LMFDB - Embedded newform 5445.2.a.bq.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5445,2,Mod(1,5445)] 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("5445.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,1,0,1,-4,0,0,-3,0,-1,0,0,-7,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.209057$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.33826 q^{2} -0.209057 q^{4} -1.00000 q^{5} +3.78339 q^{7} -2.95630 q^{8} -1.33826 q^{10} -4.93395 q^{13} +5.06316 q^{14} -3.53818 q^{16} +3.25841 q^{17} -1.37283 q^{19} +0.209057 q^{20} +1.93395 q^{23} +1.00000 q^{25} -6.60292 q^{26} -0.790943 q^{28} -1.73968 q^{29} -5.92173 q^{31} +1.17758 q^{32} +4.36060 q^{34} -3.78339 q^{35} +4.69789 q^{37} -1.83720 q^{38} +2.95630 q^{40} +9.43757 q^{41} -11.4086 q^{43} +2.58814 q^{46} -0.805727 q^{47} +7.31401 q^{49} +1.33826 q^{50} +1.03148 q^{52} -10.6960 q^{53} -11.1848 q^{56} -2.32815 q^{58} -12.1634 q^{59} -8.59102 q^{61} -7.92482 q^{62} +8.65227 q^{64} +4.93395 q^{65} -2.77425 q^{67} -0.681193 q^{68} -5.06316 q^{70} +2.73968 q^{71} +9.37717 q^{73} +6.28700 q^{74} +0.286999 q^{76} -12.7667 q^{79} +3.53818 q^{80} +12.6299 q^{82} -3.41304 q^{83} -3.25841 q^{85} -15.2677 q^{86} +7.85817 q^{89} -18.6671 q^{91} -0.404307 q^{92} -1.07827 q^{94} +1.37283 q^{95} -13.6196 q^{97} +9.78806 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{4} - 4 q^{5} - 3 q^{8} - q^{10} - 7 q^{13} + 5 q^{14} - 9 q^{16} + 8 q^{17} - 11 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 12 q^{26} - 5 q^{28} + 17 q^{29} - 5 q^{31} + 17 q^{34} + 15 q^{37}+ \cdots + 13 q^{98}+O(q^{100}) $ 4 * q + q^2 + q^4 - 4 * q^5 - 3 * q^8 - q^10 - 7 * q^13 + 5 * q^14 - 9 * q^16 + 8 * q^17 - 11 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 12 * q^26 - 5 * q^28 + 17 * q^29 - 5 * q^31 + 17 * q^34 + 15 * q^37 + q^38 + 3 * q^40 + 10 * q^41 + 4 * q^43 - 15 * q^46 + 8 * q^47 - 8 * q^49 + q^50 + 7 * q^52 - 10 * q^53 - 10 * q^56 - 7 * q^58 - 9 * q^59 - 37 * q^61 - 20 * q^62 - 7 * q^64 + 7 * q^65 + 3 * q^67 + 17 * q^68 - 5 * q^70 - 13 * q^71 - 15 * q^73 - 5 * q^74 - 29 * q^76 - 20 * q^79 + 9 * q^80 + 5 * q^82 - 17 * q^83 - 8 * q^85 - 34 * q^86 + 24 * q^89 - 20 * q^91 - 10 * q^92 - 23 * q^94 + 11 * q^95 - 32 * q^97 + 13 * q^98

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 $

$2$	1.33826	0.946294	0.473147	−	0.880984i	$-0.343118\pi$
$2$	1.33826	0.946294	0.473147	+	0.880984i	$0.343118\pi$
$3$	0	0
$3$	0	0
$4$	−0.209057	−0.104528
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.78339	1.42999	0.714993	−	0.699132i	$-0.246430\pi$
$7$	3.78339	1.42999	0.714993	+	0.699132i	$0.246430\pi$
$8$	−2.95630	−1.04521
$9$	0	0
$10$	−1.33826	−0.423195
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.93395	−1.36843	−0.684216	−	0.729279i	$-0.739856\pi$
$13$	−4.93395	−1.36843	−0.684216	+	0.729279i	$0.739856\pi$
$14$	5.06316	1.35319
$15$	0	0
$16$	−3.53818	−0.884545
$17$	3.25841	0.790280	0.395140	−	0.918621i	$-0.370696\pi$
$17$	3.25841	0.790280	0.395140	+	0.918621i	$0.370696\pi$
$18$	0	0
$19$	−1.37283	−0.314949	−0.157474	−	0.987523i	$-0.550335\pi$
$19$	−1.37283	−0.314949	−0.157474	+	0.987523i	$0.550335\pi$
$20$	0.209057	0.0467465
$21$	0	0
$22$	0	0
$23$	1.93395	0.403257	0.201629	−	0.979462i	$-0.435377\pi$
$23$	1.93395	0.403257	0.201629	+	0.979462i	$0.435377\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.60292	−1.29494
$27$	0	0
$28$	−0.790943	−0.149474
$29$	−1.73968	−0.323051	−0.161525	−	0.986869i	$-0.551641\pi$
$29$	−1.73968	−0.323051	−0.161525	+	0.986869i	$0.551641\pi$
$30$	0	0
$31$	−5.92173	−1.06357	−0.531787	−	0.846878i	$-0.678479\pi$
$31$	−5.92173	−1.06357	−0.531787	+	0.846878i	$0.678479\pi$
$32$	1.17758	0.208169
$33$	0	0
$34$	4.36060	0.747837
$35$	−3.78339	−0.639509
$36$	0	0
$37$	4.69789	0.772328	0.386164	−	0.922430i	$-0.373800\pi$
$37$	4.69789	0.772328	0.386164	+	0.922430i	$0.373800\pi$
$38$	−1.83720	−0.298034
$39$	0	0
$40$	2.95630	0.467431
$41$	9.43757	1.47390	0.736950	−	0.675947i	$-0.236265\pi$
$41$	9.43757	1.47390	0.736950	+	0.675947i	$0.236265\pi$
$42$	0	0
$43$	−11.4086	−1.73980	−0.869901	−	0.493226i	$-0.835818\pi$
$43$	−11.4086	−1.73980	−0.869901	+	0.493226i	$0.835818\pi$
$44$	0	0
$45$	0	0
$46$	2.58814	0.381600
$47$	−0.805727	−0.117527	−0.0587637	−	0.998272i	$-0.518716\pi$
$47$	−0.805727	−0.117527	−0.0587637	+	0.998272i	$0.518716\pi$
$48$	0	0
$49$	7.31401	1.04486
$50$	1.33826	0.189259
$51$	0	0
$52$	1.03148	0.143040
$53$	−10.6960	−1.46921	−0.734603	−	0.678498i	$-0.762631\pi$
$53$	−10.6960	−1.46921	−0.734603	+	0.678498i	$0.762631\pi$
$54$	0	0
$55$	0	0
$56$	−11.1848	−1.49463
$57$	0	0
$58$	−2.32815	−0.305701
$59$	−12.1634	−1.58355	−0.791773	−	0.610816i	$-0.790842\pi$
$59$	−12.1634	−1.58355	−0.791773	+	0.610816i	$0.790842\pi$
$60$	0	0
$61$	−8.59102	−1.09997	−0.549984	−	0.835175i	$-0.685366\pi$
$61$	−8.59102	−1.09997	−0.549984	+	0.835175i	$0.685366\pi$
$62$	−7.92482	−1.00645
$63$	0	0
$64$	8.65227	1.08153
$65$	4.93395	0.611982
$66$	0	0
$67$	−2.77425	−0.338928	−0.169464	−	0.985536i	$-0.554204\pi$
$67$	−2.77425	−0.338928	−0.169464	+	0.985536i	$0.554204\pi$
$68$	−0.681193	−0.0826068
$69$	0	0
$70$	−5.06316	−0.605163
$71$	2.73968	0.325140	0.162570	−	0.986697i	$-0.448022\pi$
$71$	2.73968	0.325140	0.162570	+	0.986697i	$0.448022\pi$
$72$	0	0
$73$	9.37717	1.09751	0.548757	−	0.835982i	$-0.315101\pi$
$73$	9.37717	1.09751	0.548757	+	0.835982i	$0.315101\pi$
$74$	6.28700	0.730849
$75$	0	0
$76$	0.286999	0.0329211
$77$	0	0
$78$	0	0
$79$	−12.7667	−1.43637	−0.718183	−	0.695855i	$-0.755026\pi$
$79$	−12.7667	−1.43637	−0.718183	+	0.695855i	$0.755026\pi$
$80$	3.53818	0.395581
$81$	0	0
$82$	12.6299	1.39474
$83$	−3.41304	−0.374630	−0.187315	−	0.982300i	$-0.559979\pi$
$83$	−3.41304	−0.374630	−0.187315	+	0.982300i	$0.559979\pi$
$84$	0	0
$85$	−3.25841	−0.353424
$86$	−15.2677	−1.64636
$87$	0	0
$88$	0	0
$89$	7.85817	0.832964	0.416482	−	0.909144i	$-0.363263\pi$
$89$	7.85817	0.832964	0.416482	+	0.909144i	$0.363263\pi$
$90$	0	0
$91$	−18.6671	−1.95684
$92$	−0.404307	−0.0421519
$93$	0	0
$94$	−1.07827	−0.111215
$95$	1.37283	0.140849
$96$	0	0
$97$	−13.6196	−1.38286	−0.691431	−	0.722442i	$-0.743019\pi$
$97$	−13.6196	−1.38286	−0.691431	+	0.722442i	$0.743019\pi$
$98$	9.78806	0.988743
$99$	0	0
$100$	−0.209057	−0.0209057
$101$	7.93842	0.789902	0.394951	−	0.918702i	$-0.370761\pi$
$101$	7.93842	0.789902	0.394951	+	0.918702i	$0.370761\pi$
$102$	0	0
$103$	−12.2052	−1.20262	−0.601309	−	0.799017i	$-0.705354\pi$
$103$	−12.2052	−1.20262	−0.601309	+	0.799017i	$0.705354\pi$
$104$	14.5862	1.43030
$105$	0	0
$106$	−14.3140	−1.39030
$107$	2.33537	0.225769	0.112885	−	0.993608i	$-0.463991\pi$
$107$	2.33537	0.225769	0.112885	+	0.993608i	$0.463991\pi$
$108$	0	0
$109$	−6.39419	−0.612453	−0.306226	−	0.951959i	$-0.599066\pi$
$109$	−6.39419	−0.612453	−0.306226	+	0.951959i	$0.599066\pi$
$110$	0	0
$111$	0	0
$112$	−13.3863	−1.26489
$113$	−18.1316	−1.70568	−0.852840	−	0.522172i	$-0.825122\pi$
$113$	−18.1316	−1.70568	−0.852840	+	0.522172i	$0.825122\pi$
$114$	0	0
$115$	−1.93395	−0.180342
$116$	0.363692	0.0337680
$117$	0	0
$118$	−16.2779	−1.49850
$119$	12.3278	1.13009
$120$	0	0
$121$	0	0
$122$	−11.4970	−1.04089
$123$	0	0
$124$	1.23798	0.111174
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	22.3123	1.97990	0.989948	−	0.141429i	$-0.0451696\pi$
$127$	22.3123	1.97990	0.989948	+	0.141429i	$0.0451696\pi$
$128$	9.22384	0.815280
$129$	0	0
$130$	6.60292	0.579114
$131$	−0.562102	−0.0491111	−0.0245555	−	0.999698i	$-0.507817\pi$
$131$	−0.562102	−0.0491111	−0.0245555	+	0.999698i	$0.507817\pi$
$132$	0	0
$133$	−5.19394	−0.450372
$134$	−3.71267	−0.320726
$135$	0	0
$136$	−9.63282	−0.826007
$137$	18.2719	1.56107	0.780536	−	0.625110i	$-0.214946\pi$
$137$	18.2719	1.56107	0.780536	+	0.625110i	$0.214946\pi$
$138$	0	0
$139$	−14.2600	−1.20952	−0.604758	−	0.796409i	$-0.706730\pi$
$139$	−14.2600	−1.20952	−0.604758	+	0.796409i	$0.706730\pi$
$140$	0.790943	0.0668469
$141$	0	0
$142$	3.66641	0.307678
$143$	0	0
$144$	0	0
$145$	1.73968	0.144473
$146$	12.5491	1.03857
$147$	0	0
$148$	−0.982126	−0.0807302
$149$	10.6039	0.868705	0.434353	−	0.900743i	$-0.356977\pi$
$149$	10.6039	0.868705	0.434353	+	0.900743i	$0.356977\pi$
$150$	0	0
$151$	−12.5734	−1.02320	−0.511602	−	0.859222i	$-0.670948\pi$
$151$	−12.5734	−1.02320	−0.511602	+	0.859222i	$0.670948\pi$
$152$	4.05849	0.329187
$153$	0	0
$154$	0	0
$155$	5.92173	0.475645
$156$	0	0
$157$	−12.5332	−1.00026	−0.500128	−	0.865951i	$-0.666714\pi$
$157$	−12.5332	−1.00026	−0.500128	+	0.865951i	$0.666714\pi$
$158$	−17.0852	−1.35922
$159$	0	0
$160$	−1.17758	−0.0930958
$161$	7.31690	0.576652
$162$	0	0
$163$	−3.10317	−0.243059	−0.121530	−	0.992588i	$-0.538780\pi$
$163$	−3.10317	−0.243059	−0.121530	+	0.992588i	$0.538780\pi$
$164$	−1.97299	−0.154065
$165$	0	0
$166$	−4.56754	−0.354510
$167$	−8.34898	−0.646063	−0.323032	−	0.946388i	$-0.604702\pi$
$167$	−8.34898	−0.646063	−0.323032	+	0.946388i	$0.604702\pi$
$168$	0	0
$169$	11.3439	0.872608
$170$	−4.36060	−0.334443
$171$	0	0
$172$	2.38506	0.181859
$173$	−6.84208	−0.520194	−0.260097	−	0.965583i	$-0.583754\pi$
$173$	−6.84208	−0.520194	−0.260097	+	0.965583i	$0.583754\pi$
$174$	0	0
$175$	3.78339	0.285997
$176$	0	0
$177$	0	0
$178$	10.5163	0.788229
$179$	−15.5085	−1.15916	−0.579579	−	0.814916i	$-0.696783\pi$
$179$	−15.5085	−1.15916	−0.579579	+	0.814916i	$0.696783\pi$
$180$	0	0
$181$	−2.26096	−0.168056	−0.0840281	−	0.996463i	$-0.526779\pi$
$181$	−2.26096	−0.168056	−0.0840281	+	0.996463i	$0.526779\pi$
$182$	−24.9814	−1.85174
$183$	0	0
$184$	−5.71734	−0.421488
$185$	−4.69789	−0.345395
$186$	0	0
$187$	0	0
$188$	0.168443	0.0122850
$189$	0	0
$190$	1.83720	0.133285
$191$	17.7692	1.28574	0.642869	−	0.765976i	$-0.277744\pi$
$191$	17.7692	1.28574	0.642869	+	0.765976i	$0.277744\pi$
$192$	0	0
$193$	−21.1559	−1.52283	−0.761417	−	0.648262i	$-0.775496\pi$
$193$	−21.1559	−1.52283	−0.761417	+	0.648262i	$0.775496\pi$
$194$	−18.2266	−1.30859
$195$	0	0
$196$	−1.52904	−0.109217
$197$	−24.1034	−1.71730	−0.858650	−	0.512563i	$-0.828696\pi$
$197$	−24.1034	−1.71730	−0.858650	+	0.512563i	$0.828696\pi$
$198$	0	0
$199$	4.30369	0.305081	0.152540	−	0.988297i	$-0.451255\pi$
$199$	4.30369	0.305081	0.152540	+	0.988297i	$0.451255\pi$
$200$	−2.95630	−0.209042
$201$	0	0
$202$	10.6237	0.747480
$203$	−6.58189	−0.461958
$204$	0	0
$205$	−9.43757	−0.659148
$206$	−16.3338	−1.13803
$207$	0	0
$208$	17.4572	1.21044
$209$	0	0
$210$	0	0
$211$	−15.2270	−1.04827	−0.524135	−	0.851635i	$-0.675611\pi$
$211$	−15.2270	−1.04827	−0.524135	+	0.851635i	$0.675611\pi$
$212$	2.23607	0.153574
$213$	0	0
$214$	3.12534	0.213644
$215$	11.4086	0.778063
$216$	0	0
$217$	−22.4042	−1.52089
$218$	−8.55710	−0.579560
$219$	0	0
$220$	0	0
$221$	−16.0768	−1.08145
$222$	0	0
$223$	28.6450	1.91821	0.959105	−	0.283050i	$-0.0913463\pi$
$223$	28.6450	1.91821	0.959105	+	0.283050i	$0.0913463\pi$
$224$	4.45524	0.297678
$225$	0	0
$226$	−24.2649	−1.61407
$227$	3.02608	0.200848	0.100424	−	0.994945i	$-0.467980\pi$
$227$	3.02608	0.200848	0.100424	+	0.994945i	$0.467980\pi$
$228$	0	0
$229$	−3.05244	−0.201711	−0.100856	−	0.994901i	$-0.532158\pi$
$229$	−3.05244	−0.201711	−0.100856	+	0.994901i	$0.532158\pi$
$230$	−2.58814	−0.170657
$231$	0	0
$232$	5.14301	0.337655
$233$	−6.30934	−0.413339	−0.206669	−	0.978411i	$-0.566262\pi$
$233$	−6.30934	−0.413339	−0.206669	+	0.978411i	$0.566262\pi$
$234$	0	0
$235$	0.805727	0.0525598
$236$	2.54285	0.165526
$237$	0	0
$238$	16.4978	1.06940
$239$	−5.71109	−0.369420	−0.184710	−	0.982793i	$-0.559135\pi$
$239$	−5.71109	−0.369420	−0.184710	+	0.982793i	$0.559135\pi$
$240$	0	0
$241$	−4.81966	−0.310462	−0.155231	−	0.987878i	$-0.549612\pi$
$241$	−4.81966	−0.310462	−0.155231	+	0.987878i	$0.549612\pi$
$242$	0	0
$243$	0	0
$244$	1.79601	0.114978
$245$	−7.31401	−0.467275
$246$	0	0
$247$	6.77348	0.430986
$248$	17.5064	1.11166
$249$	0	0
$250$	−1.33826	−0.0846391
$251$	0.883023	0.0557359	0.0278680	−	0.999612i	$-0.491128\pi$
$251$	0.883023	0.0557359	0.0278680	+	0.999612i	$0.491128\pi$
$252$	0	0
$253$	0	0
$254$	29.8597	1.87356
$255$	0	0
$256$	−4.96064	−0.310040
$257$	−8.63651	−0.538731	−0.269365	−	0.963038i	$-0.586814\pi$
$257$	−8.63651	−0.538731	−0.269365	+	0.963038i	$0.586814\pi$
$258$	0	0
$259$	17.7739	1.10442
$260$	−1.03148	−0.0639695
$261$	0	0
$262$	−0.752239	−0.0464735
$263$	−4.73594	−0.292031	−0.146015	−	0.989282i	$-0.546645\pi$
$263$	−4.73594	−0.292031	−0.146015	+	0.989282i	$0.546645\pi$
$264$	0	0
$265$	10.6960	0.657049
$266$	−6.95085	−0.426184
$267$	0	0
$268$	0.579976	0.0354277
$269$	14.2672	0.869887	0.434944	−	0.900458i	$-0.356768\pi$
$269$	14.2672	0.869887	0.434944	+	0.900458i	$0.356768\pi$
$270$	0	0
$271$	−22.9367	−1.39331	−0.696653	−	0.717408i	$-0.745328\pi$
$271$	−22.9367	−1.39331	−0.696653	+	0.717408i	$0.745328\pi$
$272$	−11.5288	−0.699039
$273$	0	0
$274$	24.4526	1.47723
$275$	0	0
$276$	0	0
$277$	5.82302	0.349872	0.174936	−	0.984580i	$-0.444028\pi$
$277$	5.82302	0.349872	0.174936	+	0.984580i	$0.444028\pi$
$278$	−19.0836	−1.14456
$279$	0	0
$280$	11.1848	0.668420
$281$	−0.157591	−0.00940109	−0.00470055	−	0.999989i	$-0.501496\pi$
$281$	−0.157591	−0.00940109	−0.00470055	+	0.999989i	$0.501496\pi$
$282$	0	0
$283$	8.94565	0.531764	0.265882	−	0.964006i	$-0.414337\pi$
$283$	8.94565	0.531764	0.265882	+	0.964006i	$0.414337\pi$
$284$	−0.572749	−0.0339864
$285$	0	0
$286$	0	0
$287$	35.7060	2.10766
$288$	0	0
$289$	−6.38277	−0.375457
$290$	2.32815	0.136714
$291$	0	0
$292$	−1.96036	−0.114722
$293$	25.9879	1.51823	0.759113	−	0.650959i	$-0.225633\pi$
$293$	25.9879	1.51823	0.759113	+	0.650959i	$0.225633\pi$
$294$	0	0
$295$	12.1634	0.708183
$296$	−13.8883	−0.807243
$297$	0	0
$298$	14.1908	0.822050
$299$	−9.54204	−0.551831
$300$	0	0
$301$	−43.1633	−2.48789
$302$	−16.8264	−0.968252
$303$	0	0
$304$	4.85732	0.278586
$305$	8.59102	0.491921
$306$	0	0
$307$	−4.72359	−0.269590	−0.134795	−	0.990874i	$-0.543038\pi$
$307$	−4.72359	−0.269590	−0.134795	+	0.990874i	$0.543038\pi$
$308$	0	0
$309$	0	0
$310$	7.92482	0.450099
$311$	19.2517	1.09166	0.545832	−	0.837895i	$-0.316214\pi$
$311$	19.2517	1.09166	0.545832	+	0.837895i	$0.316214\pi$
$312$	0	0
$313$	2.94346	0.166374	0.0831872	−	0.996534i	$-0.473490\pi$
$313$	2.94346	0.166374	0.0831872	+	0.996534i	$0.473490\pi$
$314$	−16.7727	−0.946536
$315$	0	0
$316$	2.66897	0.150141
$317$	−1.82705	−0.102617	−0.0513086	−	0.998683i	$-0.516339\pi$
$317$	−1.82705	−0.102617	−0.0513086	+	0.998683i	$0.516339\pi$
$318$	0	0
$319$	0	0
$320$	−8.65227	−0.483677
$321$	0	0
$322$	9.79192	0.545682
$323$	−4.47324	−0.248898
$324$	0	0
$325$	−4.93395	−0.273687
$326$	−4.15285	−0.230005
$327$	0	0
$328$	−27.9002	−1.54053
$329$	−3.04838	−0.168062
$330$	0	0
$331$	17.3017	0.950985	0.475493	−	0.879720i	$-0.342270\pi$
$331$	17.3017	0.950985	0.475493	+	0.879720i	$0.342270\pi$
$332$	0.713521	0.0391595
$333$	0	0
$334$	−11.1731	−0.611366
$335$	2.77425	0.151573
$336$	0	0
$337$	10.8095	0.588829	0.294414	−	0.955678i	$-0.404875\pi$
$337$	10.8095	0.588829	0.294414	+	0.955678i	$0.404875\pi$
$338$	15.1811	0.825744
$339$	0	0
$340$	0.681193	0.0369429
$341$	0	0
$342$	0	0
$343$	1.18802	0.0641472
$344$	33.7273	1.81846
$345$	0	0
$346$	−9.15649	−0.492256
$347$	−15.1883	−0.815350	−0.407675	−	0.913127i	$-0.633660\pi$
$347$	−15.1883	−0.815350	−0.407675	+	0.913127i	$0.633660\pi$
$348$	0	0
$349$	−3.79541	−0.203164	−0.101582	−	0.994827i	$-0.532390\pi$
$349$	−3.79541	−0.203164	−0.101582	+	0.994827i	$0.532390\pi$
$350$	5.06316	0.270637
$351$	0	0
$352$	0	0
$353$	−0.519577	−0.0276543	−0.0138272	−	0.999904i	$-0.504401\pi$
$353$	−0.519577	−0.0276543	−0.0138272	+	0.999904i	$0.504401\pi$
$354$	0	0
$355$	−2.73968	−0.145407
$356$	−1.64280	−0.0870685
$357$	0	0
$358$	−20.7544	−1.09690
$359$	−13.8374	−0.730310	−0.365155	−	0.930947i	$-0.618984\pi$
$359$	−13.8374	−0.730310	−0.365155	+	0.930947i	$0.618984\pi$
$360$	0	0
$361$	−17.1153	−0.900807
$362$	−3.02576	−0.159030
$363$	0	0
$364$	3.90248	0.204545
$365$	−9.37717	−0.490823
$366$	0	0
$367$	−20.6220	−1.07646	−0.538229	−	0.842799i	$-0.680907\pi$
$367$	−20.6220	−1.07646	−0.538229	+	0.842799i	$0.680907\pi$
$368$	−6.84268	−0.356699
$369$	0	0
$370$	−6.28700	−0.326846
$371$	−40.4670	−2.10094
$372$	0	0
$373$	−8.17818	−0.423450	−0.211725	−	0.977329i	$-0.567908\pi$
$373$	−8.17818	−0.423450	−0.211725	+	0.977329i	$0.567908\pi$
$374$	0	0
$375$	0	0
$376$	2.38197	0.122841
$377$	8.58351	0.442073
$378$	0	0
$379$	26.5982	1.36626	0.683128	−	0.730299i	$-0.260619\pi$
$379$	26.5982	1.36626	0.683128	+	0.730299i	$0.260619\pi$
$380$	−0.286999	−0.0147228
$381$	0	0
$382$	23.7799	1.21669
$383$	14.5607	0.744018	0.372009	−	0.928229i	$-0.378669\pi$
$383$	14.5607	0.744018	0.372009	+	0.928229i	$0.378669\pi$
$384$	0	0
$385$	0	0
$386$	−28.3121	−1.44105
$387$	0	0
$388$	2.84727	0.144548
$389$	3.94911	0.200228	0.100114	−	0.994976i	$-0.468079\pi$
$389$	3.94911	0.200228	0.100114	+	0.994976i	$0.468079\pi$
$390$	0	0
$391$	6.30161	0.318686
$392$	−21.6224	−1.09209
$393$	0	0
$394$	−32.2567	−1.62507
$395$	12.7667	0.642362
$396$	0	0
$397$	−18.0162	−0.904209	−0.452105	−	0.891965i	$-0.649327\pi$
$397$	−18.0162	−0.904209	−0.452105	+	0.891965i	$0.649327\pi$
$398$	5.75947	0.288696
$399$	0	0
$400$	−3.53818	−0.176909
$401$	26.7798	1.33732	0.668659	−	0.743569i	$-0.266869\pi$
$401$	26.7798	1.33732	0.668659	+	0.743569i	$0.266869\pi$
$402$	0	0
$403$	29.2175	1.45543
$404$	−1.65958	−0.0825673
$405$	0	0
$406$	−8.80828	−0.437148
$407$	0	0
$408$	0	0
$409$	−28.5313	−1.41078	−0.705390	−	0.708819i	$-0.749228\pi$
$409$	−28.5313	−1.41078	−0.705390	+	0.708819i	$0.749228\pi$
$410$	−12.6299	−0.623748
$411$	0	0
$412$	2.55159	0.125708
$413$	−46.0190	−2.26445
$414$	0	0
$415$	3.41304	0.167540
$416$	−5.81012	−0.284865
$417$	0	0
$418$	0	0
$419$	−14.7459	−0.720386	−0.360193	−	0.932878i	$-0.617289\pi$
$419$	−14.7459	−0.720386	−0.360193	+	0.932878i	$0.617289\pi$
$420$	0	0
$421$	19.3034	0.940788	0.470394	−	0.882456i	$-0.344112\pi$
$421$	19.3034	0.940788	0.470394	+	0.882456i	$0.344112\pi$
$422$	−20.3777	−0.991971
$423$	0	0
$424$	31.6205	1.53563
$425$	3.25841	0.158056
$426$	0	0
$427$	−32.5032	−1.57294
$428$	−0.488226	−0.0235993
$429$	0	0
$430$	15.2677	0.736276
$431$	−10.4970	−0.505622	−0.252811	−	0.967516i	$-0.581355\pi$
$431$	−10.4970	−0.505622	−0.252811	+	0.967516i	$0.581355\pi$
$432$	0	0
$433$	23.1357	1.11183	0.555915	−	0.831239i	$-0.312368\pi$
$433$	23.1357	1.11183	0.555915	+	0.831239i	$0.312368\pi$
$434$	−29.9826	−1.43921
$435$	0	0
$436$	1.33675	0.0640187
$437$	−2.65499	−0.127005
$438$	0	0
$439$	25.2313	1.20422	0.602111	−	0.798412i	$-0.294326\pi$
$439$	25.2313	1.20422	0.602111	+	0.798412i	$0.294326\pi$
$440$	0	0
$441$	0	0
$442$	−21.5150	−1.02336
$443$	−6.68546	−0.317636	−0.158818	−	0.987308i	$-0.550768\pi$
$443$	−6.68546	−0.317636	−0.158818	+	0.987308i	$0.550768\pi$
$444$	0	0
$445$	−7.85817	−0.372513
$446$	38.3345	1.81519
$447$	0	0
$448$	32.7349	1.54658
$449$	13.1159	0.618976	0.309488	−	0.950903i	$-0.399842\pi$
$449$	13.1159	0.618976	0.309488	+	0.950903i	$0.399842\pi$
$450$	0	0
$451$	0	0
$452$	3.79054	0.178292
$453$	0	0
$454$	4.04968	0.190061
$455$	18.6671	0.875125
$456$	0	0
$457$	−35.1276	−1.64320	−0.821599	−	0.570066i	$-0.806918\pi$
$457$	−35.1276	−1.64320	−0.821599	+	0.570066i	$0.806918\pi$
$458$	−4.08497	−0.190878
$459$	0	0
$460$	0.404307	0.0188509
$461$	30.4370	1.41759	0.708797	−	0.705413i	$-0.249238\pi$
$461$	30.4370	1.41759	0.708797	+	0.705413i	$0.249238\pi$
$462$	0	0
$463$	−10.0870	−0.468783	−0.234392	−	0.972142i	$-0.575310\pi$
$463$	−10.0870	−0.468783	−0.234392	+	0.972142i	$0.575310\pi$
$464$	6.15531	0.285753
$465$	0	0
$466$	−8.44355	−0.391140
$467$	−17.0520	−0.789072	−0.394536	−	0.918881i	$-0.629095\pi$
$467$	−17.0520	−0.789072	−0.394536	+	0.918881i	$0.629095\pi$
$468$	0	0
$469$	−10.4961	−0.484663
$470$	1.07827	0.0497370
$471$	0	0
$472$	35.9587	1.65513
$473$	0	0
$474$	0	0
$475$	−1.37283	−0.0629897
$476$	−2.57722	−0.118126
$477$	0	0
$478$	−7.64293	−0.349580
$479$	8.92831	0.407945	0.203972	−	0.978977i	$-0.434615\pi$
$479$	8.92831	0.407945	0.203972	+	0.978977i	$0.434615\pi$
$480$	0	0
$481$	−23.1792	−1.05688
$482$	−6.44996	−0.293788
$483$	0	0
$484$	0	0
$485$	13.6196	0.618435
$486$	0	0
$487$	33.4638	1.51639	0.758196	−	0.652027i	$-0.226081\pi$
$487$	33.4638	1.51639	0.758196	+	0.652027i	$0.226081\pi$
$488$	25.3976	1.14970
$489$	0	0
$490$	−9.78806	−0.442179
$491$	9.07217	0.409421	0.204711	−	0.978823i	$-0.434375\pi$
$491$	9.07217	0.409421	0.204711	+	0.978823i	$0.434375\pi$
$492$	0	0
$493$	−5.66859	−0.255301
$494$	9.06468	0.407839
$495$	0	0
$496$	20.9521	0.940779
$497$	10.3653	0.464946
$498$	0	0
$499$	39.0776	1.74935	0.874677	−	0.484706i	$-0.161073\pi$
$499$	39.0776	1.74935	0.874677	+	0.484706i	$0.161073\pi$
$500$	0.209057	0.00934931
$501$	0	0
$502$	1.18172	0.0527425
$503$	22.9483	1.02322	0.511608	−	0.859219i	$-0.329050\pi$
$503$	22.9483	1.02322	0.511608	+	0.859219i	$0.329050\pi$
$504$	0	0
$505$	−7.93842	−0.353255
$506$	0	0
$507$	0	0
$508$	−4.66454	−0.206956
$509$	−40.0637	−1.77579	−0.887896	−	0.460045i	$-0.847833\pi$
$509$	−40.0637	−1.77579	−0.887896	+	0.460045i	$0.847833\pi$
$510$	0	0
$511$	35.4775	1.56943
$512$	−25.0863	−1.10867
$513$	0	0
$514$	−11.5579	−0.509797
$515$	12.2052	0.537827
$516$	0	0
$517$	0	0
$518$	23.7861	1.04510
$519$	0	0
$520$	−14.5862	−0.639648
$521$	30.3711	1.33058	0.665292	−	0.746583i	$-0.268307\pi$
$521$	30.3711	1.33058	0.665292	+	0.746583i	$0.268307\pi$
$522$	0	0
$523$	−19.0349	−0.832337	−0.416169	−	0.909287i	$-0.636627\pi$
$523$	−19.0349	−0.832337	−0.416169	+	0.909287i	$0.636627\pi$
$524$	0.117511	0.00513350
$525$	0	0
$526$	−6.33793	−0.276347
$527$	−19.2954	−0.840521
$528$	0	0
$529$	−19.2598	−0.837383
$530$	14.3140	0.621761
$531$	0	0
$532$	1.08583	0.0470767
$533$	−46.5645	−2.01693
$534$	0	0
$535$	−2.33537	−0.100967
$536$	8.20150	0.354251
$537$	0	0
$538$	19.0933	0.823169
$539$	0	0
$540$	0	0
$541$	11.6232	0.499721	0.249861	−	0.968282i	$-0.419615\pi$
$541$	11.6232	0.499721	0.249861	+	0.968282i	$0.419615\pi$
$542$	−30.6953	−1.31848
$543$	0	0
$544$	3.83704	0.164512
$545$	6.39419	0.273897
$546$	0	0
$547$	−13.6517	−0.583703	−0.291852	−	0.956464i	$-0.594271\pi$
$547$	−13.6517	−0.583703	−0.291852	+	0.956464i	$0.594271\pi$
$548$	−3.81986	−0.163177
$549$	0	0
$550$	0	0
$551$	2.38829	0.101744
$552$	0	0
$553$	−48.3013	−2.05398
$554$	7.79273	0.331081
$555$	0	0
$556$	2.98115	0.126429
$557$	5.62173	0.238200	0.119100	−	0.992882i	$-0.461999\pi$
$557$	5.62173	0.238200	0.119100	+	0.992882i	$0.461999\pi$
$558$	0	0
$559$	56.2897	2.38080
$560$	13.3863	0.565675
$561$	0	0
$562$	−0.210898	−0.00889619
$563$	−26.0494	−1.09785	−0.548926	−	0.835871i	$-0.684963\pi$
$563$	−26.0494	−1.09785	−0.548926	+	0.835871i	$0.684963\pi$
$564$	0	0
$565$	18.1316	0.762804
$566$	11.9716	0.503204
$567$	0	0
$568$	−8.09931	−0.339839
$569$	−28.7427	−1.20496	−0.602478	−	0.798135i	$-0.705820\pi$
$569$	−28.7427	−1.20496	−0.602478	+	0.798135i	$0.705820\pi$
$570$	0	0
$571$	−17.8096	−0.745310	−0.372655	−	0.927970i	$-0.621552\pi$
$571$	−17.8096	−0.745310	−0.372655	+	0.927970i	$0.621552\pi$
$572$	0	0
$573$	0	0
$574$	47.7839	1.99446
$575$	1.93395	0.0806515
$576$	0	0
$577$	−16.3416	−0.680308	−0.340154	−	0.940370i	$-0.610479\pi$
$577$	−16.3416	−0.680308	−0.340154	+	0.940370i	$0.610479\pi$
$578$	−8.54182	−0.355293
$579$	0	0
$580$	−0.363692	−0.0151015
$581$	−12.9129	−0.535716
$582$	0	0
$583$	0	0
$584$	−27.7217	−1.14713
$585$	0	0
$586$	34.7785	1.43669
$587$	−12.0714	−0.498238	−0.249119	−	0.968473i	$-0.580141\pi$
$587$	−12.0714	−0.498238	−0.249119	+	0.968473i	$0.580141\pi$
$588$	0	0
$589$	8.12952	0.334971
$590$	16.2779	0.670149
$591$	0	0
$592$	−16.6220	−0.683159
$593$	28.5085	1.17071	0.585353	−	0.810779i	$-0.300956\pi$
$593$	28.5085	1.17071	0.585353	+	0.810779i	$0.300956\pi$
$594$	0	0
$595$	−12.3278	−0.505391
$596$	−2.21682	−0.0908044
$597$	0	0
$598$	−12.7697	−0.522194
$599$	1.66060	0.0678503	0.0339252	−	0.999424i	$-0.489199\pi$
$599$	1.66060	0.0678503	0.0339252	+	0.999424i	$0.489199\pi$
$600$	0	0
$601$	11.5218	0.469982	0.234991	−	0.971997i	$-0.424494\pi$
$601$	11.5218	0.469982	0.234991	+	0.971997i	$0.424494\pi$
$602$	−57.7638	−2.35428
$603$	0	0
$604$	2.62855	0.106954
$605$	0	0
$606$	0	0
$607$	8.37099	0.339768	0.169884	−	0.985464i	$-0.445661\pi$
$607$	8.37099	0.339768	0.169884	+	0.985464i	$0.445661\pi$
$608$	−1.61662	−0.0655624
$609$	0	0
$610$	11.4970	0.465501
$611$	3.97542	0.160828
$612$	0	0
$613$	21.2085	0.856604	0.428302	−	0.903636i	$-0.359112\pi$
$613$	21.2085	0.856604	0.428302	+	0.903636i	$0.359112\pi$
$614$	−6.32140	−0.255111
$615$	0	0
$616$	0	0
$617$	−19.6085	−0.789408	−0.394704	−	0.918808i	$-0.629153\pi$
$617$	−19.6085	−0.789408	−0.394704	+	0.918808i	$0.629153\pi$
$618$	0	0
$619$	26.7786	1.07632	0.538162	−	0.842841i	$-0.319119\pi$
$619$	26.7786	1.07632	0.538162	+	0.842841i	$0.319119\pi$
$620$	−1.23798	−0.0497184
$621$	0	0
$622$	25.7638	1.03303
$623$	29.7305	1.19113
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.93912	0.157439
$627$	0	0
$628$	2.62015	0.104555
$629$	15.3076	0.610355
$630$	0	0
$631$	38.1370	1.51821	0.759106	−	0.650967i	$-0.225637\pi$
$631$	38.1370	1.51821	0.759106	+	0.650967i	$0.225637\pi$
$632$	37.7421	1.50130
$633$	0	0
$634$	−2.44507	−0.0971060
$635$	−22.3123	−0.885437
$636$	0	0
$637$	−36.0870	−1.42982
$638$	0	0
$639$	0	0
$640$	−9.22384	−0.364604
$641$	30.6279	1.20973	0.604865	−	0.796328i	$-0.293227\pi$
$641$	30.6279	1.20973	0.604865	+	0.796328i	$0.293227\pi$
$642$	0	0
$643$	−3.35053	−0.132132	−0.0660660	−	0.997815i	$-0.521045\pi$
$643$	−3.35053	−0.132132	−0.0660660	+	0.997815i	$0.521045\pi$
$644$	−1.52965	−0.0602766
$645$	0	0
$646$	−5.98636	−0.235530
$647$	24.5143	0.963755	0.481878	−	0.876239i	$-0.339955\pi$
$647$	24.5143	0.963755	0.481878	+	0.876239i	$0.339955\pi$
$648$	0	0
$649$	0	0
$650$	−6.60292	−0.258988
$651$	0	0
$652$	0.648739	0.0254066
$653$	46.1663	1.80663	0.903313	−	0.428982i	$-0.141128\pi$
$653$	46.1663	1.80663	0.903313	+	0.428982i	$0.141128\pi$
$654$	0	0
$655$	0.562102	0.0219631
$656$	−33.3918	−1.30373
$657$	0	0
$658$	−4.07952	−0.159036
$659$	−11.4010	−0.444122	−0.222061	−	0.975033i	$-0.571278\pi$
$659$	−11.4010	−0.444122	−0.222061	+	0.975033i	$0.571278\pi$
$660$	0	0
$661$	34.7413	1.35128	0.675641	−	0.737231i	$-0.263867\pi$
$661$	34.7413	1.35128	0.675641	+	0.737231i	$0.263867\pi$
$662$	23.1541	0.899911
$663$	0	0
$664$	10.0900	0.391567
$665$	5.19394	0.201412
$666$	0	0
$667$	−3.36446	−0.130273
$668$	1.74541	0.0675320
$669$	0	0
$670$	3.71267	0.143433
$671$	0	0
$672$	0	0
$673$	15.0160	0.578823	0.289411	−	0.957205i	$-0.406540\pi$
$673$	15.0160	0.578823	0.289411	+	0.957205i	$0.406540\pi$
$674$	14.4659	0.557205
$675$	0	0
$676$	−2.37152	−0.0912124
$677$	24.6817	0.948594	0.474297	−	0.880365i	$-0.342702\pi$
$677$	24.6817	0.948594	0.474297	+	0.880365i	$0.342702\pi$
$678$	0	0
$679$	−51.5283	−1.97747
$680$	9.63282	0.369402
$681$	0	0
$682$	0	0
$683$	48.9656	1.87361	0.936807	−	0.349846i	$-0.113766\pi$
$683$	48.9656	1.87361	0.936807	+	0.349846i	$0.113766\pi$
$684$	0	0
$685$	−18.2719	−0.698133
$686$	1.58989	0.0607021
$687$	0	0
$688$	40.3659	1.53893
$689$	52.7735	2.01051
$690$	0	0
$691$	45.6987	1.73846	0.869230	−	0.494408i	$-0.164615\pi$
$691$	45.6987	1.73846	0.869230	+	0.494408i	$0.164615\pi$
$692$	1.43038	0.0543750
$693$	0	0
$694$	−20.3259	−0.771561
$695$	14.2600	0.540912
$696$	0	0
$697$	30.7515	1.16479
$698$	−5.07925	−0.192252
$699$	0	0
$700$	−0.790943	−0.0298948
$701$	6.87571	0.259692	0.129846	−	0.991534i	$-0.458552\pi$
$701$	6.87571	0.259692	0.129846	+	0.991534i	$0.458552\pi$
$702$	0	0
$703$	−6.44940	−0.243244
$704$	0	0
$705$	0	0
$706$	−0.695330	−0.0261691
$707$	30.0341	1.12955
$708$	0	0
$709$	13.7735	0.517275	0.258638	−	0.965974i	$-0.416726\pi$
$709$	13.7735	0.517275	0.258638	+	0.965974i	$0.416726\pi$
$710$	−3.66641	−0.137598
$711$	0	0
$712$	−23.2311	−0.870621
$713$	−11.4524	−0.428894
$714$	0	0
$715$	0	0
$716$	3.24216	0.121165
$717$	0	0
$718$	−18.5181	−0.691088
$719$	2.09938	0.0782935	0.0391468	−	0.999233i	$-0.487536\pi$
$719$	2.09938	0.0782935	0.0391468	+	0.999233i	$0.487536\pi$
$720$	0	0
$721$	−46.1771	−1.71973
$722$	−22.9048	−0.852428
$723$	0	0
$724$	0.472670	0.0175667
$725$	−1.73968	−0.0646101
$726$	0	0
$727$	−14.9925	−0.556042	−0.278021	−	0.960575i	$-0.589678\pi$
$727$	−14.9925	−0.556042	−0.278021	+	0.960575i	$0.589678\pi$
$728$	55.1853	2.04530
$729$	0	0
$730$	−12.5491	−0.464463
$731$	−37.1740	−1.37493
$732$	0	0
$733$	−16.5017	−0.609506	−0.304753	−	0.952431i	$-0.598574\pi$
$733$	−16.5017	−0.609506	−0.304753	+	0.952431i	$0.598574\pi$
$734$	−27.5976	−1.01865
$735$	0	0
$736$	2.27739	0.0839455
$737$	0	0
$738$	0	0
$739$	−7.28575	−0.268011	−0.134005	−	0.990981i	$-0.542784\pi$
$739$	−7.28575	−0.268011	−0.134005	+	0.990981i	$0.542784\pi$
$740$	0.982126	0.0361037
$741$	0	0
$742$	−54.1554	−1.98811
$743$	−27.6106	−1.01293	−0.506467	−	0.862259i	$-0.669049\pi$
$743$	−27.6106	−1.01293	−0.506467	+	0.862259i	$0.669049\pi$
$744$	0	0
$745$	−10.6039	−0.388497
$746$	−10.9445	−0.400708
$747$	0	0
$748$	0	0
$749$	8.83562	0.322847
$750$	0	0
$751$	−2.24849	−0.0820486	−0.0410243	−	0.999158i	$-0.513062\pi$
$751$	−2.24849	−0.0820486	−0.0410243	+	0.999158i	$0.513062\pi$
$752$	2.85081	0.103958
$753$	0	0
$754$	11.4870	0.418331
$755$	12.5734	0.457591
$756$	0	0
$757$	33.9846	1.23519	0.617596	−	0.786495i	$-0.288107\pi$
$757$	33.9846	1.23519	0.617596	+	0.786495i	$0.288107\pi$
$758$	35.5953	1.29288
$759$	0	0
$760$	−4.05849	−0.147217
$761$	−27.7004	−1.00414	−0.502069	−	0.864828i	$-0.667428\pi$
$761$	−27.7004	−1.00414	−0.502069	+	0.864828i	$0.667428\pi$
$762$	0	0
$763$	−24.1917	−0.875798
$764$	−3.71478	−0.134396
$765$	0	0
$766$	19.4861	0.704060
$767$	60.0139	2.16698
$768$	0	0
$769$	−2.29270	−0.0826770	−0.0413385	−	0.999145i	$-0.513162\pi$
$769$	−2.29270	−0.0826770	−0.0413385	+	0.999145i	$0.513162\pi$
$770$	0	0
$771$	0	0
$772$	4.42278	0.159179
$773$	30.9947	1.11480	0.557400	−	0.830244i	$-0.311799\pi$
$773$	30.9947	1.11480	0.557400	+	0.830244i	$0.311799\pi$
$774$	0	0
$775$	−5.92173	−0.212715
$776$	40.2636	1.44538
$777$	0	0
$778$	5.28494	0.189474
$779$	−12.9562	−0.464203
$780$	0	0
$781$	0	0
$782$	8.43321	0.301571
$783$	0	0
$784$	−25.8783	−0.924225
$785$	12.5332	0.447328
$786$	0	0
$787$	49.1385	1.75160	0.875798	−	0.482677i	$-0.160335\pi$
$787$	49.1385	1.75160	0.875798	+	0.482677i	$0.160335\pi$
$788$	5.03899	0.179507
$789$	0	0
$790$	17.0852	0.607863
$791$	−68.5990	−2.43910
$792$	0	0
$793$	42.3877	1.50523
$794$	−24.1104	−0.855647
$795$	0	0
$796$	−0.899717	−0.0318896
$797$	−2.77965	−0.0984602	−0.0492301	−	0.998787i	$-0.515677\pi$
$797$	−2.77965	−0.0984602	−0.0492301	+	0.998787i	$0.515677\pi$
$798$	0	0
$799$	−2.62539	−0.0928795
$800$	1.17758	0.0416337
$801$	0	0
$802$	35.8383	1.26550
$803$	0	0
$804$	0	0
$805$	−7.31690	−0.257887
$806$	39.1007	1.37726
$807$	0	0
$808$	−23.4683	−0.825612
$809$	−44.0336	−1.54814	−0.774069	−	0.633101i	$-0.781782\pi$
$809$	−44.0336	−1.54814	−0.774069	+	0.633101i	$0.781782\pi$
$810$	0	0
$811$	17.9977	0.631985	0.315993	−	0.948762i	$-0.397662\pi$
$811$	17.9977	0.631985	0.315993	+	0.948762i	$0.397662\pi$
$812$	1.37599	0.0482877
$813$	0	0
$814$	0	0
$815$	3.10317	0.108699
$816$	0	0
$817$	15.6621	0.547948
$818$	−38.1823	−1.33501
$819$	0	0
$820$	1.97299	0.0688998
$821$	−13.6645	−0.476894	−0.238447	−	0.971155i	$-0.576638\pi$
$821$	−13.6645	−0.476894	−0.238447	+	0.971155i	$0.576638\pi$
$822$	0	0
$823$	−9.55474	−0.333057	−0.166529	−	0.986037i	$-0.553256\pi$
$823$	−9.55474	−0.333057	−0.166529	+	0.986037i	$0.553256\pi$
$824$	36.0823	1.25699
$825$	0	0
$826$	−61.5854	−2.14283
$827$	2.86560	0.0996467	0.0498233	−	0.998758i	$-0.484134\pi$
$827$	2.86560	0.0996467	0.0498233	+	0.998758i	$0.484134\pi$
$828$	0	0
$829$	−21.2099	−0.736649	−0.368325	−	0.929697i	$-0.620068\pi$
$829$	−21.2099	−0.736649	−0.368325	+	0.929697i	$0.620068\pi$
$830$	4.56754	0.158542
$831$	0	0
$832$	−42.6899	−1.48001
$833$	23.8320	0.825731
$834$	0	0
$835$	8.34898	0.288928
$836$	0	0
$837$	0	0
$838$	−19.7339	−0.681697
$839$	−1.40998	−0.0486779	−0.0243390	−	0.999704i	$-0.507748\pi$
$839$	−1.40998	−0.0486779	−0.0243390	+	0.999704i	$0.507748\pi$
$840$	0	0
$841$	−25.9735	−0.895638
$842$	25.8329	0.890262
$843$	0	0
$844$	3.18331	0.109574
$845$	−11.3439	−0.390242
$846$	0	0
$847$	0	0
$848$	37.8443	1.29958
$849$	0	0
$850$	4.36060	0.149567
$851$	9.08550	0.311447
$852$	0	0
$853$	13.4716	0.461257	0.230629	−	0.973042i	$-0.425922\pi$
$853$	13.4716	0.461257	0.230629	+	0.973042i	$0.425922\pi$
$854$	−43.4977	−1.48846
$855$	0	0
$856$	−6.90406	−0.235976
$857$	25.5677	0.873376	0.436688	−	0.899613i	$-0.356151\pi$
$857$	25.5677	0.873376	0.436688	+	0.899613i	$0.356151\pi$
$858$	0	0
$859$	21.7766	0.743008	0.371504	−	0.928431i	$-0.378842\pi$
$859$	21.7766	0.743008	0.371504	+	0.928431i	$0.378842\pi$
$860$	−2.38506	−0.0813298
$861$	0	0
$862$	−14.0477	−0.478467
$863$	51.9088	1.76700	0.883499	−	0.468434i	$-0.155181\pi$
$863$	51.9088	1.76700	0.883499	+	0.468434i	$0.155181\pi$
$864$	0	0
$865$	6.84208	0.232638
$866$	30.9616	1.05212
$867$	0	0
$868$	4.68375	0.158977
$869$	0	0
$870$	0	0
$871$	13.6880	0.463801
$872$	18.9031	0.640141
$873$	0	0
$874$	−3.55307	−0.120184
$875$	−3.78339	−0.127902
$876$	0	0
$877$	10.9745	0.370584	0.185292	−	0.982684i	$-0.440677\pi$
$877$	10.9745	0.370584	0.185292	+	0.982684i	$0.440677\pi$
$878$	33.7660	1.13955
$879$	0	0
$880$	0	0
$881$	8.42384	0.283806	0.141903	−	0.989881i	$-0.454678\pi$
$881$	8.42384	0.283806	0.141903	+	0.989881i	$0.454678\pi$
$882$	0	0
$883$	−13.9067	−0.467997	−0.233998	−	0.972237i	$-0.575181\pi$
$883$	−13.9067	−0.467997	−0.233998	+	0.972237i	$0.575181\pi$
$884$	3.36097	0.113042
$885$	0	0
$886$	−8.94690	−0.300577
$887$	−27.7955	−0.933283	−0.466641	−	0.884447i	$-0.654536\pi$
$887$	−27.7955	−0.933283	−0.466641	+	0.884447i	$0.654536\pi$
$888$	0	0
$889$	84.4161	2.83122
$890$	−10.5163	−0.352507
$891$	0	0
$892$	−5.98843	−0.200508
$893$	1.10613	0.0370151
$894$	0	0
$895$	15.5085	0.518392
$896$	34.8974	1.16584
$897$	0	0
$898$	17.5525	0.585733
$899$	10.3019	0.343588
$900$	0	0
$901$	−34.8519	−1.16108
$902$	0	0
$903$	0	0
$904$	53.6025	1.78279
$905$	2.26096	0.0751570
$906$	0	0
$907$	10.1506	0.337046	0.168523	−	0.985698i	$-0.446100\pi$
$907$	10.1506	0.337046	0.168523	+	0.985698i	$0.446100\pi$
$908$	−0.632622	−0.0209943
$909$	0	0
$910$	24.9814	0.828125
$911$	−45.0748	−1.49339	−0.746697	−	0.665164i	$-0.768362\pi$
$911$	−45.0748	−1.49339	−0.746697	+	0.665164i	$0.768362\pi$
$912$	0	0
$913$	0	0
$914$	−47.0099	−1.55495
$915$	0	0
$916$	0.638134	0.0210845
$917$	−2.12665	−0.0702281
$918$	0	0
$919$	−5.79094	−0.191026	−0.0955128	−	0.995428i	$-0.530449\pi$
$919$	−5.79094	−0.191026	−0.0955128	+	0.995428i	$0.530449\pi$
$920$	5.71734	0.188495
$921$	0	0
$922$	40.7327	1.34146
$923$	−13.5175	−0.444933
$924$	0	0
$925$	4.69789	0.154466
$926$	−13.4991	−0.443606
$927$	0	0
$928$	−2.04861	−0.0672490
$929$	42.8241	1.40501	0.702506	−	0.711678i	$-0.252064\pi$
$929$	42.8241	1.40501	0.702506	+	0.711678i	$0.252064\pi$
$930$	0	0
$931$	−10.0409	−0.329077
$932$	1.31901	0.0432056
$933$	0	0
$934$	−22.8200	−0.746694
$935$	0	0
$936$	0	0
$937$	11.8861	0.388302	0.194151	−	0.980972i	$-0.437805\pi$
$937$	11.8861	0.388302	0.194151	+	0.980972i	$0.437805\pi$
$938$	−14.0465	−0.458633
$939$	0	0
$940$	−0.168443	−0.00549400
$941$	40.3081	1.31401	0.657003	−	0.753888i	$-0.271824\pi$
$941$	40.3081	1.31401	0.657003	+	0.753888i	$0.271824\pi$
$942$	0	0
$943$	18.2518	0.594361
$944$	43.0365	1.40072
$945$	0	0
$946$	0	0
$947$	−53.3167	−1.73256	−0.866280	−	0.499560i	$-0.833495\pi$
$947$	−53.3167	−1.73256	−0.866280	+	0.499560i	$0.833495\pi$
$948$	0	0
$949$	−46.2665	−1.50188
$950$	−1.83720	−0.0596068
$951$	0	0
$952$	−36.4447	−1.18118
$953$	23.1339	0.749379	0.374690	−	0.927150i	$-0.377749\pi$
$953$	23.1339	0.749379	0.374690	+	0.927150i	$0.377749\pi$
$954$	0	0
$955$	−17.7692	−0.574999
$956$	1.19394	0.0386149
$957$	0	0
$958$	11.9484	0.386036
$959$	69.1296	2.23231
$960$	0	0
$961$	4.06685	0.131189
$962$	−31.0198	−1.00012
$963$	0	0
$964$	1.00758	0.0324521
$965$	21.1559	0.681032
$966$	0	0
$967$	−32.8910	−1.05770	−0.528851	−	0.848715i	$-0.677377\pi$
$967$	−32.8910	−1.05770	−0.528851	+	0.848715i	$0.677377\pi$
$968$	0	0
$969$	0	0
$970$	18.2266	0.585221
$971$	−22.5974	−0.725185	−0.362592	−	0.931948i	$-0.618108\pi$
$971$	−22.5974	−0.725185	−0.362592	+	0.931948i	$0.618108\pi$
$972$	0	0
$973$	−53.9510	−1.72959
$974$	44.7834	1.43495
$975$	0	0
$976$	30.3966	0.972971
$977$	−30.7844	−0.984882	−0.492441	−	0.870346i	$-0.663895\pi$
$977$	−30.7844	−0.984882	−0.492441	+	0.870346i	$0.663895\pi$
$978$	0	0
$979$	0	0
$980$	1.52904	0.0488435
$981$	0	0
$982$	12.1409	0.387433
$983$	−34.8963	−1.11302	−0.556509	−	0.830841i	$-0.687860\pi$
$983$	−34.8963	−1.11302	−0.556509	+	0.830841i	$0.687860\pi$
$984$	0	0
$985$	24.1034	0.768000
$986$	−7.58606	−0.241589
$987$	0	0
$988$	−1.41604	−0.0450503
$989$	−22.0638	−0.701588
$990$	0	0
$991$	−41.7439	−1.32604	−0.663019	−	0.748602i	$-0.730725\pi$
$991$	−41.7439	−1.32604	−0.663019	+	0.748602i	$0.730725\pi$
$992$	−6.97330	−0.221403
$993$	0	0
$994$	13.8714	0.439975
$995$	−4.30369	−0.136436
$996$	0	0
$997$	33.7340	1.06836	0.534182	−	0.845369i	$-0.320620\pi$
$997$	33.7340	1.06836	0.534182	+	0.845369i	$0.320620\pi$
$998$	52.2961	1.65540
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bq.1.3		4
3.2	odd	2		1815.2.a.q.1.2		4
11.2	odd	10		495.2.n.c.136.2		8
11.6	odd	10		495.2.n.c.91.2		8
11.10	odd	2		5445.2.a.bj.1.2		4
15.14	odd	2		9075.2.a.df.1.3		4
33.2	even	10		165.2.m.c.136.1	yes	8
33.17	even	10		165.2.m.c.91.1	✓	8
33.32	even	2		1815.2.a.u.1.3		4
165.2	odd	20		825.2.bx.e.499.3		16
165.17	odd	20		825.2.bx.e.124.2		16
165.68	odd	20		825.2.bx.e.499.2		16
165.83	odd	20		825.2.bx.e.124.3		16
165.134	even	10		825.2.n.j.301.2		8
165.149	even	10		825.2.n.j.751.2		8
165.164	even	2		9075.2.a.co.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.c.91.1	✓	8	33.17	even	10
165.2.m.c.136.1	yes	8	33.2	even	10
495.2.n.c.91.2		8	11.6	odd	10
495.2.n.c.136.2		8	11.2	odd	10
825.2.n.j.301.2		8	165.134	even	10
825.2.n.j.751.2		8	165.149	even	10
825.2.bx.e.124.2		16	165.17	odd	20
825.2.bx.e.124.3		16	165.83	odd	20
825.2.bx.e.499.2		16	165.68	odd	20
825.2.bx.e.499.3		16	165.2	odd	20
1815.2.a.q.1.2		4	3.2	odd	2
1815.2.a.u.1.3		4	33.32	even	2
5445.2.a.bj.1.2		4	11.10	odd	2
5445.2.a.bq.1.3		4	1.1	even	1	trivial
9075.2.a.co.1.2		4	165.164	even	2
9075.2.a.df.1.3		4	15.14	odd	2

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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+1.33826 q^{2} -0.209057 q^{4} -1.00000 q^{5} +3.78339 q^{7} -2.95630 q^{8} -1.33826 q^{10} -4.93395 q^{13} +5.06316 q^{14} -3.53818 q^{16} +3.25841 q^{17} -1.37283 q^{19} +0.209057 q^{20} +1.93395 q^{23} +1.00000 q^{25} -6.60292 q^{26} -0.790943 q^{28} -1.73968 q^{29} -5.92173 q^{31} +1.17758 q^{32} +4.36060 q^{34} -3.78339 q^{35} +4.69789 q^{37} -1.83720 q^{38} +2.95630 q^{40} +9.43757 q^{41} -11.4086 q^{43} +2.58814 q^{46} -0.805727 q^{47} +7.31401 q^{49} +1.33826 q^{50} +1.03148 q^{52} -10.6960 q^{53} -11.1848 q^{56} -2.32815 q^{58} -12.1634 q^{59} -8.59102 q^{61} -7.92482 q^{62} +8.65227 q^{64} +4.93395 q^{65} -2.77425 q^{67} -0.681193 q^{68} -5.06316 q^{70} +2.73968 q^{71} +9.37717 q^{73} +6.28700 q^{74} +0.286999 q^{76} -12.7667 q^{79} +3.53818 q^{80} +12.6299 q^{82} -3.41304 q^{83} -3.25841 q^{85} -15.2677 q^{86} +7.85817 q^{89} -18.6671 q^{91} -0.404307 q^{92} -1.07827 q^{94} +1.37283 q^{95} -13.6196 q^{97} +9.78806 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{4} - 4 q^{5} - 3 q^{8} - q^{10} - 7 q^{13} + 5 q^{14} - 9 q^{16} + 8 q^{17} - 11 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 12 q^{26} - 5 q^{28} + 17 q^{29} - 5 q^{31} + 17 q^{34} + 15 q^{37}+ \cdots + 13 q^{98}+O(q^{100}) \) 4 * q + q^2 + q^4 - 4 * q^5 - 3 * q^8 - q^10 - 7 * q^13 + 5 * q^14 - 9 * q^16 + 8 * q^17 - 11 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 12 * q^26 - 5 * q^28 + 17 * q^29 - 5 * q^31 + 17 * q^34 + 15 * q^37 + q^38 + 3 * q^40 + 10 * q^41 + 4 * q^43 - 15 * q^46 + 8 * q^47 - 8 * q^49 + q^50 + 7 * q^52 - 10 * q^53 - 10 * q^56 - 7 * q^58 - 9 * q^59 - 37 * q^61 - 20 * q^62 - 7 * q^64 + 7 * q^65 + 3 * q^67 + 17 * q^68 - 5 * q^70 - 13 * q^71 - 15 * q^73 - 5 * q^74 - 29 * q^76 - 20 * q^79 + 9 * q^80 + 5 * q^82 - 17 * q^83 - 8 * q^85 - 34 * q^86 + 24 * q^89 - 20 * q^91 - 10 * q^92 - 23 * q^94 + 11 * q^95 - 32 * q^97 + 13 * q^98

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