LMFDB - Embedded newform 400.9.b.e.351.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,9,Mod(351,400)] 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("400.351"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	400.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,12738,0,0,0,-38276] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$162.951444024$
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:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	400.351
Dual form			400.9.b.e.351.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	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
$5$	0	0
$5$	0	0
$6$	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	$-0.709703\pi$
$11$	− 23154.1i	− 1.58145i	0.612170	−	0.790727i	$-0.290297\pi$
$12$	0	0
$13$	−19138.0	−0.670075	−0.335037	−	0.942205i	$-0.608749\pi$
$13$	−19138.0	−0.670075	−0.335037	+	0.942205i	$0.608749\pi$
$14$	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
$21$	−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
$25$	0	0
$26$	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	$-0.807281\pi$
$31$	− 1.05120e6i	− 1.13826i	0.822249	−	0.569128i	$-0.192719\pi$
$32$	0	0
$33$	320832.	0.270534
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.54861e6	−1.35987	−0.679934	−	0.733274i	$-0.737991\pi$
$37$	−2.54861e6	−1.35987	−0.679934	+	0.733274i	$0.737991\pi$
$38$	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
$45$	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
$51$	813177.i	0.120200i
$52$	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
$55$	0	0
$56$	0	0
$57$	−2.11411e6	−0.200276
$58$	0	0
$59$	− 338249.i	− 0.0279144i	−0.999903	−	0.0139572i	$-0.995557\pi$
$59$	− 338249.i	− 0.0279144i	0.999903	−	0.0139572i	$-0.00444286\pi$
$60$	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
$65$	0	0
$66$	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	$0.207838\pi$
$71$	3.08765e7i	1.21505i	−0.794299	+	0.607526i	$0.792162\pi$
$72$	0	0
$73$	−1.27359e7	−0.448474	−0.224237	−	0.974535i	$-0.571989\pi$
$73$	−1.27359e7	−0.448474	−0.224237	+	0.974535i	$0.571989\pi$
$74$	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
$81$	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
$85$	0	0
$86$	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
$95$	0	0
$96$	0	0
$97$	−1.20995e8	−1.36672	−0.683361	−	0.730081i	$-0.739482\pi$
$97$	−1.20995e8	−1.36672	−0.683361	+	0.730081i	$0.739482\pi$
$98$	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
$105$	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
$111$	− 3.53146e7i	− 0.232628i
$112$	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
$115$	0	0
$116$	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
$125$	0	0
$126$	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	$0.0291523\pi$
$131$	5.38679e7i	0.182913i	−0.995809	+	0.0914566i	$0.970848\pi$
$132$	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
$141$	1.00226e8	0.253574
$142$	0	0
$143$	4.43122e8i	1.05969i
$144$	0	0
$145$	0	0
$146$	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	$-0.944753\pi$
$151$	− 1.79562e8i	− 0.345388i	0.984976	−	0.172694i	$-0.0552471\pi$
$152$	0	0
$153$	3.73771e8	0.682087
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.56860e8	1.41030	0.705150	−	0.709058i	$-0.250880\pi$
$157$	8.56860e8	1.41030	0.705150	+	0.709058i	$0.250880\pi$
$158$	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
$165$	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
$171$	9.71737e8i	1.13649i
$172$	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
$175$	0	0
$176$	0	0
$177$	4.68691e6	0.00477522
$178$	0	0
$179$	− 1.45845e9i	− 1.42062i	−0.703889	−	0.710310i	$-0.748555\pi$
$179$	− 1.45845e9i	− 1.42062i	0.703889	−	0.710310i	$-0.251445\pi$
$180$	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
$185$	0	0
$186$	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	$-0.762711\pi$
$191$	− 1.80549e9i	− 1.35663i	0.734772	−	0.678314i	$-0.237289\pi$
$192$	0	0
$193$	−1.80728e9	−1.30255	−0.651277	−	0.758840i	$-0.725766\pi$
$193$	−1.80728e9	−1.30255	−0.651277	+	0.758840i	$0.725766\pi$
$194$	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
$201$	−2.41444e8	−0.147922
$202$	0	0
$203$	2.77736e9i	1.63549i
$204$	0	0
$205$	0	0
$206$	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	$-0.982703\pi$
$211$	− 2.15314e8i	− 0.108628i	0.998524	−	0.0543141i	$-0.0172972\pi$
$212$	0	0
$213$	−4.27838e8	−0.207855
$214$	0	0
$215$	0	0
$216$	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
$225$	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
$231$	1.05805e9i	0.371584i
$232$	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
$235$	0	0
$236$	0	0
$237$	−8.71104e7	−0.0276106
$238$	0	0
$239$	− 3.04443e9i	− 0.933071i	−0.884503	−	0.466535i	$-0.845502\pi$
$239$	− 3.04443e9i	− 0.933071i	0.884503	−	0.466535i	$-0.154498\pi$
$240$	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
$245$	0	0
$246$	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	$-0.923275\pi$
$251$	− 1.89495e9i	− 0.477422i	0.971091	−	0.238711i	$-0.0767248\pi$
$252$	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
$261$	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
$265$	0	0
$266$	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	$-0.948750\pi$
$271$	− 1.72933e9i	− 0.320627i	0.987066	−	0.160313i	$-0.0512505\pi$
$272$	0	0
$273$	8.74530e8	0.157443
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.04182e9	0.516671	0.258336	−	0.966055i	$-0.416826\pi$
$277$	3.04182e9	0.516671	0.258336	+	0.966055i	$0.416826\pi$
$278$	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
$285$	0	0
$286$	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	$-0.231337\pi$
$293$	1.10157e10	1.49466	0.747328	+	0.664456i	$0.231337\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	0	0
$306$	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	$0.189915\pi$
$311$	1.05124e10i	1.12372i	−0.827231	+	0.561862i	$0.810085\pi$
$312$	0	0
$313$	−1.08431e10	−1.12974	−0.564868	−	0.825181i	$-0.691073\pi$
$313$	−1.08431e10	−1.12974	−0.564868	+	0.825181i	$0.691073\pi$
$314$	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
$321$	1.82172e9	0.171578
$322$	0	0
$323$	8.95389e9i	0.822625i
$324$	0	0
$325$	0	0
$326$	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	$-0.937811\pi$
$331$	− 4.66058e9i	− 0.388265i	0.980975	−	0.194132i	$-0.0621891\pi$
$332$	0	0
$333$	−1.62321e10	−1.32007
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.98174e9	0.153648	0.0768239	−	0.997045i	$-0.475522\pi$
$337$	1.98174e9	0.153648	0.0768239	+	0.997045i	$0.475522\pi$
$338$	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
$345$	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
$351$	− 3.42883e9i	− 0.225901i
$352$	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
$355$	0	0
$356$	0	0
$357$	−2.68172e9	−0.165097
$358$	0	0
$359$	9.78764e9i	0.589251i	0.955613	+	0.294626i	$0.0951949\pi$
$359$	9.78764e9i	0.589251i	−0.955613	+	0.294626i	$0.904805\pi$
$360$	0	0
$361$	−6.29492e9	−0.370648
$362$	0	0
$363$	− 4.45832e9i	− 0.256770i
$364$	0	0
$365$	0	0
$366$	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	$-0.363752\pi$
$373$	1.60695e10	0.830168	0.415084	+	0.909783i	$0.363752\pi$
$374$	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
$381$	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
$385$	0	0
$386$	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
$395$	0	0
$396$	0	0
$397$	−2.55255e10	−1.02757	−0.513786	−	0.857918i	$-0.671757\pi$
$397$	−2.55255e10	−1.02757	−0.513786	+	0.857918i	$0.671757\pi$
$398$	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
$405$	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
$411$	5.41544e9i	0.189787i
$412$	0	0
$413$	1.11549e9	0.0383410
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.56276e9	0.183970
$418$	0	0
$419$	− 2.64123e10i	− 0.856941i	−0.903556	−	0.428470i	$-0.859053\pi$
$419$	− 2.64123e10i	− 0.856941i	0.903556	−	0.428470i	$-0.140947\pi$
$420$	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
$425$	0	0
$426$	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	$-0.883096\pi$
$431$	− 2.47805e10i	− 0.718127i	0.933313	−	0.359063i	$-0.116904\pi$
$432$	0	0
$433$	1.97724e10	0.562482	0.281241	−	0.959637i	$-0.409254\pi$
$433$	1.97724e10	0.562482	0.281241	+	0.959637i	$0.409254\pi$
$434$	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
$441$	−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
$445$	0	0
$446$	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
$455$	0	0
$456$	0	0
$457$	1.89812e10	0.435171	0.217585	−	0.976041i	$-0.430182\pi$
$457$	1.89812e10	0.435171	0.217585	+	0.976041i	$0.430182\pi$
$458$	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
$465$	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
$471$	1.18730e10i	0.241255i
$472$	0	0
$473$	−4.71652e10	−0.942274
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.59308e9	−0.146671
$478$	0	0
$479$	7.76849e10i	1.47569i	0.674971	+	0.737844i	$0.264156\pi$
$479$	7.76849e10i	1.47569i	−0.674971	+	0.737844i	$0.735844\pi$
$480$	0	0
$481$	4.87753e10	0.911212
$482$	0	0
$483$	1.23433e10i	0.226799i
$484$	0	0
$485$	0	0
$486$	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	$-0.855634\pi$
$491$	− 5.09306e10i	− 0.876299i	0.898902	−	0.438150i	$-0.144366\pi$
$492$	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
$501$	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
$505$	0	0
$506$	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
$515$	0	0
$516$	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
$525$	0	0
$526$	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
$535$	0	0
$536$	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
$545$	0	0
$546$	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
$555$	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
$561$	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
$565$	0	0
$566$	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	$-0.725727\pi$
$571$	− 1.61351e11i	− 1.51784i	0.651182	−	0.758922i	$-0.274273\pi$
$572$	0	0
$573$	2.50176e10	0.232074
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.65816e10	0.510472	0.255236	−	0.966879i	$-0.417847\pi$
$577$	5.65816e10	0.510472	0.255236	+	0.966879i	$0.417847\pi$
$578$	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
$585$	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
$591$	− 3.16576e9i	− 0.0259494i
$592$	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
$595$	0	0
$596$	0	0
$597$	1.70865e10	0.134511
$598$	0	0
$599$	− 2.52874e10i	− 0.196425i	−0.995165	−	0.0982126i	$-0.968687\pi$
$599$	− 2.52874e10i	− 0.196425i	0.995165	−	0.0982126i	$-0.0313125\pi$
$600$	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
$605$	0	0
$606$	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	$-0.531182\pi$
$613$	−2.76203e10	−0.195608	−0.0978040	+	0.995206i	$0.531182\pi$
$614$	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
$621$	4.83950e10	0.325413
$622$	0	0
$623$	− 5.54127e10i	− 0.367839i
$624$	0	0
$625$	0	0
$626$	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	$0.162870\pi$
$631$	1.55246e11i	0.979273i	−0.871927	+	0.489636i	$0.837130\pi$
$632$	0	0
$633$	2.98348e9	0.0185827
$634$	0	0
$635$	0	0
$636$	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
$645$	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
$651$	4.80357e10i	0.267449i
$652$	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
$655$	0	0
$656$	0	0
$657$	−8.11148e10	−0.435350
$658$	0	0
$659$	− 6.20284e10i	− 0.328889i	−0.986386	−	0.164444i	$-0.947417\pi$
$659$	− 6.20284e10i	− 0.328889i	0.986386	−	0.164444i	$-0.0525831\pi$
$660$	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
$665$	0	0
$666$	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	$-0.641486\pi$
$673$	−1.76424e11	−0.859998	−0.429999	+	0.902829i	$0.641486\pi$
$674$	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
$681$	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
$685$	0	0
$686$	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	$-0.881729\pi$
$691$	− 1.65551e11i	− 0.726139i	0.931762	−	0.363069i	$-0.118271\pi$
$692$	0	0
$693$	4.86324e11	2.10859
$694$	0	0
$695$	0	0
$696$	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
$705$	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
$711$	4.00397e10i	0.156680i
$712$	0	0
$713$	−2.83947e11	−1.09870
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.21849e10	0.159617
$718$	0	0
$719$	− 1.65608e11i	− 0.619676i	−0.950789	−	0.309838i	$-0.899725\pi$
$719$	− 1.65608e11i	− 0.619676i	0.950789	−	0.309838i	$-0.100275\pi$
$720$	0	0
$721$	−4.44732e11	−1.64573
$722$	0	0
$723$	− 4.29880e10i	− 0.157324i
$724$	0	0
$725$	0	0
$726$	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	$-0.403036\pi$
$733$	1.73168e11	0.599863	0.299932	+	0.953961i	$0.403036\pi$
$734$	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
$741$	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
$745$	0	0
$746$	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	$-0.932099\pi$
$751$	− 1.34684e11i	− 0.423405i	0.977334	−	0.211703i	$-0.0679008\pi$
$752$	0	0
$753$	2.62572e10	0.0816710
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.86211e11	1.78513	0.892567	−	0.450915i	$-0.148902\pi$
$757$	5.86211e11	1.78513	0.892567	+	0.450915i	$0.148902\pi$
$758$	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
$765$	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
$771$	4.12120e10i	0.116629i
$772$	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
$775$	0	0
$776$	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
$785$	0	0
$786$	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
$795$	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
$801$	−1.07017e11	−0.259970
$802$	0	0
$803$	2.94887e11i	0.709241i
$804$	0	0
$805$	0	0
$806$	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	$0.0631136\pi$
$811$	1.70426e11i	0.393961i	−0.980407	+	0.196980i	$0.936886\pi$
$812$	0	0
$813$	2.39623e10	0.0548486
$814$	0	0
$815$	0	0
$816$	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
$825$	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
$831$	4.21487e10i	0.0883853i
$832$	0	0
$833$	−2.99935e11	−0.622942
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.88337e11	0.383737
$838$	0	0
$839$	− 2.28645e10i	− 0.0461439i	−0.999734	−	0.0230720i	$-0.992655\pi$
$839$	− 2.28645e10i	− 0.0461439i	0.999734	−	0.0230720i	$-0.00734469\pi$
$840$	0	0
$841$	2.09017e11	0.417829
$842$	0	0
$843$	5.20750e10i	0.103114i
$844$	0	0
$845$	0	0
$846$	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	$-0.177226\pi$
$853$	8.98910e11	1.69793	0.848965	+	0.528449i	$0.177226\pi$
$854$	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
$861$	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
$865$	0	0
$866$	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
$875$	0	0
$876$	0	0
$877$	−5.09974e11	−0.862084	−0.431042	−	0.902332i	$-0.641854\pi$
$877$	−5.09974e11	−0.862084	−0.431042	+	0.902332i	$0.641854\pi$
$878$	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
$885$	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
$891$	− 9.10057e11i	− 1.44397i
$892$	0	0
$893$	1.10359e12	1.73541
$894$	0	0
$895$	0	0
$896$	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
$905$	0	0
$906$	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	$-0.944814\pi$
$911$	− 2.37631e11i	− 0.345008i	0.985009	−	0.172504i	$-0.0551858\pi$
$912$	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
$921$	6.75646e10	0.0939033
$922$	0	0
$923$	− 5.90915e11i	− 0.814176i
$924$	0	0
$925$	0	0
$926$	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
$935$	0	0
$936$	0	0
$937$	6.83385e11	0.886557	0.443279	−	0.896384i	$-0.353815\pi$
$937$	6.83385e11	0.886557	0.443279	+	0.896384i	$0.353815\pi$
$938$	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
$945$	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
$951$	1.98790e11i	0.243037i
$952$	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
$955$	0	0
$956$	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
$965$	0	0
$966$	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	$0.0604485\pi$
$971$	3.35605e11i	0.377530i	−0.982022	+	0.188765i	$0.939552\pi$
$972$	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
$981$	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
$985$	0	0
$986$	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	$0.266825\pi$
$991$	1.43414e12i	1.48696i	−0.668761	+	0.743478i	$0.733175\pi$
$992$	0	0
$993$	6.45789e10	0.0664192
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.38321e12	−1.39993	−0.699965	−	0.714177i	$-0.746801\pi$
$997$	−1.38321e12	−1.39993	−0.699965	+	0.714177i	$0.746801\pi$
$998$	0	0
$999$	− 4.56617e11i	− 0.458449i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.9.b.e.351.2		2
4.3	odd	2	inner	400.9.b.e.351.1		2
5.2	odd	4		400.9.h.a.399.3		4
5.3	odd	4		400.9.h.a.399.1		4
5.4	even	2		16.9.c.b.15.1	✓	2
15.14	odd	2		144.9.g.d.127.1		2
20.3	even	4		400.9.h.a.399.4		4
20.7	even	4		400.9.h.a.399.2		4
20.19	odd	2		16.9.c.b.15.2	yes	2
40.19	odd	2		64.9.c.c.63.1		2
40.29	even	2		64.9.c.c.63.2		2
60.59	even	2		144.9.g.d.127.2		2
80.19	odd	4		256.9.d.d.127.1		4
80.29	even	4		256.9.d.d.127.3		4
80.59	odd	4		256.9.d.d.127.4		4
80.69	even	4		256.9.d.d.127.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.9.c.b.15.1	✓	2	5.4	even	2
16.9.c.b.15.2	yes	2	20.19	odd	2
64.9.c.c.63.1		2	40.19	odd	2
64.9.c.c.63.2		2	40.29	even	2
144.9.g.d.127.1		2	15.14	odd	2
144.9.g.d.127.2		2	60.59	even	2
256.9.d.d.127.1		4	80.19	odd	4
256.9.d.d.127.2		4	80.69	even	4
256.9.d.d.127.3		4	80.29	even	4
256.9.d.d.127.4		4	80.59	odd	4
400.9.b.e.351.1		2	4.3	odd	2	inner
400.9.b.e.351.2		2	1.1	even	1	trivial
400.9.h.a.399.1		4	5.3	odd	4
400.9.h.a.399.2		4	20.7	even	4
400.9.h.a.399.3		4	5.2	odd	4
400.9.h.a.399.4		4	20.3	even	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	400.b (of order \(2\), degree \(1\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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