LMFDB - Embedded newform 16.9.c.b.15.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [16,9,Mod(15,16)] 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("16.15"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(16, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	16.c (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:	$6.51805776098$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	16.15
Dual form			16.9.c.b.15.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+13.8564i q^{3} +258.000 q^{5} +3297.82i q^{7} +6369.00 q^{9} +23154.1i q^{11} +19138.0 q^{13} +3574.95i q^{15} -58686.0 q^{17} -152573. i q^{19} -45696.0 q^{21} -270117. i q^{23} -324061. q^{25} +179163. i q^{27} +842178. q^{29} +1.05120e6i q^{31} -320832. q^{33} +850839. i q^{35} +2.54861e6 q^{37} +265184. i q^{39} -4.32416e6 q^{41} -2.03702e6i q^{43} +1.64320e6 q^{45} -7.23321e6i q^{47} -5.11085e6 q^{49} -813177. i q^{51} +1.19219e6 q^{53} +5.97375e6i q^{55} +2.11411e6 q^{57} +338249. i q^{59} +8.41479e6 q^{61} +2.10038e7i q^{63} +4.93760e6 q^{65} +1.74247e7i q^{67} +3.74285e6 q^{69} -3.08765e7i q^{71} +1.27359e7 q^{73} -4.49032e6i q^{75} -7.63580e7 q^{77} -6.28665e6i q^{79} +3.93044e7 q^{81} -8.30740e7i q^{83} -1.51410e7 q^{85} +1.16696e7i q^{87} -1.68028e7 q^{89} +6.31138e7i q^{91} -1.45659e7 q^{93} -3.93638e7i q^{95} +1.20995e8 q^{97} +1.47468e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 516 q^{5} + 12738 q^{9} + 38276 q^{13} - 117372 q^{17} - 91392 q^{21} - 648122 q^{25} + 1684356 q^{29} - 641664 q^{33} + 5097220 q^{37} - 8648316 q^{41} + 3286404 q^{45} - 10221694 q^{49} + 2384388 q^{53}+ \cdots + 241989764 q^{97}+O(q^{100}) $ 2 * q + 516 * q^5 + 12738 * q^9 + 38276 * q^13 - 117372 * q^17 - 91392 * q^21 - 648122 * q^25 + 1684356 * q^29 - 641664 * q^33 + 5097220 * q^37 - 8648316 * q^41 + 3286404 * q^45 - 10221694 * q^49 + 2384388 * q^53 + 4228224 * q^57 + 16829572 * q^61 + 9875208 * q^65 + 7485696 * q^69 + 25471748 * q^73 - 152716032 * q^77 + 78608898 * q^81 - 30281976 * q^85 - 33605628 * q^89 - 29131776 * q^93 + 241989764 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$.

$n$	$5$	$15$
$\chi(n)$	$1$	$-1$

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
$3$	13.8564i	0.171067i	0.996335	+	0.0855334i	$0.0272594\pi$
$3$	13.8564i	0.171067i	−0.996335	+	0.0855334i	$0.972741\pi$
$4$	0	0
$5$	258.000	0.412800	0.206400	−	0.978468i	$-0.433825\pi$
$5$	258.000	0.412800	0.206400	+	0.978468i	$0.433825\pi$
$6$	0	0
$7$	3297.82i	1.37352i	0.726884	+	0.686761i	$0.240968\pi$
$7$	3297.82i	1.37352i	−0.726884	+	0.686761i	$0.759032\pi$
$8$	0	0
$9$	6369.00	0.970736
$10$	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
$15$	3574.95i	0.0706164i
$16$	0	0
$17$	−58686.0	−0.702650	−0.351325	−	0.936254i	$-0.614269\pi$
$17$	−58686.0	−0.702650	−0.351325	+	0.936254i	$0.614269\pi$
$18$	0	0
$19$	− 152573.i	− 1.17075i	−0.810764	−	0.585373i	$-0.800948\pi$
$19$	− 152573.i	− 1.17075i	0.810764	−	0.585373i	$-0.199052\pi$
$20$	0	0
$21$	−45696.0	−0.234964
$22$	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
$27$	179163.i	0.337127i
$28$	0	0
$29$	842178.	1.19073	0.595363	−	0.803457i	$-0.297008\pi$
$29$	842178.	1.19073	0.595363	+	0.803457i	$0.297008\pi$
$30$	0	0
$31$	1.05120e6i	1.13826i	0.822249	+	0.569128i	$0.192719\pi$
$31$	1.05120e6i	1.13826i	−0.822249	+	0.569128i	$0.807281\pi$
$32$	0	0
$33$	−320832.	−0.270534
$34$	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
$39$	265184.i	0.114627i
$40$	0	0
$41$	−4.32416e6	−1.53026	−0.765132	−	0.643874i	$-0.777326\pi$
$41$	−4.32416e6	−1.53026	−0.765132	+	0.643874i	$0.777326\pi$
$42$	0	0
$43$	− 2.03702e6i	− 0.595828i	−0.954593	−	0.297914i	$-0.903709\pi$
$43$	− 2.03702e6i	− 0.595828i	0.954593	−	0.297914i	$-0.0962908\pi$
$44$	0	0
$45$	1.64320e6	0.400720
$46$	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
$51$	− 813177.i	− 0.120200i
$52$	0	0
$53$	1.19219e6	0.151093	0.0755463	−	0.997142i	$-0.475930\pi$
$53$	1.19219e6	0.151093	0.0755463	+	0.997142i	$0.475930\pi$
$54$	0	0
$55$	5.97375e6i	0.652824i
$56$	0	0
$57$	2.11411e6	0.200276
$58$	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
$63$	2.10038e7i	1.33333i
$64$	0	0
$65$	4.93760e6	0.276607
$66$	0	0
$67$	1.74247e7i	0.864702i	0.901705	+	0.432351i	$0.142316\pi$
$67$	1.74247e7i	0.864702i	−0.901705	+	0.432351i	$0.857684\pi$
$68$	0	0
$69$	3.74285e6	0.165122
$70$	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
$75$	− 4.49032e6i	− 0.141916i
$76$	0	0
$77$	−7.63580e7	−2.17216
$78$	0	0
$79$	− 6.28665e6i	− 0.161403i	−0.996738	−	0.0807014i	$-0.974284\pi$
$79$	− 6.28665e6i	− 0.161403i	0.996738	−	0.0807014i	$-0.0257160\pi$
$80$	0	0
$81$	3.93044e7	0.913065
$82$	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
$87$	1.16696e7i	0.203694i
$88$	0	0
$89$	−1.68028e7	−0.267807	−0.133904	−	0.990994i	$-0.542751\pi$
$89$	−1.68028e7	−0.267807	−0.133904	+	0.990994i	$0.542751\pi$
$90$	0	0
$91$	6.31138e7i	0.920362i
$92$	0	0
$93$	−1.45659e7	−0.194718
$94$	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
$99$	1.47468e8i	1.53517i
$100$	0	0
$101$	1.16328e7	0.111789	0.0558943	−	0.998437i	$-0.482199\pi$
$101$	1.16328e7	0.111789	0.0558943	+	0.998437i	$0.482199\pi$
$102$	0	0
$103$	1.34856e8i	1.19818i	0.800682	+	0.599090i	$0.204471\pi$
$103$	1.34856e8i	1.19818i	−0.800682	+	0.599090i	$0.795529\pi$
$104$	0	0
$105$	−1.17896e7	−0.0969931
$106$	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
$111$	3.53146e7i	0.232628i
$112$	0	0
$113$	−4.64134e7	−0.284662	−0.142331	−	0.989819i	$-0.545460\pi$
$113$	−4.64134e7	−0.284662	−0.142331	+	0.989819i	$0.545460\pi$
$114$	0	0
$115$	− 6.96901e7i	− 0.398456i
$116$	0	0
$117$	1.21890e8	0.650466
$118$	0	0
$119$	− 1.93536e8i	− 0.965104i
$120$	0	0
$121$	−3.21751e8	−1.50099
$122$	0	0
$123$	− 5.99173e7i	− 0.261777i
$124$	0	0
$125$	−1.84389e8	−0.755257
$126$	0	0
$127$	− 1.85766e8i	− 0.714087i	−0.934088	−	0.357043i	$-0.883785\pi$
$127$	− 1.85766e8i	− 0.714087i	0.934088	−	0.357043i	$-0.116215\pi$
$128$	0	0
$129$	2.82257e7	0.101926
$130$	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
$135$	4.62241e7i	0.139166i
$136$	0	0
$137$	−3.90826e8	−1.10943	−0.554716	−	0.832040i	$-0.687173\pi$
$137$	−3.90826e8	−1.10943	−0.554716	+	0.832040i	$0.687173\pi$
$138$	0	0
$139$	4.01458e8i	1.07543i	0.843128	+	0.537713i	$0.180712\pi$
$139$	4.01458e8i	1.07543i	−0.843128	+	0.537713i	$0.819288\pi$
$140$	0	0
$141$	1.00226e8	0.253574
$142$	0	0
$143$	4.43122e8i	1.05969i
$144$	0	0
$145$	2.17282e8	0.491532
$146$	0	0
$147$	− 7.08180e7i	− 0.151661i
$148$	0	0
$149$	7.54777e8	1.53135	0.765674	−	0.643229i	$-0.222406\pi$
$149$	7.54777e8	1.53135	0.765674	+	0.643229i	$0.222406\pi$
$150$	0	0
$151$	1.79562e8i	0.345388i	0.984976	+	0.172694i	$0.0552471\pi$
$151$	1.79562e8i	0.345388i	−0.984976	+	0.172694i	$0.944753\pi$
$152$	0	0
$153$	−3.73771e8	−0.682087
$154$	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
$159$	1.65195e7i	0.0258469i
$160$	0	0
$161$	8.90798e8	1.32579
$162$	0	0
$163$	− 4.60153e8i	− 0.651856i	−0.945395	−	0.325928i	$-0.894323\pi$
$163$	− 4.60153e8i	− 0.651856i	0.945395	−	0.325928i	$-0.105677\pi$
$164$	0	0
$165$	−8.27747e7	−0.111676
$166$	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
$171$	− 9.71737e8i	− 1.13649i
$172$	0	0
$173$	−9.80770e8	−1.09492	−0.547461	−	0.836832i	$-0.684405\pi$
$173$	−9.80770e8	−1.09492	−0.547461	+	0.836832i	$0.684405\pi$
$174$	0	0
$175$	− 1.06870e9i	− 1.13947i
$176$	0	0
$177$	−4.68691e6	−0.00477522
$178$	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
$183$	1.16599e8i	0.103966i
$184$	0	0
$185$	6.57541e8	0.561353
$186$	0	0
$187$	− 1.35882e9i	− 1.11121i
$188$	0	0
$189$	−5.90849e8	−0.463052
$190$	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
$195$	6.84174e7i	0.0473182i
$196$	0	0
$197$	2.28469e8	0.151692	0.0758459	−	0.997120i	$-0.475834\pi$
$197$	2.28469e8	0.151692	0.0758459	+	0.997120i	$0.475834\pi$
$198$	0	0
$199$	1.23311e9i	0.786305i	0.919473	+	0.393153i	$0.128616\pi$
$199$	1.23311e9i	0.786305i	−0.919473	+	0.393153i	$0.871384\pi$
$200$	0	0
$201$	−2.41444e8	−0.147922
$202$	0	0
$203$	2.77736e9i	1.63549i
$204$	0	0
$205$	−1.11563e9	−0.631693
$206$	0	0
$207$	− 1.72037e9i	− 0.937004i
$208$	0	0
$209$	3.53268e9	1.85148
$210$	0	0
$211$	2.15314e8i	0.108628i	0.998524	+	0.0543141i	$0.0172972\pi$
$211$	2.15314e8i	0.108628i	−0.998524	+	0.0543141i	$0.982703\pi$
$212$	0	0
$213$	4.27838e8	0.207855
$214$	0	0
$215$	− 5.25550e8i	− 0.245958i
$216$	0	0
$217$	−3.46668e9	−1.56342
$218$	0	0
$219$	1.76473e8i	0.0767190i
$220$	0	0
$221$	−1.12313e9	−0.470828
$222$	0	0
$223$	− 4.09616e8i	− 0.165637i	−0.996565	−	0.0828185i	$-0.973608\pi$
$223$	− 4.09616e8i	− 0.165637i	0.996565	−	0.0828185i	$-0.0263922\pi$
$224$	0	0
$225$	−2.06394e9	−0.805319
$226$	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
$231$	− 1.05805e9i	− 0.371584i
$232$	0	0
$233$	−4.91979e8	−0.166925	−0.0834627	−	0.996511i	$-0.526598\pi$
$233$	−4.91979e8	−0.166925	−0.0834627	+	0.996511i	$0.526598\pi$
$234$	0	0
$235$	− 1.86617e9i	− 0.611898i
$236$	0	0
$237$	8.71104e7	0.0276106
$238$	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
$243$	1.72011e9i	0.493322i
$244$	0	0
$245$	−1.31860e9	−0.365972
$246$	0	0
$247$	− 2.91994e9i	− 0.784488i
$248$	0	0
$249$	1.15111e9	0.299446
$250$	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
$255$	− 2.09800e8i	− 0.0496186i
$256$	0	0
$257$	−2.97422e9	−0.681774	−0.340887	−	0.940104i	$-0.610727\pi$
$257$	−2.97422e9	−0.681774	−0.340887	+	0.940104i	$0.610727\pi$
$258$	0	0
$259$	8.40487e9i	1.86781i
$260$	0	0
$261$	5.36383e9	1.15588
$262$	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
$267$	− 2.32827e8i	− 0.0458129i
$268$	0	0
$269$	1.48778e7	0.00284137	0.00142069	−	0.999999i	$-0.499548\pi$
$269$	1.48778e7	0.00284137	0.00142069	+	0.999999i	$0.499548\pi$
$270$	0	0
$271$	1.72933e9i	0.320627i	0.987066	+	0.160313i	$0.0512505\pi$
$271$	1.72933e9i	0.320627i	−0.987066	+	0.160313i	$0.948750\pi$
$272$	0	0
$273$	−8.74530e8	−0.157443
$274$	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
$279$	6.69511e9i	1.10495i
$280$	0	0
$281$	3.75819e9	0.602773	0.301386	−	0.953502i	$-0.402551\pi$
$281$	3.75819e9	0.602773	0.301386	+	0.953502i	$0.402551\pi$
$282$	0	0
$283$	− 1.14220e9i	− 0.178072i	−0.996028	−	0.0890358i	$-0.971621\pi$
$283$	− 1.14220e9i	− 0.178072i	0.996028	−	0.0890358i	$-0.0283786\pi$
$284$	0	0
$285$	5.45441e8	0.0826739
$286$	0	0
$287$	− 1.42603e10i	− 2.10185i
$288$	0	0
$289$	−3.53171e9	−0.506283
$290$	0	0
$291$	1.67655e9i	0.233801i
$292$	0	0
$293$	−1.10157e10	−1.49466	−0.747328	−	0.664456i	$-0.768663\pi$
$293$	−1.10157e10	−1.49466	−0.747328	+	0.664456i	$0.768663\pi$
$294$	0	0
$295$	8.72682e7i	0.0115231i
$296$	0	0
$297$	−4.14836e9	−0.533151
$298$	0	0
$299$	− 5.16950e9i	− 0.646790i
$300$	0	0
$301$	6.71772e9	0.818382
$302$	0	0
$303$	1.61188e8i	0.0191233i
$304$	0	0
$305$	2.17101e9	0.250878
$306$	0	0
$307$	− 4.87606e9i	− 0.548928i	−0.961597	−	0.274464i	$-0.911500\pi$
$307$	− 4.87606e9i	− 0.548928i	0.961597	−	0.274464i	$-0.0885004\pi$
$308$	0	0
$309$	−1.86862e9	−0.204969
$310$	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
$315$	5.41899e9i	0.550397i
$316$	0	0
$317$	−1.43464e10	−1.42071	−0.710356	−	0.703842i	$-0.751466\pi$
$317$	−1.43464e10	−1.42071	−0.710356	+	0.703842i	$0.751466\pi$
$318$	0	0
$319$	1.94998e10i	1.88308i
$320$	0	0
$321$	1.82172e9	0.171578
$322$	0	0
$323$	8.95389e9i	0.822625i
$324$	0	0
$325$	−6.20188e9	−0.555891
$326$	0	0
$327$	2.15914e8i	0.0188838i
$328$	0	0
$329$	2.38539e10	2.03599
$330$	0	0
$331$	4.66058e9i	0.388265i	0.980975	+	0.194132i	$0.0621891\pi$
$331$	4.66058e9i	0.388265i	−0.980975	+	0.194132i	$0.937811\pi$
$332$	0	0
$333$	1.62321e10	1.32007
$334$	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
$339$	− 6.43123e8i	− 0.0486963i
$340$	0	0
$341$	−2.43396e10	−1.80010
$342$	0	0
$343$	2.15663e9i	0.155811i
$344$	0	0
$345$	9.65655e8	0.0681625
$346$	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
$351$	3.42883e9i	0.225901i
$352$	0	0
$353$	−4.45970e9	−0.287214	−0.143607	−	0.989635i	$-0.545870\pi$
$353$	−4.45970e9	−0.287214	−0.143607	+	0.989635i	$0.545870\pi$
$354$	0	0
$355$	− 7.96615e9i	− 0.501574i
$356$	0	0
$357$	2.68172e9	0.165097
$358$	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
$363$	− 4.45832e9i	− 0.256770i
$364$	0	0
$365$	3.28586e9	0.185130
$366$	0	0
$367$	− 3.12246e10i	− 1.72121i	−0.509276	−	0.860603i	$-0.670087\pi$
$367$	− 3.12246e10i	− 1.72121i	0.509276	−	0.860603i	$-0.329913\pi$
$368$	0	0
$369$	−2.75406e10	−1.48548
$370$	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
$375$	− 2.55497e9i	− 0.129199i
$376$	0	0
$377$	1.61176e10	0.797875
$378$	0	0
$379$	− 3.79438e9i	− 0.183901i	−0.995764	−	0.0919506i	$-0.970690\pi$
$379$	− 3.79438e9i	− 0.183901i	0.995764	−	0.0919506i	$-0.0293102\pi$
$380$	0	0
$381$	2.57405e9	0.122157
$382$	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
$387$	− 1.29738e10i	− 0.578392i
$388$	0	0
$389$	2.16828e10	0.946926	0.473463	−	0.880814i	$-0.343004\pi$
$389$	2.16828e10	0.946926	0.473463	+	0.880814i	$0.343004\pi$
$390$	0	0
$391$	1.58521e10i	0.678233i
$392$	0	0
$393$	7.46416e8	0.0312904
$394$	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
$399$	6.97197e9i	0.275083i
$400$	0	0
$401$	8.44054e9	0.326432	0.163216	−	0.986590i	$-0.447813\pi$
$401$	8.44054e9	0.326432	0.163216	+	0.986590i	$0.447813\pi$
$402$	0	0
$403$	2.01179e10i	0.762716i
$404$	0	0
$405$	1.01405e10	0.376913
$406$	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
$411$	− 5.41544e9i	− 0.189787i
$412$	0	0
$413$	−1.11549e9	−0.0383410
$414$	0	0
$415$	− 2.14331e10i	− 0.722591i
$416$	0	0
$417$	−5.56276e9	−0.183970
$418$	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
$423$	− 4.60683e10i	− 1.43893i
$424$	0	0
$425$	1.90178e10	0.582915
$426$	0	0
$427$	2.77505e10i	0.834755i
$428$	0	0
$429$	−6.14008e9	−0.181278
$430$	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
$435$	3.01075e9i	0.0840847i
$436$	0	0
$437$	−4.12125e10	−1.13006
$438$	0	0
$439$	− 5.78110e10i	− 1.55651i	−0.627948	−	0.778255i	$-0.716105\pi$
$439$	− 5.78110e10i	− 1.55651i	0.627948	−	0.778255i	$-0.283895\pi$
$440$	0	0
$441$	−3.25510e10	−0.860617
$442$	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
$447$	1.04585e10i	0.261963i
$448$	0	0
$449$	4.37962e10	1.07758	0.538792	−	0.842439i	$-0.318881\pi$
$449$	4.37962e10	1.07758	0.538792	+	0.842439i	$0.318881\pi$
$450$	0	0
$451$	− 1.00122e11i	− 2.42004i
$452$	0	0
$453$	−2.48809e9	−0.0590844
$454$	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
$459$	− 1.05144e10i	− 0.236882i
$460$	0	0
$461$	−2.20604e10	−0.488438	−0.244219	−	0.969720i	$-0.578532\pi$
$461$	−2.20604e10	−0.488438	−0.244219	+	0.969720i	$0.578532\pi$
$462$	0	0
$463$	2.37920e10i	0.517734i	0.965913	+	0.258867i	$0.0833491\pi$
$463$	2.37920e10i	0.517734i	−0.965913	+	0.258867i	$0.916651\pi$
$464$	0	0
$465$	−3.75800e9	−0.0803794
$466$	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
$471$	− 1.18730e10i	− 0.241255i
$472$	0	0
$473$	4.71652e10	0.942274
$474$	0	0
$475$	4.94429e10i	0.971247i
$476$	0	0
$477$	7.59308e9	0.146671
$478$	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
$483$	1.23433e10i	0.226799i
$484$	0	0
$485$	3.12167e10	0.564183
$486$	0	0
$487$	7.39381e10i	1.31448i	0.753683	+	0.657238i	$0.228275\pi$
$487$	7.39381e10i	1.31448i	−0.753683	+	0.657238i	$0.771725\pi$
$488$	0	0
$489$	6.37607e9	0.111511
$490$	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
$495$	3.80468e10i	0.633720i
$496$	0	0
$497$	1.01825e11	1.66890
$498$	0	0
$499$	− 9.24638e9i	− 0.149132i	−0.997216	−	0.0745658i	$-0.976243\pi$
$499$	− 9.24638e9i	− 0.149132i	0.997216	−	0.0745658i	$-0.0237571\pi$
$500$	0	0
$501$	7.45720e9	0.118365
$502$	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
$507$	− 6.22801e9i	− 0.0942578i
$508$	0	0
$509$	−1.34179e11	−1.99900	−0.999500	−	0.0316279i	$-0.989931\pi$
$509$	−1.34179e11	−1.99900	−0.999500	+	0.0316279i	$0.989931\pi$
$510$	0	0
$511$	4.20007e10i	0.615989i
$512$	0	0
$513$	2.73355e10	0.394691
$514$	0	0
$515$	3.47929e10i	0.494609i
$516$	0	0
$517$	1.67478e11	2.34421
$518$	0	0
$519$	− 1.35899e10i	− 0.187305i
$520$	0	0
$521$	−1.26281e11	−1.71391	−0.856954	−	0.515392i	$-0.827646\pi$
$521$	−1.26281e11	−1.71391	−0.856954	+	0.515392i	$0.827646\pi$
$522$	0	0
$523$	− 1.13332e11i	− 1.51476i	−0.652973	−	0.757381i	$-0.726479\pi$
$523$	− 1.13332e11i	− 1.51476i	0.652973	−	0.757381i	$-0.273521\pi$
$524$	0	0
$525$	1.48083e10	0.194925
$526$	0	0
$527$	− 6.16909e10i	− 0.799794i
$528$	0	0
$529$	5.34791e9	0.0682906
$530$	0	0
$531$	2.15431e9i	0.0270975i
$532$	0	0
$533$	−8.27557e10	−1.02539
$534$	0	0
$535$	− 3.39197e10i	− 0.414034i
$536$	0	0
$537$	−2.02088e10	−0.243021
$538$	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
$543$	− 1.49663e9i	− 0.0172153i
$544$	0	0
$545$	4.02023e9	0.0455685
$546$	0	0
$547$	1.56718e11i	1.75053i	0.483648	+	0.875263i	$0.339311\pi$
$547$	1.56718e11i	1.75053i	−0.483648	+	0.875263i	$0.660689\pi$
$548$	0	0
$549$	5.35938e10	0.589963
$550$	0	0
$551$	− 1.28494e11i	− 1.39404i
$552$	0	0
$553$	2.07323e10	0.221690
$554$	0	0
$555$	9.11116e9i	0.0960289i
$556$	0	0
$557$	7.50567e10	0.779774	0.389887	−	0.920863i	$-0.372514\pi$
$557$	7.50567e10	0.779774	0.389887	+	0.920863i	$0.372514\pi$
$558$	0	0
$559$	− 3.89844e10i	− 0.399249i
$560$	0	0
$561$	1.88283e10	0.190091
$562$	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
$567$	1.29619e11i	1.25411i
$568$	0	0
$569$	−1.64531e11	−1.56964	−0.784818	−	0.619727i	$-0.787243\pi$
$569$	−1.64531e11	−1.56964	−0.784818	+	0.619727i	$0.787243\pi$
$570$	0	0
$571$	1.61351e11i	1.51784i	0.651182	+	0.758922i	$0.274273\pi$
$571$	1.61351e11i	1.51784i	−0.651182	+	0.758922i	$0.725727\pi$
$572$	0	0
$573$	−2.50176e10	−0.232074
$574$	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
$579$	2.50424e10i	0.222824i
$580$	0	0
$581$	2.73963e11	2.40430
$582$	0	0
$583$	2.76041e10i	0.238946i
$584$	0	0
$585$	3.14476e10	0.268512
$586$	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
$591$	3.16576e9i	0.0259494i
$592$	0	0
$593$	−2.12807e10	−0.172095	−0.0860474	−	0.996291i	$-0.527424\pi$
$593$	−2.12807e10	−0.172095	−0.0860474	+	0.996291i	$0.527424\pi$
$594$	0	0
$595$	− 4.99323e10i	− 0.398395i
$596$	0	0
$597$	−1.70865e10	−0.134511
$598$	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
$603$	1.10978e11i	0.839398i
$604$	0	0
$605$	−8.30119e10	−0.619610
$606$	0	0
$607$	1.14604e11i	0.844200i	0.906549	+	0.422100i	$0.138707\pi$
$607$	1.14604e11i	0.844200i	−0.906549	+	0.422100i	$0.861293\pi$
$608$	0	0
$609$	−3.84842e10	−0.279778
$610$	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
$615$	− 1.54587e10i	− 0.108062i
$616$	0	0
$617$	−9.65475e10	−0.666193	−0.333097	−	0.942893i	$-0.608094\pi$
$617$	−9.65475e10	−0.666193	−0.333097	+	0.942893i	$0.608094\pi$
$618$	0	0
$619$	− 1.47548e11i	− 1.00501i	−0.864575	−	0.502505i	$-0.832412\pi$
$619$	− 1.47548e11i	− 1.00501i	0.864575	−	0.502505i	$-0.167588\pi$
$620$	0	0
$621$	4.83950e10	0.325413
$622$	0	0
$623$	− 5.54127e10i	− 0.367839i
$624$	0	0
$625$	7.90140e10	0.517826
$626$	0	0
$627$	4.89503e10i	0.316727i
$628$	0	0
$629$	−1.49568e11	−0.955510
$630$	0	0
$631$	− 1.55246e11i	− 0.979273i	−0.871927	−	0.489636i	$-0.837130\pi$
$631$	− 1.55246e11i	− 0.979273i	0.871927	−	0.489636i	$-0.162870\pi$
$632$	0	0
$633$	−2.98348e9	−0.0185827
$634$	0	0
$635$	− 4.79276e10i	− 0.294775i
$636$	0	0
$637$	−9.78114e10	−0.594062
$638$	0	0
$639$	− 1.96653e11i	− 1.17950i
$640$	0	0
$641$	−7.80081e10	−0.462070	−0.231035	−	0.972945i	$-0.574211\pi$
$641$	−7.80081e10	−0.462070	−0.231035	+	0.972945i	$0.574211\pi$
$642$	0	0
$643$	1.17490e11i	0.687317i	0.939095	+	0.343659i	$0.111666\pi$
$643$	1.17490e11i	0.687317i	−0.939095	+	0.343659i	$0.888334\pi$
$644$	0	0
$645$	7.28224e9	0.0420752
$646$	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
$651$	− 4.80357e10i	− 0.267449i
$652$	0	0
$653$	2.22742e10	0.122504	0.0612520	−	0.998122i	$-0.480491\pi$
$653$	2.22742e10	0.122504	0.0612520	+	0.998122i	$0.480491\pi$
$654$	0	0
$655$	− 1.38979e10i	− 0.0755066i
$656$	0	0
$657$	8.11148e10	0.435350
$658$	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
$663$	− 1.55626e10i	− 0.0805430i
$664$	0	0
$665$	1.29815e11	0.663801
$666$	0	0
$667$	− 2.27486e11i	− 1.14935i
$668$	0	0
$669$	5.67581e9	0.0283350
$670$	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
$675$	− 5.80598e10i	− 0.279680i
$676$	0	0
$677$	2.46667e11	1.17424	0.587119	−	0.809501i	$-0.300262\pi$
$677$	2.46667e11	1.17424	0.587119	+	0.809501i	$0.300262\pi$
$678$	0	0
$679$	3.99020e11i	1.87722i
$680$	0	0
$681$	7.23766e10	0.336519
$682$	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
$687$	3.97152e10i	0.178291i
$688$	0	0
$689$	2.28162e10	0.101243
$690$	0	0
$691$	1.65551e11i	0.726139i	0.931762	+	0.363069i	$0.118271\pi$
$691$	1.65551e11i	0.726139i	−0.931762	+	0.363069i	$0.881729\pi$
$692$	0	0
$693$	−4.86324e11	−2.10859
$694$	0	0
$695$	1.03576e11i	0.443936i
$696$	0	0
$697$	2.53768e11	1.07524
$698$	0	0
$699$	− 6.81706e9i	− 0.0285554i
$700$	0	0
$701$	2.76257e11	1.14404	0.572020	−	0.820240i	$-0.306160\pi$
$701$	2.76257e11	1.14404	0.572020	+	0.820240i	$0.306160\pi$
$702$	0	0
$703$	− 3.88849e11i	− 1.59206i
$704$	0	0
$705$	2.58584e10	0.104675
$706$	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
$711$	− 4.00397e10i	− 0.156680i
$712$	0	0
$713$	2.83947e11	1.09870
$714$	0	0
$715$	1.14326e11i	0.437441i
$716$	0	0
$717$	−4.21849e10	−0.159617
$718$	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
$723$	− 4.29880e10i	− 0.157324i
$724$	0	0
$725$	−2.72917e11	−0.987822
$726$	0	0
$727$	− 3.35426e11i	− 1.20077i	−0.799712	−	0.600383i	$-0.795015\pi$
$727$	− 3.35426e11i	− 1.20077i	0.799712	−	0.600383i	$-0.204985\pi$
$728$	0	0
$729$	2.34042e11	0.828674
$730$	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
$735$	− 1.82710e10i	− 0.0626057i
$736$	0	0
$737$	−4.03453e11	−1.36749
$738$	0	0
$739$	− 5.36956e11i	− 1.80036i	−0.435513	−	0.900182i	$-0.643433\pi$
$739$	− 5.36956e11i	− 1.80036i	0.435513	−	0.900182i	$-0.356567\pi$
$740$	0	0
$741$	4.04599e10	0.134200
$742$	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
$747$	− 5.29098e11i	− 1.69924i
$748$	0	0
$749$	4.33570e11	1.37763
$750$	0	0
$751$	1.34684e11i	0.423405i	0.977334	+	0.211703i	$0.0679008\pi$
$751$	1.34684e11i	0.423405i	−0.977334	+	0.211703i	$0.932099\pi$
$752$	0	0
$753$	−2.62572e10	−0.0816710
$754$	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
$759$	8.66621e10i	0.261133i
$760$	0	0
$761$	−2.93475e11	−0.875049	−0.437525	−	0.899206i	$-0.644145\pi$
$761$	−2.93475e11	−0.875049	−0.437525	+	0.899206i	$0.644145\pi$
$762$	0	0
$763$	5.13876e10i	0.151621i
$764$	0	0
$765$	−9.64330e10	−0.281566
$766$	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
$771$	− 4.12120e10i	− 0.116629i
$772$	0	0
$773$	5.32516e11	1.49147	0.745735	−	0.666243i	$-0.232099\pi$
$773$	5.32516e11	1.49147	0.745735	+	0.666243i	$0.232099\pi$
$774$	0	0
$775$	− 3.40654e11i	− 0.944292i
$776$	0	0
$777$	−1.16461e11	−0.319520
$778$	0	0
$779$	6.59749e11i	1.79155i
$780$	0	0
$781$	7.14917e11	1.92155
$782$	0	0
$783$	1.50887e11i	0.401426i
$784$	0	0
$785$	−2.21070e11	−0.582172
$786$	0	0
$787$	3.20509e11i	0.835490i	0.908564	+	0.417745i	$0.137180\pi$
$787$	3.20509e11i	0.835490i	−0.908564	+	0.417745i	$0.862820\pi$
$788$	0	0
$789$	−4.36862e10	−0.112729
$790$	0	0
$791$	− 1.53063e11i	− 0.390990i
$792$	0	0
$793$	1.61042e11	0.407237
$794$	0	0
$795$	4.26204e9i	0.0106696i
$796$	0	0
$797$	2.88402e11	0.714768	0.357384	−	0.933958i	$-0.383669\pi$
$797$	2.88402e11	0.714768	0.357384	+	0.933958i	$0.383669\pi$
$798$	0	0
$799$	4.24488e11i	1.04155i
$800$	0	0
$801$	−1.07017e11	−0.259970
$802$	0	0
$803$	2.94887e11i	0.709241i
$804$	0	0
$805$	2.29826e11	0.547287
$806$	0	0
$807$	2.06152e8i	0 0.000486065i
$808$	0	0
$809$	−2.95879e11	−0.690748	−0.345374	−	0.938465i	$-0.612248\pi$
$809$	−2.95879e11	−0.690748	−0.345374	+	0.938465i	$0.612248\pi$
$810$	0	0
$811$	− 1.70426e11i	− 0.393961i	−0.980407	−	0.196980i	$-0.936886\pi$
$811$	− 1.70426e11i	− 0.393961i	0.980407	−	0.196980i	$-0.0631136\pi$
$812$	0	0
$813$	−2.39623e10	−0.0548486
$814$	0	0
$815$	− 1.18719e11i	− 0.269086i
$816$	0	0
$817$	−3.10793e11	−0.697563
$818$	0	0
$819$	4.01972e11i	0.893428i
$820$	0	0
$821$	2.71581e11	0.597760	0.298880	−	0.954291i	$-0.403387\pi$
$821$	2.71581e11	0.597760	0.298880	+	0.954291i	$0.403387\pi$
$822$	0	0
$823$	7.55264e11i	1.64626i	0.567851	+	0.823131i	$0.307775\pi$
$823$	7.55264e11i	1.64626i	−0.567851	+	0.823131i	$0.692225\pi$
$824$	0	0
$825$	1.03969e11	0.224434
$826$	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
$831$	− 4.21487e10i	− 0.0883853i
$832$	0	0
$833$	2.99935e11	0.622942
$834$	0	0
$835$	− 1.38850e11i	− 0.285627i
$836$	0	0
$837$	−1.88337e11	−0.383737
$838$	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
$843$	5.20750e10i	0.103114i
$844$	0	0
$845$	−1.15963e11	−0.227453
$846$	0	0
$847$	− 1.06108e12i	− 2.06165i
$848$	0	0
$849$	1.58267e10	0.0304621
$850$	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
$855$	− 2.50708e11i	− 0.469142i
$856$	0	0
$857$	−8.50807e11	−1.57728	−0.788638	−	0.614857i	$-0.789214\pi$
$857$	−8.50807e11	−1.57728	−0.788638	+	0.614857i	$0.789214\pi$
$858$	0	0
$859$	6.19591e11i	1.13797i	0.822346	+	0.568987i	$0.192665\pi$
$859$	6.19591e11i	1.13797i	−0.822346	+	0.568987i	$0.807335\pi$
$860$	0	0
$861$	1.97597e11	0.359556
$862$	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
$867$	− 4.89368e10i	− 0.0866083i
$868$	0	0
$869$	1.45561e11	0.255251
$870$	0	0
$871$	3.33474e11i	0.579415i
$872$	0	0
$873$	7.70616e11	1.32673
$874$	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
$879$	− 1.52638e11i	− 0.255686i
$880$	0	0
$881$	−5.15781e11	−0.856173	−0.428087	−	0.903738i	$-0.640812\pi$
$881$	−5.15781e11	−0.856173	−0.428087	+	0.903738i	$0.640812\pi$
$882$	0	0
$883$	− 4.89454e11i	− 0.805136i	−0.915390	−	0.402568i	$-0.868118\pi$
$883$	− 4.89454e11i	− 0.805136i	0.915390	−	0.402568i	$-0.131882\pi$
$884$	0	0
$885$	−1.20922e9	−0.00197121
$886$	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
$891$	9.10057e11i	1.44397i
$892$	0	0
$893$	−1.10359e12	−1.73541
$894$	0	0
$895$	3.76279e11i	0.586432i
$896$	0	0
$897$	7.16306e10	0.110644
$898$	0	0
$899$	8.85300e11i	1.35535i
$900$	0	0
$901$	−6.99651e10	−0.106165
$902$	0	0
$903$	9.30835e10i	0.139998i
$904$	0	0
$905$	−2.78665e10	−0.0415421
$906$	0	0
$907$	1.42541e11i	0.210626i	0.994439	+	0.105313i	$0.0335844\pi$
$907$	1.42541e11i	0.210626i	−0.994439	+	0.105313i	$0.966416\pi$
$908$	0	0
$909$	7.40891e10	0.108517
$910$	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
$915$	3.00825e10i	0.0429170i
$916$	0	0
$917$	1.77647e11	0.251235
$918$	0	0
$919$	− 1.08609e11i	− 0.152266i	−0.997098	−	0.0761332i	$-0.975743\pi$
$919$	− 1.08609e11i	− 0.152266i	0.997098	−	0.0761332i	$-0.0242574\pi$
$920$	0	0
$921$	6.75646e10	0.0939033
$922$	0	0
$923$	− 5.90915e11i	− 0.814176i
$924$	0	0
$925$	−8.25905e11	−1.12814
$926$	0	0
$927$	8.58899e11i	1.16312i
$928$	0	0
$929$	8.87938e11	1.19212	0.596060	−	0.802940i	$-0.296732\pi$
$929$	8.87938e11	1.19212	0.596060	+	0.802940i	$0.296732\pi$
$930$	0	0
$931$	7.79777e11i	1.03794i
$932$	0	0
$933$	1.45664e11	0.192232
$934$	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
$939$	1.50247e11i	0.193260i
$940$	0	0
$941$	−3.37589e11	−0.430556	−0.215278	−	0.976553i	$-0.569066\pi$
$941$	−3.37589e11	−0.430556	−0.215278	+	0.976553i	$0.569066\pi$
$942$	0	0
$943$	1.16803e12i	1.47709i
$944$	0	0
$945$	−1.52439e11	−0.191148
$946$	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
$951$	− 1.98790e11i	− 0.243037i
$952$	0	0
$953$	−1.63793e11	−0.198574	−0.0992871	−	0.995059i	$-0.531656\pi$
$953$	−1.63793e11	−0.198574	−0.0992871	+	0.995059i	$0.531656\pi$
$954$	0	0
$955$	4.65816e11i	0.560016i
$956$	0	0
$957$	−2.70198e11	−0.322132
$958$	0	0
$959$	− 1.28887e12i	− 1.52383i
$960$	0	0
$961$	−2.52135e11	−0.295625
$962$	0	0
$963$	− 8.37343e11i	− 0.973639i
$964$	0	0
$965$	4.66278e11	0.537694
$966$	0	0
$967$	− 8.83990e11i	− 1.01098i	−0.862833	−	0.505489i	$-0.831312\pi$
$967$	− 8.83990e11i	− 1.01098i	0.862833	−	0.505489i	$-0.168688\pi$
$968$	0	0
$969$	−1.24069e11	−0.140724
$970$	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
$975$	− 8.59358e10i	− 0.0950945i
$976$	0	0
$977$	2.95199e11	0.323994	0.161997	−	0.986791i	$-0.448206\pi$
$977$	2.95199e11	0.323994	0.161997	+	0.986791i	$0.448206\pi$
$978$	0	0
$979$	− 3.89053e11i	− 0.423524i
$980$	0	0
$981$	9.92435e10	0.107158
$982$	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
$987$	3.30529e11i	0.348290i
$988$	0	0
$989$	−5.50232e11	−0.575123
$990$	0	0
$991$	− 1.43414e12i	− 1.48696i	−0.668761	−	0.743478i	$-0.733175\pi$
$991$	− 1.43414e12i	− 1.48696i	0.668761	−	0.743478i	$-0.266825\pi$
$992$	0	0
$993$	−6.45789e10	−0.0664192
$994$	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
$999$	4.56617e11i	0.458449i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.9.c.b.15.2	yes	2
3.2	odd	2		144.9.g.d.127.2		2
4.3	odd	2	inner	16.9.c.b.15.1	✓	2
5.2	odd	4		400.9.h.a.399.4		4
5.3	odd	4		400.9.h.a.399.2		4
5.4	even	2		400.9.b.e.351.1		2
8.3	odd	2		64.9.c.c.63.2		2
8.5	even	2		64.9.c.c.63.1		2
12.11	even	2		144.9.g.d.127.1		2
16.3	odd	4		256.9.d.d.127.3		4
16.5	even	4		256.9.d.d.127.4		4
16.11	odd	4		256.9.d.d.127.2		4
16.13	even	4		256.9.d.d.127.1		4
20.3	even	4		400.9.h.a.399.3		4
20.7	even	4		400.9.h.a.399.1		4
20.19	odd	2		400.9.b.e.351.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.9.c.b.15.1	✓	2	4.3	odd	2	inner
16.9.c.b.15.2	yes	2	1.1	even	1	trivial
64.9.c.c.63.1		2	8.5	even	2
64.9.c.c.63.2		2	8.3	odd	2
144.9.g.d.127.1		2	12.11	even	2
144.9.g.d.127.2		2	3.2	odd	2
256.9.d.d.127.1		4	16.13	even	4
256.9.d.d.127.2		4	16.11	odd	4
256.9.d.d.127.3		4	16.3	odd	4
256.9.d.d.127.4		4	16.5	even	4
400.9.b.e.351.1		2	5.4	even	2
400.9.b.e.351.2		2	20.19	odd	2
400.9.h.a.399.1		4	20.7	even	4
400.9.h.a.399.2		4	5.3	odd	4
400.9.h.a.399.3		4	20.3	even	4
400.9.h.a.399.4		4	5.2	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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	16.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+13.8564i q^{3} +258.000 q^{5} +3297.82i q^{7} +6369.00 q^{9} +23154.1i q^{11} +19138.0 q^{13} +3574.95i q^{15} -58686.0 q^{17} -152573. i q^{19} -45696.0 q^{21} -270117. i q^{23} -324061. q^{25} +179163. i q^{27} +842178. q^{29} +1.05120e6i q^{31} -320832. q^{33} +850839. i q^{35} +2.54861e6 q^{37} +265184. i q^{39} -4.32416e6 q^{41} -2.03702e6i q^{43} +1.64320e6 q^{45} -7.23321e6i q^{47} -5.11085e6 q^{49} -813177. i q^{51} +1.19219e6 q^{53} +5.97375e6i q^{55} +2.11411e6 q^{57} +338249. i q^{59} +8.41479e6 q^{61} +2.10038e7i q^{63} +4.93760e6 q^{65} +1.74247e7i q^{67} +3.74285e6 q^{69} -3.08765e7i q^{71} +1.27359e7 q^{73} -4.49032e6i q^{75} -7.63580e7 q^{77} -6.28665e6i q^{79} +3.93044e7 q^{81} -8.30740e7i q^{83} -1.51410e7 q^{85} +1.16696e7i q^{87} -1.68028e7 q^{89} +6.31138e7i q^{91} -1.45659e7 q^{93} -3.93638e7i q^{95} +1.20995e8 q^{97} +1.47468e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 516 q^{5} + 12738 q^{9} + 38276 q^{13} - 117372 q^{17} - 91392 q^{21} - 648122 q^{25} + 1684356 q^{29} - 641664 q^{33} + 5097220 q^{37} - 8648316 q^{41} + 3286404 q^{45} - 10221694 q^{49} + 2384388 q^{53}+ \cdots + 241989764 q^{97}+O(q^{100}) \) 2 * q + 516 * q^5 + 12738 * q^9 + 38276 * q^13 - 117372 * q^17 - 91392 * q^21 - 648122 * q^25 + 1684356 * q^29 - 641664 * q^33 + 5097220 * q^37 - 8648316 * q^41 + 3286404 * q^45 - 10221694 * q^49 + 2384388 * q^53 + 4228224 * q^57 + 16829572 * q^61 + 9875208 * q^65 + 7485696 * q^69 + 25471748 * q^73 - 152716032 * q^77 + 78608898 * q^81 - 30281976 * q^85 - 33605628 * q^89 - 29131776 * q^93 + 241989764 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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