352i	0.726884	−	0.686761i	$-0.240968\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	23154.1i	1.58145i	0.612170	+	0.790727i	$0.290297\pi$
$11$	23154.1i	1.58145i	−0.612170	+	0.790727i	$0.709703\pi$
$12$	0	0
$13$	19138.0	0.670075	0.335037	−	0.942205i	$-0.391251\pi$
$13$	19138.0	0.670075	0.335037	+	0.942205i	$0.391251\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	58686.0	0.702650	0.351325	−	0.936254i	$-0.385731\pi$
$17$	58686.0	0.702650	0.351325	+	0.936254i	$0.385731\pi$
$18$	0	0
$19$	152573.i	1.17075i	0.810764	+	0.585373i	$0.199052\pi$
$19$	152573.i	1.17075i	−0.810764	+	0.585373i	$0.800948\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 270117.i	− 0.965251i	−0.875827	−	0.482625i	$-0.839683\pi$
$23$	− 270117.i	− 0.965251i	0.875827	−	0.482625i	$-0.160317\pi$
$24$	0	0
$25$	−324061.	−0.829596
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−842178.	−1.19073	−0.595363	−	0.803457i	$-0.702992\pi$
$29$	−842178.	−1.19073	−0.595363	+	0.803457i	$0.702992\pi$
$30$	0	0
$31$	− 1.05120e6i	− 1.13826i	−0.822249	−	0.569128i	$-0.807281\pi$
$31$	− 1.05120e6i	− 1.13826i	0.822249	−	0.569128i	$-0.192719\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	850839.i	0.566990i
$36$	0	0
$37$	2.54861e6	1.35987	0.679934	−	0.733274i	$-0.262009\pi$
$37$	2.54861e6	1.35987	0.679934	+	0.733274i	$0.262009\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.32416e6	1.53026	0.765132	−	0.643874i	$-0.222674\pi$
$41$	4.32416e6	1.53026	0.765132	+	0.643874i	$0.222674\pi$
$42$	0	0
$43$	2.03702e6i	0.595828i	0.954593	+	0.297914i	$0.0962908\pi$
$43$	2.03702e6i	0.595828i	−0.954593	+	0.297914i	$0.903709\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.23321e6i	− 1.48231i	−0.671333	−	0.741156i	$-0.734278\pi$
$47$	− 7.23321e6i	− 1.48231i	0.671333	−	0.741156i	$-0.265722\pi$
$48$	0	0
$49$	−5.11085e6	−0.886561
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.19219e6	−0.151093	−0.0755463	−	0.997142i	$-0.524070\pi$
$53$	−1.19219e6	−0.151093	−0.0755463	+	0.997142i	$0.524070\pi$
$54$	0	0
$55$	− 5.97375e6i	− 0.652824i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	338249.i	0.0279144i	0.999903	+	0.0139572i	$0.00444286\pi$
$59$	338249.i	0.0279144i	−0.999903	+	0.0139572i	$0.995557\pi$
$60$	0	0
$61$	8.41479e6	0.607748	0.303874	−	0.952712i	$-0.401720\pi$
$61$	8.41479e6	0.607748	0.303874	+	0.952712i	$0.401720\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.93760e6	−0.276607
$66$	0	0
$67$	− 1.74247e7i	− 0.864702i	−0.901705	−	0.432351i	$-0.857684\pi$
$67$	− 1.74247e7i	− 0.864702i	0.901705	−	0.432351i	$-0.142316\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 3.08765e7i	− 1.21505i	−0.794299	−	0.607526i	$-0.792162\pi$
$71$	− 3.08765e7i	− 1.21505i	0.794299	−	0.607526i	$-0.207838\pi$
$72$	0	0
$73$	1.27359e7	0.448474	0.224237	−	0.974535i	$-0.428011\pi$
$73$	1.27359e7	0.448474	0.224237	+	0.974535i	$0.428011\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.63580e7	2.17216
$78$	0	0
$79$	6.28665e6i	0.161403i	0.996738	+	0.0807014i	$0.0257160\pi$
$79$	6.28665e6i	0.161403i	−0.996738	+	0.0807014i	$0.974284\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.30740e7i	− 1.75046i	−0.483706	−	0.875231i	$-0.660709\pi$
$83$	− 8.30740e7i	− 1.75046i	0.483706	−	0.875231i	$-0.339291\pi$
$84$	0	0
$85$	−1.51410e7	−0.290054
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.68028e7	0.267807	0.133904	−	0.990994i	$-0.457249\pi$
$89$	1.68028e7	0.267807	0.133904	+	0.990994i	$0.457249\pi$
$90$	0	0
$91$	− 6.31138e7i	− 0.920362i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 3.93638e7i	− 0.483284i
$96$	0	0
$97$	1.20995e8	1.36672	0.683361	−	0.730081i	$-0.260518\pi$
$97$	1.20995e8	1.36672	0.683361	+	0.730081i	$0.260518\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.16328e7	−0.111789	−0.0558943	−	0.998437i	$-0.517801\pi$
$101$	−1.16328e7	−0.111789	−0.0558943	+	0.998437i	$0.517801\pi$
$102$	0	0
$103$	− 1.34856e8i	− 1.19818i	−0.800682	−	0.599090i	$-0.795529\pi$
$103$	− 1.34856e8i	− 1.19818i	0.800682	−	0.599090i	$-0.204471\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.31472e8i	− 1.00299i	−0.865160	−	0.501495i	$-0.832783\pi$
$107$	− 1.31472e8i	− 1.00299i	0.865160	−	0.501495i	$-0.167217\pi$
$108$	0	0
$109$	1.55823e7	0.110389	0.0551944	−	0.998476i	$-0.482422\pi$
$109$	1.55823e7	0.110389	0.0551944	+	0.998476i	$0.482422\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.64134e7	0.284662	0.142331	−	0.989819i	$-0.454540\pi$
$113$	4.64134e7	0.284662	0.142331	+	0.989819i	$0.454540\pi$
$114$	0	0
$115$	6.96901e7i	0.398456i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.93536e8i	− 0.965104i
$120$	0	0
$121$	−3.21751e8	−1.50099
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.84389e8	0.755257
$126$	0	0
$127$	1.85766e8i	0.714087i	0.934088	+	0.357043i	$0.116215\pi$
$127$	1.85766e8i	0.714087i	−0.934088	+	0.357043i	$0.883785\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 5.38679e7i	− 0.182913i	−0.995809	−	0.0914566i	$-0.970848\pi$
$131$	− 5.38679e7i	− 0.182913i	0.995809	−	0.0914566i	$-0.0291523\pi$
$132$	0	0
$133$	5.03159e8	1.60805
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.90826e8	1.10943	0.554716	−	0.832040i	$-0.312827\pi$
$137$	3.90826e8	1.10943	0.554716	+	0.832040i	$0.312827\pi$
$138$	0	0
$139$	− 4.01458e8i	− 1.07543i	−0.843128	−	0.537713i	$-0.819288\pi$
$139$	− 4.01458e8i	− 1.07543i	0.843128	−	0.537713i	$-0.180712\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.43122e8i	1.05969i
$144$	0	0
$145$	2.17282e8	0.491532
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.54777e8	−1.53135	−0.765674	−	0.643229i	$-0.777594\pi$
$149$	−7.54777e8	−1.53135	−0.765674	+	0.643229i	$0.777594\pi$
$150$	0	0
$151$	− 1.79562e8i	− 0.345388i	−0.984976	−	0.172694i	$-0.944753\pi$
$151$	− 1.79562e8i	− 0.345388i	0.984976	−	0.172694i	$-0.0552471\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.71210e8i	0.469872i
$156$	0	0
$157$	−8.56860e8	−1.41030	−0.705150	−	0.709058i	$-0.749120\pi$
$157$	−8.56860e8	−1.41030	−0.705150	+	0.709058i	$0.749120\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.90798e8	−1.32579
$162$	0	0
$163$	4.60153e8i	0.651856i	0.945395	+	0.325928i	$0.105677\pi$
$163$	4.60153e8i	0.651856i	−0.945395	+	0.325928i	$0.894323\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.38177e8i	− 0.691925i	−0.938248	−	0.345963i	$-0.887552\pi$
$167$	− 5.38177e8i	− 0.691925i	0.938248	−	0.345963i	$-0.112448\pi$
$168$	0	0
$169$	−4.49468e8	−0.551000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.80770e8	1.09492	0.547461	−	0.836832i	$-0.315595\pi$
$173$	9.80770e8	1.09492	0.547461	+	0.836832i	$0.315595\pi$
$174$	0	0
$175$	1.06870e9i	1.13947i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.45845e9i	1.42062i	0.703889	+	0.710310i	$0.251445\pi$
$179$	1.45845e9i	1.42062i	−0.703889	+	0.710310i	$0.748555\pi$
$180$	0	0
$181$	−1.08010e8	−0.100635	−0.0503175	−	0.998733i	$-0.516023\pi$
$181$	−1.08010e8	−0.100635	−0.0503175	+	0.998733i	$0.516023\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.57541e8	−0.561353
$186$	0	0
$187$	1.35882e9i	1.11121i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.80549e9i	1.35663i	0.734772	+	0.678314i	$0.237289\pi$
$191$	1.80549e9i	1.35663i	−0.734772	+	0.678314i	$0.762711\pi$
$192$	0	0
$193$	1.80728e9	1.30255	0.651277	−	0.758840i	$-0.274234\pi$
$193$	1.80728e9	1.30255	0.651277	+	0.758840i	$0.274234\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.28469e8	−0.151692	−0.0758459	−	0.997120i	$-0.524166\pi$
$197$	−2.28469e8	−0.151692	−0.0758459	+	0.997120i	$0.524166\pi$
$198$	0	0
$199$	− 1.23311e9i	− 0.786305i	−0.919473	−	0.393153i	$-0.871384\pi$
$199$	− 1.23311e9i	− 0.786305i	0.919473	−	0.393153i	$-0.128616\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.77736e9i	1.63549i
$204$	0	0
$205$	−1.11563e9	−0.631693
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.53268e9	−1.85148
$210$	0	0
$211$	− 2.15314e8i	− 0.108628i	−0.998524	−	0.0543141i	$-0.982703\pi$
$211$	− 2.15314e8i	− 0.108628i	0.998524	−	0.0543141i	$-0.0172972\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 5.25550e8i	− 0.245958i
$216$	0	0
$217$	−3.46668e9	−1.56342
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.12313e9	0.470828
$222$	0	0
$223$	4.09616e8i	0.165637i	0.996565	+	0.0828185i	$0.0263922\pi$
$223$	4.09616e8i	0.165637i	−0.996565	+	0.0828185i	$0.973608\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5.22333e9i	− 1.96718i	−0.180415	−	0.983591i	$-0.557744\pi$
$227$	− 5.22333e9i	− 1.96718i	0.180415	−	0.983591i	$-0.442256\pi$
$228$	0	0
$229$	2.86620e9	1.04223	0.521116	−	0.853486i	$-0.325516\pi$
$229$	2.86620e9	1.04223	0.521116	+	0.853486i	$0.325516\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.91979e8	0.166925	0.0834627	−	0.996511i	$-0.473402\pi$
$233$	4.91979e8	0.166925	0.0834627	+	0.996511i	$0.473402\pi$
$234$	0	0
$235$	1.86617e9i	0.611898i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.04443e9i	0.933071i	0.884503	+	0.466535i	$0.154498\pi$
$239$	3.04443e9i	0.933071i	−0.884503	+	0.466535i	$0.845502\pi$
$240$	0	0
$241$	−3.10239e9	−0.919662	−0.459831	−	0.888006i	$-0.652090\pi$
$241$	−3.10239e9	−0.919662	−0.459831	+	0.888006i	$0.652090\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.31860e9	0.365972
$246$	0	0
$247$	2.91994e9i	0.784488i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.89495e9i	0.477422i	0.971091	+	0.238711i	$0.0767248\pi$
$251$	1.89495e9i	0.477422i	−0.971091	+	0.238711i	$0.923275\pi$
$252$	0	0
$253$	6.25430e9	1.52650
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.97422e9	0.681774	0.340887	−	0.940104i	$-0.389273\pi$
$257$	2.97422e9	0.681774	0.340887	+	0.940104i	$0.389273\pi$
$258$	0	0
$259$	− 8.40487e9i	− 1.86781i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.15278e9i	0.658977i	0.944160	+	0.329488i	$0.106876\pi$
$263$	3.15278e9i	0.658977i	−0.944160	+	0.329488i	$0.893124\pi$
$264$	0	0
$265$	3.07586e8	0.0623711
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.48778e7	−0.00284137	−0.00142069	−	0.999999i	$-0.500452\pi$
$269$	−1.48778e7	−0.00284137	−0.00142069	+	0.999999i	$0.500452\pi$
$270$	0	0
$271$	− 1.72933e9i	− 0.320627i	−0.987066	−	0.160313i	$-0.948750\pi$
$271$	− 1.72933e9i	− 0.320627i	0.987066	−	0.160313i	$-0.0512505\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 7.50333e9i	− 1.31197i
$276$	0	0
$277$	−3.04182e9	−0.516671	−0.258336	−	0.966055i	$-0.583174\pi$
$277$	−3.04182e9	−0.516671	−0.258336	+	0.966055i	$0.583174\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.75819e9	−0.602773	−0.301386	−	0.953502i	$-0.597449\pi$
$281$	−3.75819e9	−0.602773	−0.301386	+	0.953502i	$0.597449\pi$
$282$	0	0
$283$	1.14220e9i	0.178072i	0.996028	+	0.0890358i	$0.0283786\pi$
$283$	1.14220e9i	0.178072i	−0.996028	+	0.0890358i	$0.971621\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.42603e10i	− 2.10185i
$288$	0	0
$289$	−3.53171e9	−0.506283
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.10157e10	1.49466	0.747328	−	0.664456i	$-0.231337\pi$
$293$	1.10157e10	1.49466	0.747328	+	0.664456i	$0.231337\pi$
$294$	0	0
$295$	− 8.72682e7i	− 0.0115231i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 5.16950e9i	− 0.646790i
$300$	0	0
$301$	6.71772e9	0.818382
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.17101e9	−0.250878
$306$	0	0
$307$	4.87606e9i	0.548928i	0.961597	+	0.274464i	$0.0885004\pi$
$307$	4.87606e9i	0.548928i	−0.961597	+	0.274464i	$0.911500\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 1.05124e10i	− 1.12372i	−0.827231	−	0.561862i	$-0.810085\pi$
$311$	− 1.05124e10i	− 1.12372i	0.827231	−	0.561862i	$-0.189915\pi$
$312$	0	0
$313$	1.08431e10	1.12974	0.564868	−	0.825181i	$-0.308927\pi$
$313$	1.08431e10	1.12974	0.564868	+	0.825181i	$0.308927\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.43464e10	1.42071	0.710356	−	0.703842i	$-0.248534\pi$
$317$	1.43464e10	1.42071	0.710356	+	0.703842i	$0.248534\pi$
$318$	0	0
$319$	− 1.94998e10i	− 1.88308i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.95389e9i	0.822625i
$324$	0	0
$325$	−6.20188e9	−0.555891
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.38539e10	−2.03599
$330$	0	0
$331$	− 4.66058e9i	− 0.388265i	−0.980975	−	0.194132i	$-0.937811\pi$
$331$	− 4.66058e9i	− 0.388265i	0.980975	−	0.194132i	$-0.0621891\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.49558e9i	0.356949i
$336$	0	0
$337$	−1.98174e9	−0.153648	−0.0768239	−	0.997045i	$-0.524478\pi$
$337$	−1.98174e9	−0.153648	−0.0768239	+	0.997045i	$0.524478\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.43396e10	1.80010
$342$	0	0
$343$	− 2.15663e9i	− 0.155811i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.31164e8i	− 0.0573283i	−0.999589	−	0.0286641i	$-0.990875\pi$
$347$	− 8.31164e8i	− 0.0573283i	0.999589	−	0.0286641i	$-0.00912533\pi$
$348$	0	0
$349$	1.24018e10	0.835958	0.417979	−	0.908457i	$-0.362739\pi$
$349$	1.24018e10	0.835958	0.417979	+	0.908457i	$0.362739\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.45970e9	0.287214	0.143607	−	0.989635i	$-0.454130\pi$
$353$	4.45970e9	0.287214	0.143607	+	0.989635i	$0.454130\pi$
$354$	0	0
$355$	7.96615e9i	0.501574i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 9.78764e9i	− 0.589251i	−0.955613	−	0.294626i	$-0.904805\pi$
$359$	− 9.78764e9i	− 0.589251i	0.955613	−	0.294626i	$-0.0951949\pi$
$360$	0	0
$361$	−6.29492e9	−0.370648
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.28586e9	−0.185130
$366$	0	0
$367$	3.12246e10i	1.72121i	0.509276	+	0.860603i	$0.329913\pi$
$367$	3.12246e10i	1.72121i	−0.509276	+	0.860603i	$0.670087\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.93165e9i	0.207529i
$372$	0	0
$373$	−1.60695e10	−0.830168	−0.415084	−	0.909783i	$-0.636248\pi$
$373$	−1.60695e10	−0.830168	−0.415084	+	0.909783i	$0.636248\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.61176e10	−0.797875
$378$	0	0
$379$	3.79438e9i	0.183901i	0.995764	+	0.0919506i	$0.0293102\pi$
$379$	3.79438e9i	0.183901i	−0.995764	+	0.0919506i	$0.970690\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.89325e9i	0.320353i	0.987088	+	0.160177i	$0.0512064\pi$
$383$	6.89325e9i	0.320353i	−0.987088	+	0.160177i	$0.948794\pi$
$384$	0	0
$385$	−1.97004e10	−0.896667
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.16828e10	−0.946926	−0.473463	−	0.880814i	$-0.656996\pi$
$389$	−2.16828e10	−0.946926	−0.473463	+	0.880814i	$0.656996\pi$
$390$	0	0
$391$	− 1.58521e10i	− 0.678233i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.62196e9i	− 0.0666271i
$396$	0	0
$397$	2.55255e10	1.02757	0.513786	−	0.857918i	$-0.328243\pi$
$397$	2.55255e10	1.02757	0.513786	+	0.857918i	$0.328243\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.44054e9	−0.326432	−0.163216	−	0.986590i	$-0.552187\pi$
$401$	−8.44054e9	−0.326432	−0.163216	+	0.986590i	$0.552187\pi$
$402$	0	0
$403$	− 2.01179e10i	− 0.762716i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.90107e10i	2.15057i
$408$	0	0
$409$	−3.07378e10	−1.09845	−0.549225	−	0.835675i	$-0.685077\pi$
$409$	−3.07378e10	−1.09845	−0.549225	+	0.835675i	$0.685077\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.11549e9	0.0383410
$414$	0	0
$415$	2.14331e10i	0.722591i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.64123e10i	0.856941i	0.903556	+	0.428470i	$0.140947\pi$
$419$	2.64123e10i	0.856941i	−0.903556	+	0.428470i	$0.859053\pi$
$420$	0	0
$421$	5.03302e10	1.60214	0.801070	−	0.598570i	$-0.204264\pi$
$421$	5.03302e10	1.60214	0.801070	+	0.598570i	$0.204264\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.90178e10	−0.582915
$426$	0	0
$427$	− 2.77505e10i	− 0.834755i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.47805e10i	0.718127i	0.933313	+	0.359063i	$0.116904\pi$
$431$	2.47805e10i	0.718127i	−0.933313	+	0.359063i	$0.883096\pi$
$432$	0	0
$433$	−1.97724e10	−0.562482	−0.281241	−	0.959637i	$-0.590746\pi$
$433$	−1.97724e10	−0.562482	−0.281241	+	0.959637i	$0.590746\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.12125e10	1.13006
$438$	0	0
$439$	5.78110e10i	1.55651i	0.627948	+	0.778255i	$0.283895\pi$
$439$	5.78110e10i	1.55651i	−0.627948	+	0.778255i	$0.716105\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 2.77961e10i	− 0.721719i	−0.932620	−	0.360860i	$-0.882483\pi$
$443$	− 2.77961e10i	− 0.721719i	0.932620	−	0.360860i	$-0.117517\pi$
$444$	0	0
$445$	−4.33513e9	−0.110551
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.37962e10	−1.07758	−0.538792	−	0.842439i	$-0.681119\pi$
$449$	−4.37962e10	−1.07758	−0.538792	+	0.842439i	$0.681119\pi$
$450$	0	0
$451$	1.00122e11i	2.42004i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.62834e10i	0.379925i
$456$	0	0
$457$	−1.89812e10	−0.435171	−0.217585	−	0.976041i	$-0.569818\pi$
$457$	−1.89812e10	−0.435171	−0.217585	+	0.976041i	$0.569818\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.20604e10	0.488438	0.244219	−	0.969720i	$-0.421468\pi$
$461$	2.20604e10	0.488438	0.244219	+	0.969720i	$0.421468\pi$
$462$	0	0
$463$	− 2.37920e10i	− 0.517734i	−0.965913	−	0.258867i	$-0.916651\pi$
$463$	− 2.37920e10i	− 0.517734i	0.965913	−	0.258867i	$-0.0833491\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.51836e10i	0.739729i	0.929086	+	0.369864i	$0.120596\pi$
$467$	3.51836e10i	0.739729i	−0.929086	+	0.369864i	$0.879404\pi$
$468$	0	0
$469$	−5.74637e10	−1.18769
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.71652e10	−0.942274
$474$	0	0
$475$	− 4.94429e10i	− 0.971247i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 7.76849e10i	− 1.47569i	−0.674971	−	0.737844i	$-0.735844\pi$
$479$	− 7.76849e10i	− 1.47569i	0.674971	−	0.737844i	$-0.264156\pi$
$480$	0	0
$481$	4.87753e10	0.911212
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.12167e10	−0.564183
$486$	0	0
$487$	− 7.39381e10i	− 1.31448i	−0.753683	−	0.657238i	$-0.771725\pi$
$487$	− 7.39381e10i	− 1.31448i	0.753683	−	0.657238i	$-0.228275\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.09306e10i	0.876299i	0.898902	+	0.438150i	$0.144366\pi$
$491$	5.09306e10i	0.876299i	−0.898902	+	0.438150i	$0.855634\pi$
$492$	0	0
$493$	−4.94241e10	−0.836663
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.01825e11	−1.66890
$498$	0	0
$499$	9.24638e9i	0.149132i	0.997216	+	0.0745658i	$0.0237571\pi$
$499$	9.24638e9i	0.149132i	−0.997216	+	0.0745658i	$0.976243\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.60365e10i	0.250517i	0.992124	+	0.125259i	$0.0399760\pi$
$503$	1.60365e10i	0.250517i	−0.992124	+	0.125259i	$0.960024\pi$
$504$	0	0
$505$	3.00125e9	0.0461463
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.34179e11	1.99900	0.999500	−	0.0316279i	$-0.0100692\pi$
$509$	1.34179e11	1.99900	0.999500	+	0.0316279i	$0.0100692\pi$
$510$	0	0
$511$	− 4.20007e10i	− 0.615989i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.47929e10i	0.494609i
$516$	0	0
$517$	1.67478e11	2.34421
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.26281e11	1.71391	0.856954	−	0.515392i	$-0.172354\pi$
$521$	1.26281e11	1.71391	0.856954	+	0.515392i	$0.172354\pi$
$522$	0	0
$523$	1.13332e11i	1.51476i	0.652973	+	0.757381i	$0.273521\pi$
$523$	1.13332e11i	1.51476i	−0.652973	+	0.757381i	$0.726479\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6.16909e10i	− 0.799794i
$528$	0	0
$529$	5.34791e9	0.0682906
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.27557e10	1.02539
$534$	0	0
$535$	3.39197e10i	0.414034i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.18337e11i	− 1.40205i
$540$	0	0
$541$	−1.02482e11	−1.19635	−0.598174	−	0.801366i	$-0.704107\pi$
$541$	−1.02482e11	−1.19635	−0.598174	+	0.801366i	$0.704107\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.02023e9	−0.0455685
$546$	0	0
$547$	− 1.56718e11i	− 1.75053i	−0.483648	−	0.875263i	$-0.660689\pi$
$547$	− 1.56718e11i	− 1.75053i	0.483648	−	0.875263i	$-0.339311\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1.28494e11i	− 1.39404i
$552$	0	0
$553$	2.07323e10	0.221690
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.50567e10	−0.779774	−0.389887	−	0.920863i	$-0.627486\pi$
$557$	−7.50567e10	−0.779774	−0.389887	+	0.920863i	$0.627486\pi$
$558$	0	0
$559$	3.89844e10i	0.399249i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1.69090e11i	− 1.68300i	−0.540255	−	0.841502i	$-0.681672\pi$
$563$	− 1.69090e11i	− 1.68300i	0.540255	−	0.841502i	$-0.318328\pi$
$564$	0	0
$565$	−1.19747e10	−0.117509
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.64531e11	1.56964	0.784818	−	0.619727i	$-0.212757\pi$
$569$	1.64531e11	1.56964	0.784818	+	0.619727i	$0.212757\pi$
$570$	0	0
$571$	− 1.61351e11i	− 1.51784i	−0.651182	−	0.758922i	$-0.725727\pi$
$571$	− 1.61351e11i	− 1.51784i	0.651182	−	0.758922i	$-0.274273\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.75343e10i	0.800768i
$576$	0	0
$577$	−5.65816e10	−0.510472	−0.255236	−	0.966879i	$-0.582153\pi$
$577$	−5.65816e10	−0.510472	−0.255236	+	0.966879i	$0.582153\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.73963e11	−2.40430
$582$	0	0
$583$	− 2.76041e10i	− 0.238946i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.21103e10i	0.691584i	0.938311	+	0.345792i	$0.112390\pi$
$587$	8.21103e10i	0.691584i	−0.938311	+	0.345792i	$0.887610\pi$
$588$	0	0
$589$	1.60385e11	1.33261
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.12807e10	0.172095	0.0860474	−	0.996291i	$-0.472576\pi$
$593$	2.12807e10	0.172095	0.0860474	+	0.996291i	$0.472576\pi$
$594$	0	0
$595$	4.99323e10i	0.398395i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.52874e10i	0.196425i	0.995165	+	0.0982126i	$0.0313125\pi$
$599$	2.52874e10i	0.196425i	−0.995165	+	0.0982126i	$0.968687\pi$
$600$	0	0
$601$	1.98139e9	0.0151870	0.00759349	−	0.999971i	$-0.497583\pi$
$601$	1.98139e9	0.0151870	0.00759349	+	0.999971i	$0.497583\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.30119e10	0.619610
$606$	0	0
$607$	− 1.14604e11i	− 0.844200i	−0.906549	−	0.422100i	$-0.861293\pi$
$607$	− 1.14604e11i	− 0.844200i	0.906549	−	0.422100i	$-0.138707\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.38429e11i	− 0.993260i
$612$	0	0
$613$	2.76203e10	0.195608	0.0978040	−	0.995206i	$-0.468818\pi$
$613$	2.76203e10	0.195608	0.0978040	+	0.995206i	$0.468818\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.65475e10	0.666193	0.333097	−	0.942893i	$-0.391906\pi$
$617$	9.65475e10	0.666193	0.333097	+	0.942893i	$0.391906\pi$
$618$	0	0
$619$	1.47548e11i	1.00501i	0.864575	+	0.502505i	$0.167588\pi$
$619$	1.47548e11i	1.00501i	−0.864575	+	0.502505i	$0.832412\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 5.54127e10i	− 0.367839i
$624$	0	0
$625$	7.90140e10	0.517826
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.49568e11	0.955510
$630$	0	0
$631$	1.55246e11i	0.979273i	0.871927	+	0.489636i	$0.162870\pi$
$631$	1.55246e11i	0.979273i	−0.871927	+	0.489636i	$0.837130\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 4.79276e10i	− 0.294775i
$636$	0	0
$637$	−9.78114e10	−0.594062
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.80081e10	0.462070	0.231035	−	0.972945i	$-0.425789\pi$
$641$	7.80081e10	0.462070	0.231035	+	0.972945i	$0.425789\pi$
$642$	0	0
$643$	− 1.17490e11i	− 0.687317i	−0.939095	−	0.343659i	$-0.888334\pi$
$643$	− 1.17490e11i	− 0.687317i	0.939095	−	0.343659i	$-0.111666\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 5.16493e10i	− 0.294745i	−0.989081	−	0.147373i	$-0.952918\pi$
$647$	− 5.16493e10i	− 0.294745i	0.989081	−	0.147373i	$-0.0470817\pi$
$648$	0	0
$649$	−7.83183e9	−0.0441453
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.22742e10	−0.122504	−0.0612520	−	0.998122i	$-0.519509\pi$
$653$	−2.22742e10	−0.122504	−0.0612520	+	0.998122i	$0.519509\pi$
$654$	0	0
$655$	1.38979e10i	0.0755066i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.20284e10i	0.328889i	0.986386	+	0.164444i	$0.0525831\pi$
$659$	6.20284e10i	0.328889i	−0.986386	+	0.164444i	$0.947417\pi$
$660$	0	0
$661$	−2.63096e11	−1.37819	−0.689093	−	0.724672i	$-0.741991\pi$
$661$	−2.63096e11	−1.37819	−0.689093	+	0.724672i	$0.741991\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.29815e11	−0.663801
$666$	0	0
$667$	2.27486e11i	1.14935i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.94836e11i	0.961125i
$672$	0	0
$673$	1.76424e11	0.859998	0.429999	−	0.902829i	$-0.358514\pi$
$673$	1.76424e11	0.859998	0.429999	+	0.902829i	$0.358514\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.46667e11	−1.17424	−0.587119	−	0.809501i	$-0.699738\pi$
$677$	−2.46667e11	−1.17424	−0.587119	+	0.809501i	$0.699738\pi$
$678$	0	0
$679$	− 3.99020e11i	− 1.87722i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.76508e11i	1.27065i	0.772245	+	0.635324i	$0.219134\pi$
$683$	2.76508e11i	1.27065i	−0.772245	+	0.635324i	$0.780866\pi$
$684$	0	0
$685$	−1.00833e11	−0.457974
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.28162e10	−0.101243
$690$	0	0
$691$	− 1.65551e11i	− 0.726139i	−0.931762	−	0.363069i	$-0.881729\pi$
$691$	− 1.65551e11i	− 0.726139i	0.931762	−	0.363069i	$-0.118271\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.03576e11i	0.443936i
$696$	0	0
$697$	2.53768e11	1.07524
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.76257e11	−1.14404	−0.572020	−	0.820240i	$-0.693840\pi$
$701$	−2.76257e11	−1.14404	−0.572020	+	0.820240i	$0.693840\pi$
$702$	0	0
$703$	3.88849e11i	1.59206i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.83628e10i	0.153544i
$708$	0	0
$709$	9.90119e10	0.391834	0.195917	−	0.980620i	$-0.437232\pi$
$709$	9.90119e10	0.391834	0.195917	+	0.980620i	$0.437232\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.83947e11	−1.09870
$714$	0	0
$715$	− 1.14326e11i	− 0.437441i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.65608e11i	0.619676i	0.950789	+	0.309838i	$0.100275\pi$
$719$	1.65608e11i	0.619676i	−0.950789	+	0.309838i	$0.899725\pi$
$720$	0	0
$721$	−4.44732e11	−1.64573
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.72917e11	0.987822
$726$	0	0
$727$	3.35426e11i	1.20077i	0.799712	+	0.600383i	$0.204985\pi$
$727$	3.35426e11i	1.20077i	−0.799712	+	0.600383i	$0.795015\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.19544e11i	0.418658i
$732$	0	0
$733$	−1.73168e11	−0.599863	−0.299932	−	0.953961i	$-0.596964\pi$
$733$	−1.73168e11	−0.599863	−0.299932	+	0.953961i	$0.596964\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.03453e11	1.36749
$738$	0	0
$739$	5.36956e11i	1.80036i	0.435513	+	0.900182i	$0.356567\pi$
$739$	5.36956e11i	1.80036i	−0.435513	+	0.900182i	$0.643433\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 5.97670e9i	− 0.0196113i	−0.999952	−	0.00980564i	$-0.996879\pi$
$743$	− 5.97670e9i	− 0.0196113i	0.999952	−	0.00980564i	$-0.00312128\pi$
$744$	0	0
$745$	1.94733e11	0.632140
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.33570e11	−1.37763
$750$	0	0
$751$	− 1.34684e11i	− 0.423405i	−0.977334	−	0.211703i	$-0.932099\pi$
$751$	− 1.34684e11i	− 0.423405i	0.977334	−	0.211703i	$-0.0679008\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.63270e10i	0.142576i
$756$	0	0
$757$	−5.86211e11	−1.78513	−0.892567	−	0.450915i	$-0.851098\pi$
$757$	−5.86211e11	−1.78513	−0.892567	+	0.450915i	$0.851098\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.93475e11	0.875049	0.437525	−	0.899206i	$-0.355855\pi$
$761$	2.93475e11	0.875049	0.437525	+	0.899206i	$0.355855\pi$
$762$	0	0
$763$	− 5.13876e10i	− 0.151621i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.47340e9i	0.0187047i
$768$	0	0
$769$	−2.61135e10	−0.0746724	−0.0373362	−	0.999303i	$-0.511887\pi$
$769$	−2.61135e10	−0.0746724	−0.0373362	+	0.999303i	$0.511887\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5.32516e11	−1.49147	−0.745735	−	0.666243i	$-0.767901\pi$
$773$	−5.32516e11	−1.49147	−0.745735	+	0.666243i	$0.767901\pi$
$774$	0	0
$775$	3.40654e11i	0.944292i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.59749e11i	1.79155i
$780$	0	0
$781$	7.14917e11	1.92155
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.21070e11	0.582172
$786$	0	0
$787$	− 3.20509e11i	− 0.835490i	−0.908564	−	0.417745i	$-0.862820\pi$
$787$	− 3.20509e11i	− 0.835490i	0.908564	−	0.417745i	$-0.137180\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1.53063e11i	− 0.390990i
$792$	0	0
$793$	1.61042e11	0.407237
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.88402e11	−0.714768	−0.357384	−	0.933958i	$-0.616331\pi$
$797$	−2.88402e11	−0.714768	−0.357384	+	0.933958i	$0.616331\pi$
$798$	0	0
$799$	− 4.24488e11i	− 1.04155i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.94887e11i	0.709241i
$804$	0	0
$805$	2.29826e11	0.547287
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.95879e11	0.690748	0.345374	−	0.938465i	$-0.387752\pi$
$809$	2.95879e11	0.690748	0.345374	+	0.938465i	$0.387752\pi$
$810$	0	0
$811$	1.70426e11i	0.393961i	0.980407	+	0.196980i	$0.0631136\pi$
$811$	1.70426e11i	0.393961i	−0.980407	+	0.196980i	$0.936886\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1.18719e11i	− 0.269086i
$816$	0	0
$817$	−3.10793e11	−0.697563
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.71581e11	−0.597760	−0.298880	−	0.954291i	$-0.596613\pi$
$821$	−2.71581e11	−0.597760	−0.298880	+	0.954291i	$0.596613\pi$
$822$	0	0
$823$	− 7.55264e11i	− 1.64626i	−0.567851	−	0.823131i	$-0.692225\pi$
$823$	− 7.55264e11i	− 1.64626i	0.567851	−	0.823131i	$-0.307775\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.91170e11i	− 1.90519i	−0.304240	−	0.952596i	$-0.598402\pi$
$827$	− 8.91170e11i	− 1.90519i	0.304240	−	0.952596i	$-0.401598\pi$
$828$	0	0
$829$	−5.33503e10	−0.112958	−0.0564792	−	0.998404i	$-0.517987\pi$
$829$	−5.33503e10	−0.112958	−0.0564792	+	0.998404i	$0.517987\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.99935e11	−0.622942
$834$	0	0
$835$	1.38850e11i	0.285627i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.28645e10i	0.0461439i	0.999734	+	0.0230720i	$0.00734469\pi$
$839$	2.28645e10i	0.0461439i	−0.999734	+	0.0230720i	$0.992655\pi$
$840$	0	0
$841$	2.09017e11	0.417829
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.15963e11	0.227453
$846$	0	0
$847$	1.06108e12i	2.06165i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 6.88422e11i	− 1.31261i
$852$	0	0
$853$	−8.98910e11	−1.69793	−0.848965	−	0.528449i	$-0.822774\pi$
$853$	−8.98910e11	−1.69793	−0.848965	+	0.528449i	$0.822774\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.50807e11	1.57728	0.788638	−	0.614857i	$-0.210786\pi$
$857$	8.50807e11	1.57728	0.788638	+	0.614857i	$0.210786\pi$
$858$	0	0
$859$	− 6.19591e11i	− 1.13797i	−0.822346	−	0.568987i	$-0.807335\pi$
$859$	− 6.19591e11i	− 1.13797i	0.822346	−	0.568987i	$-0.192665\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 3.14088e10i	− 0.0566250i	−0.999599	−	0.0283125i	$-0.990987\pi$
$863$	− 3.14088e10i	− 0.0566250i	0.999599	−	0.0283125i	$-0.00901335\pi$
$864$	0	0
$865$	−2.53039e11	−0.451983
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.45561e11	−0.255251
$870$	0	0
$871$	− 3.33474e11i	− 0.579415i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 6.08083e11i	− 1.03736i
$876$	0	0
$877$	5.09974e11	0.862084	0.431042	−	0.902332i	$-0.358146\pi$
$877$	5.09974e11	0.862084	0.431042	+	0.902332i	$0.358146\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.15781e11	0.856173	0.428087	−	0.903738i	$-0.359188\pi$
$881$	5.15781e11	0.856173	0.428087	+	0.903738i	$0.359188\pi$
$882$	0	0
$883$	4.89454e11i	0.805136i	0.915390	+	0.402568i	$0.131882\pi$
$883$	4.89454e11i	0.805136i	−0.915390	+	0.402568i	$0.868118\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.92939e11i	− 0.311692i	−0.987781	−	0.155846i	$-0.950190\pi$
$887$	− 1.92939e11i	− 0.311692i	0.987781	−	0.155846i	$-0.0498103\pi$
$888$	0	0
$889$	6.12623e11	0.980813
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.10359e12	1.73541
$894$	0	0
$895$	− 3.76279e11i	− 0.586432i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.85300e11i	1.35535i
$900$	0	0
$901$	−6.99651e10	−0.106165
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.78665e10	0.0415421
$906$	0	0
$907$	− 1.42541e11i	− 0.210626i	−0.994439	−	0.105313i	$-0.966416\pi$
$907$	− 1.42541e11i	− 0.210626i	0.994439	−	0.105313i	$-0.0335844\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.37631e11i	0.345008i	0.985009	+	0.172504i	$0.0551858\pi$
$911$	2.37631e11i	0.345008i	−0.985009	+	0.172504i	$0.944814\pi$
$912$	0	0
$913$	1.92350e12	2.76827
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.77647e11	−0.251235
$918$	0	0
$919$	1.08609e11i	0.152266i	0.997098	+	0.0761332i	$0.0242574\pi$
$919$	1.08609e11i	0.152266i	−0.997098	+	0.0761332i	$0.975743\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 5.90915e11i	− 0.814176i
$924$	0	0
$925$	−8.25905e11	−1.12814
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8.87938e11	−1.19212	−0.596060	−	0.802940i	$-0.703268\pi$
$929$	−8.87938e11	−1.19212	−0.596060	+	0.802940i	$0.703268\pi$
$930$	0	0
$931$	− 7.79777e11i	− 1.03794i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 3.50575e11i	− 0.458706i
$936$	0	0
$937$	−6.83385e11	−0.886557	−0.443279	−	0.896384i	$-0.646185\pi$
$937$	−6.83385e11	−0.886557	−0.443279	+	0.896384i	$0.646185\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.37589e11	0.430556	0.215278	−	0.976553i	$-0.430934\pi$
$941$	3.37589e11	0.430556	0.215278	+	0.976553i	$0.430934\pi$
$942$	0	0
$943$	− 1.16803e12i	− 1.47709i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.44705e11i	− 1.05028i	−0.851016	−	0.525140i	$-0.824013\pi$
$947$	− 8.44705e11i	− 1.05028i	0.851016	−	0.525140i	$-0.175987\pi$
$948$	0	0
$949$	2.43739e11	0.300511
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.63793e11	0.198574	0.0992871	−	0.995059i	$-0.468344\pi$
$953$	1.63793e11	0.198574	0.0992871	+	0.995059i	$0.468344\pi$
$954$	0	0
$955$	− 4.65816e11i	− 0.560016i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.28887e12i	− 1.52383i
$960$	0	0
$961$	−2.52135e11	−0.295625
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.66278e11	−0.537694
$966$	0	0
$967$	8.83990e11i	1.01098i	0.862833	+	0.505489i	$0.168688\pi$
$967$	8.83990e11i	1.01098i	−0.862833	+	0.505489i	$0.831312\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 3.35605e11i	− 0.377530i	−0.982022	−	0.188765i	$-0.939552\pi$
$971$	− 3.35605e11i	− 0.377530i	0.982022	−	0.188765i	$-0.0604485\pi$
$972$	0	0
$973$	−1.32394e12	−1.47712
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.95199e11	−0.323994	−0.161997	−	0.986791i	$-0.551794\pi$
$977$	−2.95199e11	−0.323994	−0.161997	+	0.986791i	$0.551794\pi$
$978$	0	0
$979$	3.89053e11i	0.423524i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.59775e11i	0.492415i	0.969217	+	0.246208i	$0.0791845\pi$
$983$	4.59775e11i	0.492415i	−0.969217	+	0.246208i	$0.920816\pi$
$984$	0	0
$985$	5.89449e10	0.0626183
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.50232e11	0.575123
$990$	0	0
$991$	1.43414e12i	1.48696i	0.668761	+	0.743478i	$0.266825\pi$
$991$	1.43414e12i	1.48696i	−0.668761	+	0.743478i	$0.733175\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.18144e11i	0.324587i
$996$	0	0
$997$	1.38321e12	1.39993	0.699965	−	0.714177i	$-0.253199\pi$
$997$	1.38321e12	1.39993	0.699965	+	0.714177i	$0.253199\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.9.g.d.127.1		2
3.2	odd	2		16.9.c.b.15.1	✓	2
4.3	odd	2	inner	144.9.g.d.127.2		2
12.11	even	2		16.9.c.b.15.2	yes	2
15.2	even	4		400.9.h.a.399.1		4
15.8	even	4		400.9.h.a.399.3		4
15.14	odd	2		400.9.b.e.351.2		2
24.5	odd	2		64.9.c.c.63.2		2
24.11	even	2		64.9.c.c.63.1		2
48.5	odd	4		256.9.d.d.127.2		4
48.11	even	4		256.9.d.d.127.4		4
48.29	odd	4		256.9.d.d.127.3		4
48.35	even	4		256.9.d.d.127.1		4
60.23	odd	4		400.9.h.a.399.2		4
60.47	odd	4		400.9.h.a.399.4		4
60.59	even	2		400.9.b.e.351.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.9.c.b.15.1	✓	2	3.2	odd	2
16.9.c.b.15.2	yes	2	12.11	even	2
64.9.c.c.63.1		2	24.11	even	2
64.9.c.c.63.2		2	24.5	odd	2
144.9.g.d.127.1		2	1.1	even	1	trivial
144.9.g.d.127.2		2	4.3	odd	2	inner
256.9.d.d.127.1		4	48.35	even	4
256.9.d.d.127.2		4	48.5	odd	4
256.9.d.d.127.3		4	48.29	odd	4
256.9.d.d.127.4		4	48.11	even	4
400.9.b.e.351.1		2	60.59	even	2
400.9.b.e.351.2		2	15.14	odd	2
400.9.h.a.399.1		4	15.2	even	4
400.9.h.a.399.2		4	60.23	odd	4
400.9.h.a.399.3		4	15.8	even	4
400.9.h.a.399.4		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	144.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-258.000 q^{5} -3297.82i q^{7} +23154.1i q^{11} +19138.0 q^{13} +58686.0 q^{17} +152573. i q^{19} -270117. i q^{23} -324061. q^{25} -842178. q^{29} -1.05120e6i q^{31} +850839. i q^{35} +2.54861e6 q^{37} +4.32416e6 q^{41} +2.03702e6i q^{43} -7.23321e6i q^{47} -5.11085e6 q^{49} -1.19219e6 q^{53} -5.97375e6i q^{55} +338249. i q^{59} +8.41479e6 q^{61} -4.93760e6 q^{65} -1.74247e7i q^{67} -3.08765e7i q^{71} +1.27359e7 q^{73} +7.63580e7 q^{77} +6.28665e6i q^{79} -8.30740e7i q^{83} -1.51410e7 q^{85} +1.68028e7 q^{89} -6.31138e7i q^{91} -3.93638e7i q^{95} +1.20995e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 516 q^{5} + 38276 q^{13} + 117372 q^{17} - 648122 q^{25} - 1684356 q^{29} + 5097220 q^{37} + 8648316 q^{41} - 10221694 q^{49} - 2384388 q^{53} + 16829572 q^{61} - 9875208 q^{65} + 25471748 q^{73}+ \cdots + 241989764 q^{97}+O(q^{100}) \) 2 * q - 516 * q^5 + 38276 * q^13 + 117372 * q^17 - 648122 * q^25 - 1684356 * q^29 + 5097220 * q^37 + 8648316 * q^41 - 10221694 * q^49 - 2384388 * q^53 + 16829572 * q^61 - 9875208 * q^65 + 25471748 * q^73 + 152716032 * q^77 - 30281976 * q^85 + 33605628 * q^89 + 241989764 * q^97

Introduction

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