LMFDB - Embedded newform 5733.2.a.by.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$45.7782354788$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 18x^{8} + 110x^{6} - 265x^{4} + 243x^{2} - 63 $ x^10 - 18x^8 + 110x^6 - 265x^4 + 243x^2 - 63
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 819)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.12683$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.12683 q^{2} -0.730265 q^{4} +3.42178 q^{5} -3.07653 q^{8} +3.85575 q^{10} -6.22870 q^{11} +1.00000 q^{13} -2.00618 q^{16} +0.602628 q^{17} +5.29924 q^{19} -2.49881 q^{20} -7.01866 q^{22} -4.71533 q^{23} +6.70856 q^{25} +1.12683 q^{26} -6.48608 q^{29} +10.7147 q^{31} +3.89245 q^{32} +0.679056 q^{34} +0.269735 q^{37} +5.97132 q^{38} -10.5272 q^{40} -0.266816 q^{41} +12.0187 q^{43} +4.54860 q^{44} -5.31335 q^{46} +8.75196 q^{47} +7.55938 q^{50} -0.730265 q^{52} +3.63786 q^{53} -21.3132 q^{55} -7.30868 q^{58} +4.28179 q^{59} -3.57983 q^{61} +12.0736 q^{62} +8.39847 q^{64} +3.42178 q^{65} +1.37515 q^{67} -0.440078 q^{68} +0.740276 q^{71} +4.18461 q^{73} +0.303944 q^{74} -3.86985 q^{76} +8.43557 q^{79} -6.86471 q^{80} -0.300654 q^{82} +0.116941 q^{83} +2.06206 q^{85} +13.5429 q^{86} +19.1628 q^{88} +9.87599 q^{89} +3.44344 q^{92} +9.86193 q^{94} +18.1328 q^{95} +10.8465 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 16 q^{4} - 4 q^{10} + 10 q^{13} + 32 q^{16} + 6 q^{19} + 10 q^{22} + 24 q^{25} + 12 q^{31} + 34 q^{34} + 26 q^{37} - 70 q^{40} + 40 q^{43} - 6 q^{46} + 16 q^{52} - 24 q^{55} + 36 q^{58} - 22 q^{61}+ \cdots + 18 q^{97}+O(q^{100}) $ 10 * q + 16 * q^4 - 4 * q^10 + 10 * q^13 + 32 * q^16 + 6 * q^19 + 10 * q^22 + 24 * q^25 + 12 * q^31 + 34 * q^34 + 26 * q^37 - 70 * q^40 + 40 * q^43 - 6 * q^46 + 16 * q^52 - 24 * q^55 + 36 * q^58 - 22 * q^61 + 78 * q^64 + 26 * q^67 + 24 * q^73 + 4 * q^76 + 28 * q^79 - 28 * q^82 - 20 * q^85 + 12 * q^88 + 4 * q^94 + 18 * q^97

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 $

$2$	1.12683	0.796786	0.398393	−	0.917215i	$-0.369568\pi$
$2$	1.12683	0.796786	0.398393	+	0.917215i	$0.369568\pi$
$3$	0	0
$3$	0	0
$4$	−0.730265	−0.365133
$5$	3.42178	1.53027	0.765133	−	0.643873i	$-0.222673\pi$
$5$	3.42178	1.53027	0.765133	+	0.643873i	$0.222673\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.07653	−1.08772
$9$	0	0
$10$	3.85575	1.21929
$11$	−6.22870	−1.87802	−0.939012	−	0.343884i	$-0.888257\pi$
$11$	−6.22870	−1.87802	−0.939012	+	0.343884i	$0.888257\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	−2.00618	−0.501546
$17$	0.602628	0.146159	0.0730794	−	0.997326i	$-0.476717\pi$
$17$	0.602628	0.146159	0.0730794	+	0.997326i	$0.476717\pi$
$18$	0	0
$19$	5.29924	1.21573	0.607865	−	0.794041i	$-0.292026\pi$
$19$	5.29924	1.21573	0.607865	+	0.794041i	$0.292026\pi$
$20$	−2.49881	−0.558750
$21$	0	0
$22$	−7.01866	−1.49638
$23$	−4.71533	−0.983214	−0.491607	−	0.870817i	$-0.663590\pi$
$23$	−4.71533	−0.983214	−0.491607	+	0.870817i	$0.663590\pi$
$24$	0	0
$25$	6.70856	1.34171
$26$	1.12683	0.220989
$27$	0	0
$28$	0	0
$29$	−6.48608	−1.20444	−0.602218	−	0.798332i	$-0.705716\pi$
$29$	−6.48608	−1.20444	−0.602218	+	0.798332i	$0.705716\pi$
$30$	0	0
$31$	10.7147	1.92443	0.962213	−	0.272299i	$-0.0877840\pi$
$31$	10.7147	1.92443	0.962213	+	0.272299i	$0.0877840\pi$
$32$	3.89245	0.688094
$33$	0	0
$34$	0.679056	0.116457
$35$	0	0
$36$	0	0
$37$	0.269735	0.0443441	0.0221721	−	0.999754i	$-0.492942\pi$
$37$	0.269735	0.0443441	0.0221721	+	0.999754i	$0.492942\pi$
$38$	5.97132	0.968676
$39$	0	0
$40$	−10.5272	−1.66450
$41$	−0.266816	−0.0416696	−0.0208348	−	0.999783i	$-0.506632\pi$
$41$	−0.266816	−0.0416696	−0.0208348	+	0.999783i	$0.506632\pi$
$42$	0	0
$43$	12.0187	1.83283	0.916414	−	0.400232i	$-0.131070\pi$
$43$	12.0187	1.83283	0.916414	+	0.400232i	$0.131070\pi$
$44$	4.54860	0.685728
$45$	0	0
$46$	−5.31335	−0.783411
$47$	8.75196	1.27660	0.638302	−	0.769786i	$-0.279637\pi$
$47$	8.75196	1.27660	0.638302	+	0.769786i	$0.279637\pi$
$48$	0	0
$49$	0	0
$50$	7.55938	1.06906
$51$	0	0
$52$	−0.730265	−0.101270
$53$	3.63786	0.499698	0.249849	−	0.968285i	$-0.419619\pi$
$53$	3.63786	0.499698	0.249849	+	0.968285i	$0.419619\pi$
$54$	0	0
$55$	−21.3132	−2.87388
$56$	0	0
$57$	0	0
$58$	−7.30868	−0.959677
$59$	4.28179	0.557441	0.278721	−	0.960372i	$-0.410090\pi$
$59$	4.28179	0.557441	0.278721	+	0.960372i	$0.410090\pi$
$60$	0	0
$61$	−3.57983	−0.458350	−0.229175	−	0.973385i	$-0.573603\pi$
$61$	−3.57983	−0.458350	−0.229175	+	0.973385i	$0.573603\pi$
$62$	12.0736	1.53335
$63$	0	0
$64$	8.39847	1.04981
$65$	3.42178	0.424419
$66$	0	0
$67$	1.37515	0.168001	0.0840004	−	0.996466i	$-0.473230\pi$
$67$	1.37515	0.168001	0.0840004	+	0.996466i	$0.473230\pi$
$68$	−0.440078	−0.0533673
$69$	0	0
$70$	0	0
$71$	0.740276	0.0878546	0.0439273	−	0.999035i	$-0.486013\pi$
$71$	0.740276	0.0878546	0.0439273	+	0.999035i	$0.486013\pi$
$72$	0	0
$73$	4.18461	0.489772	0.244886	−	0.969552i	$-0.421249\pi$
$73$	4.18461	0.489772	0.244886	+	0.969552i	$0.421249\pi$
$74$	0.303944	0.0353328
$75$	0	0
$76$	−3.86985	−0.443902
$77$	0	0
$78$	0	0
$79$	8.43557	0.949076	0.474538	−	0.880235i	$-0.342615\pi$
$79$	8.43557	0.949076	0.474538	+	0.880235i	$0.342615\pi$
$80$	−6.86471	−0.767498
$81$	0	0
$82$	−0.300654	−0.0332017
$83$	0.116941	0.0128359	0.00641795	−	0.999979i	$-0.497957\pi$
$83$	0.116941	0.0128359	0.00641795	+	0.999979i	$0.497957\pi$
$84$	0	0
$85$	2.06206	0.223662
$86$	13.5429	1.46037
$87$	0	0
$88$	19.1628	2.04276
$89$	9.87599	1.04685	0.523427	−	0.852071i	$-0.324653\pi$
$89$	9.87599	1.04685	0.523427	+	0.852071i	$0.324653\pi$
$90$	0	0
$91$	0	0
$92$	3.44344	0.359003
$93$	0	0
$94$	9.86193	1.01718
$95$	18.1328	1.86039
$96$	0	0
$97$	10.8465	1.10130	0.550649	−	0.834737i	$-0.314380\pi$
$97$	10.8465	1.10130	0.550649	+	0.834737i	$0.314380\pi$
$98$	0	0
$99$	0	0
$100$	−4.89903	−0.489903
$101$	4.47017	0.444799	0.222399	−	0.974956i	$-0.428611\pi$
$101$	4.47017	0.444799	0.222399	+	0.974956i	$0.428611\pi$
$102$	0	0
$103$	11.7272	1.15552	0.577759	−	0.816208i	$-0.303927\pi$
$103$	11.7272	1.15552	0.577759	+	0.816208i	$0.303927\pi$
$104$	−3.07653	−0.301679
$105$	0	0
$106$	4.09923	0.398152
$107$	3.51007	0.339331	0.169666	−	0.985502i	$-0.445731\pi$
$107$	3.51007	0.339331	0.169666	+	0.985502i	$0.445731\pi$
$108$	0	0
$109$	−12.4233	−1.18994	−0.594969	−	0.803749i	$-0.702836\pi$
$109$	−12.4233	−1.18994	−0.594969	+	0.803749i	$0.702836\pi$
$110$	−24.0163	−2.28986
$111$	0	0
$112$	0	0
$113$	−5.31516	−0.500008	−0.250004	−	0.968245i	$-0.580432\pi$
$113$	−5.31516	−0.500008	−0.250004	+	0.968245i	$0.580432\pi$
$114$	0	0
$115$	−16.1348	−1.50458
$116$	4.73656	0.439779
$117$	0	0
$118$	4.82483	0.444161
$119$	0	0
$120$	0	0
$121$	27.7967	2.52697
$122$	−4.03384	−0.365207
$123$	0	0
$124$	−7.82461	−0.702670
$125$	5.84633	0.522911
$126$	0	0
$127$	−6.42331	−0.569977	−0.284988	−	0.958531i	$-0.591990\pi$
$127$	−6.42331	−0.569977	−0.284988	+	0.958531i	$0.591990\pi$
$128$	1.67872	0.148379
$129$	0	0
$130$	3.85575	0.338171
$131$	17.6580	1.54279	0.771393	−	0.636359i	$-0.219560\pi$
$131$	17.6580	1.54279	0.771393	+	0.636359i	$0.219560\pi$
$132$	0	0
$133$	0	0
$134$	1.54955	0.133861
$135$	0	0
$136$	−1.85400	−0.158980
$137$	−6.17892	−0.527901	−0.263950	−	0.964536i	$-0.585025\pi$
$137$	−6.17892	−0.527901	−0.263950	+	0.964536i	$0.585025\pi$
$138$	0	0
$139$	−7.82775	−0.663941	−0.331971	−	0.943290i	$-0.607713\pi$
$139$	−7.82775	−0.663941	−0.331971	+	0.943290i	$0.607713\pi$
$140$	0	0
$141$	0	0
$142$	0.834162	0.0700013
$143$	−6.22870	−0.520870
$144$	0	0
$145$	−22.1939	−1.84311
$146$	4.71533	0.390243
$147$	0	0
$148$	−0.196978	−0.0161915
$149$	−8.32411	−0.681938	−0.340969	−	0.940075i	$-0.610755\pi$
$149$	−8.32411	−0.681938	−0.340969	+	0.940075i	$0.610755\pi$
$150$	0	0
$151$	−20.3687	−1.65758	−0.828792	−	0.559557i	$-0.810971\pi$
$151$	−20.3687	−1.65758	−0.828792	+	0.559557i	$0.810971\pi$
$152$	−16.3033	−1.32237
$153$	0	0
$154$	0	0
$155$	36.6635	2.94488
$156$	0	0
$157$	4.57679	0.365268	0.182634	−	0.983181i	$-0.441538\pi$
$157$	4.57679	0.365268	0.182634	+	0.983181i	$0.441538\pi$
$158$	9.50542	0.756210
$159$	0	0
$160$	13.3191	1.05297
$161$	0	0
$162$	0	0
$163$	9.11463	0.713913	0.356956	−	0.934121i	$-0.383814\pi$
$163$	9.11463	0.713913	0.356956	+	0.934121i	$0.383814\pi$
$164$	0.194846	0.0152149
$165$	0	0
$166$	0.131772	0.0102275
$167$	−13.9793	−1.08175	−0.540874	−	0.841104i	$-0.681906\pi$
$167$	−13.9793	−1.08175	−0.540874	+	0.841104i	$0.681906\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.32358	0.178210
$171$	0	0
$172$	−8.77681	−0.669225
$173$	−13.2681	−1.00875	−0.504376	−	0.863484i	$-0.668277\pi$
$173$	−13.2681	−1.00875	−0.504376	+	0.863484i	$0.668277\pi$
$174$	0	0
$175$	0	0
$176$	12.4959	0.941915
$177$	0	0
$178$	11.1285	0.834118
$179$	19.6988	1.47236	0.736178	−	0.676788i	$-0.236629\pi$
$179$	19.6988	1.47236	0.736178	+	0.676788i	$0.236629\pi$
$180$	0	0
$181$	−5.10682	−0.379587	−0.189794	−	0.981824i	$-0.560782\pi$
$181$	−5.10682	−0.379587	−0.189794	+	0.981824i	$0.560782\pi$
$182$	0	0
$183$	0	0
$184$	14.5069	1.06946
$185$	0.922973	0.0678583
$186$	0	0
$187$	−3.75359	−0.274490
$188$	−6.39125	−0.466130
$189$	0	0
$190$	20.4325	1.48233
$191$	17.8745	1.29335	0.646677	−	0.762764i	$-0.276158\pi$
$191$	17.8745	1.29335	0.646677	+	0.762764i	$0.276158\pi$
$192$	0	0
$193$	−0.615256	−0.0442871	−0.0221435	−	0.999755i	$-0.507049\pi$
$193$	−0.615256	−0.0442871	−0.0221435	+	0.999755i	$0.507049\pi$
$194$	12.2221	0.877498
$195$	0	0
$196$	0	0
$197$	2.01129	0.143298	0.0716492	−	0.997430i	$-0.477174\pi$
$197$	2.01129	0.143298	0.0716492	+	0.997430i	$0.477174\pi$
$198$	0	0
$199$	−18.0416	−1.27894	−0.639468	−	0.768818i	$-0.720845\pi$
$199$	−18.0416	−1.27894	−0.639468	+	0.768818i	$0.720845\pi$
$200$	−20.6391	−1.45941
$201$	0	0
$202$	5.03710	0.354409
$203$	0	0
$204$	0	0
$205$	−0.912984	−0.0637655
$206$	13.2145	0.920700
$207$	0	0
$208$	−2.00618	−0.139104
$209$	−33.0074	−2.28317
$210$	0	0
$211$	18.9585	1.30516	0.652578	−	0.757721i	$-0.273687\pi$
$211$	18.9585	1.30516	0.652578	+	0.757721i	$0.273687\pi$
$212$	−2.65660	−0.182456
$213$	0	0
$214$	3.95524	0.270374
$215$	41.1252	2.80471
$216$	0	0
$217$	0	0
$218$	−13.9989	−0.948125
$219$	0	0
$220$	15.5643	1.04935
$221$	0.602628	0.0405371
$222$	0	0
$223$	−17.7722	−1.19012	−0.595059	−	0.803682i	$-0.702871\pi$
$223$	−17.7722	−1.19012	−0.595059	+	0.803682i	$0.702871\pi$
$224$	0	0
$225$	0	0
$226$	−5.98926	−0.398400
$227$	−22.6030	−1.50021	−0.750107	−	0.661317i	$-0.769998\pi$
$227$	−22.6030	−1.50021	−0.750107	+	0.661317i	$0.769998\pi$
$228$	0	0
$229$	14.4710	0.956271	0.478135	−	0.878286i	$-0.341313\pi$
$229$	14.4710	0.956271	0.478135	+	0.878286i	$0.341313\pi$
$230$	−18.1811	−1.19883
$231$	0	0
$232$	19.9546	1.31009
$233$	−22.8477	−1.49680	−0.748402	−	0.663245i	$-0.769179\pi$
$233$	−22.8477	−1.49680	−0.748402	+	0.663245i	$0.769179\pi$
$234$	0	0
$235$	29.9473	1.95354
$236$	−3.12684	−0.203540
$237$	0	0
$238$	0	0
$239$	−5.37010	−0.347363	−0.173682	−	0.984802i	$-0.555566\pi$
$239$	−5.37010	−0.347363	−0.173682	+	0.984802i	$0.555566\pi$
$240$	0	0
$241$	5.94287	0.382814	0.191407	−	0.981511i	$-0.438695\pi$
$241$	5.94287	0.382814	0.191407	+	0.981511i	$0.438695\pi$
$242$	31.3220	2.01346
$243$	0	0
$244$	2.61422	0.167359
$245$	0	0
$246$	0	0
$247$	5.29924	0.337183
$248$	−32.9643	−2.09323
$249$	0	0
$250$	6.58779	0.416648
$251$	−8.52086	−0.537832	−0.268916	−	0.963164i	$-0.586665\pi$
$251$	−8.52086	−0.537832	−0.268916	+	0.963164i	$0.586665\pi$
$252$	0	0
$253$	29.3704	1.84650
$254$	−7.23795	−0.454149
$255$	0	0
$256$	−14.9053	−0.931583
$257$	16.6720	1.03997	0.519986	−	0.854175i	$-0.325937\pi$
$257$	16.6720	1.03997	0.519986	+	0.854175i	$0.325937\pi$
$258$	0	0
$259$	0	0
$260$	−2.49881	−0.154969
$261$	0	0
$262$	19.8975	1.22927
$263$	−27.7847	−1.71328	−0.856640	−	0.515915i	$-0.827452\pi$
$263$	−27.7847	−1.71328	−0.856640	+	0.515915i	$0.827452\pi$
$264$	0	0
$265$	12.4479	0.764671
$266$	0	0
$267$	0	0
$268$	−1.00422	−0.0613426
$269$	2.35000	0.143282	0.0716409	−	0.997430i	$-0.477176\pi$
$269$	2.35000	0.143282	0.0716409	+	0.997430i	$0.477176\pi$
$270$	0	0
$271$	−7.04388	−0.427885	−0.213942	−	0.976846i	$-0.568631\pi$
$271$	−7.04388	−0.427885	−0.213942	+	0.976846i	$0.568631\pi$
$272$	−1.20898	−0.0733053
$273$	0	0
$274$	−6.96256	−0.420624
$275$	−41.7856	−2.51977
$276$	0	0
$277$	−16.3083	−0.979871	−0.489935	−	0.871759i	$-0.662980\pi$
$277$	−16.3083	−0.979871	−0.489935	+	0.871759i	$0.662980\pi$
$278$	−8.82051	−0.529019
$279$	0	0
$280$	0	0
$281$	22.5518	1.34533	0.672665	−	0.739947i	$-0.265150\pi$
$281$	22.5518	1.34533	0.672665	+	0.739947i	$0.265150\pi$
$282$	0	0
$283$	24.8569	1.47759	0.738794	−	0.673931i	$-0.235396\pi$
$283$	24.8569	1.47759	0.738794	+	0.673931i	$0.235396\pi$
$284$	−0.540598	−0.0320786
$285$	0	0
$286$	−7.01866	−0.415022
$287$	0	0
$288$	0	0
$289$	−16.6368	−0.978638
$290$	−25.0087	−1.46856
$291$	0	0
$292$	−3.05588	−0.178832
$293$	24.2328	1.41569	0.707847	−	0.706365i	$-0.249666\pi$
$293$	24.2328	1.41569	0.707847	+	0.706365i	$0.249666\pi$
$294$	0	0
$295$	14.6513	0.853033
$296$	−0.829848	−0.0482339
$297$	0	0
$298$	−9.37981	−0.543358
$299$	−4.71533	−0.272694
$300$	0	0
$301$	0	0
$302$	−22.9520	−1.32074
$303$	0	0
$304$	−10.6312	−0.609744
$305$	−12.2494	−0.701397
$306$	0	0
$307$	−14.9020	−0.850503	−0.425252	−	0.905075i	$-0.639814\pi$
$307$	−14.9020	−0.850503	−0.425252	+	0.905075i	$0.639814\pi$
$308$	0	0
$309$	0	0
$310$	41.3133	2.34644
$311$	17.6979	1.00355	0.501777	−	0.864997i	$-0.332680\pi$
$311$	17.6979	1.00355	0.501777	+	0.864997i	$0.332680\pi$
$312$	0	0
$313$	18.6671	1.05512	0.527562	−	0.849516i	$-0.323106\pi$
$313$	18.6671	1.05512	0.527562	+	0.849516i	$0.323106\pi$
$314$	5.15724	0.291040
$315$	0	0
$316$	−6.16020	−0.346539
$317$	0.00279276	0.000156857 0	7.84284e−5	−	1.00000i	$-0.499975\pi$
$317$	0.00279276	0.000156857 0	7.84284e−5		1.00000i	$0.499975\pi$
$318$	0	0
$319$	40.3999	2.26196
$320$	28.7377	1.60649
$321$	0	0
$322$	0	0
$323$	3.19347	0.177690
$324$	0	0
$325$	6.70856	0.372124
$326$	10.2706	0.568836
$327$	0	0
$328$	0.820866	0.0453248
$329$	0	0
$330$	0	0
$331$	33.5953	1.84657	0.923283	−	0.384120i	$-0.125495\pi$
$331$	33.5953	1.84657	0.923283	+	0.384120i	$0.125495\pi$
$332$	−0.0853977	−0.00468681
$333$	0	0
$334$	−15.7522	−0.861921
$335$	4.70544	0.257086
$336$	0	0
$337$	4.38767	0.239012	0.119506	−	0.992833i	$-0.461869\pi$
$337$	4.38767	0.239012	0.119506	+	0.992833i	$0.461869\pi$
$338$	1.12683	0.0612912
$339$	0	0
$340$	−1.50585	−0.0816662
$341$	−66.7390	−3.61412
$342$	0	0
$343$	0	0
$344$	−36.9758	−1.99360
$345$	0	0
$346$	−14.9508	−0.803759
$347$	0.0713316	0.00382928	0.00191464	−	0.999998i	$-0.499391\pi$
$347$	0.0713316	0.00382928	0.00191464	+	0.999998i	$0.499391\pi$
$348$	0	0
$349$	3.77491	0.202066	0.101033	−	0.994883i	$-0.467785\pi$
$349$	3.77491	0.202066	0.101033	+	0.994883i	$0.467785\pi$
$350$	0	0
$351$	0	0
$352$	−24.2449	−1.29226
$353$	6.65376	0.354144	0.177072	−	0.984198i	$-0.443337\pi$
$353$	6.65376	0.354144	0.177072	+	0.984198i	$0.443337\pi$
$354$	0	0
$355$	2.53306	0.134441
$356$	−7.21209	−0.382240
$357$	0	0
$358$	22.1971	1.17315
$359$	−18.8163	−0.993088	−0.496544	−	0.868011i	$-0.665398\pi$
$359$	−18.8163	−0.993088	−0.496544	+	0.868011i	$0.665398\pi$
$360$	0	0
$361$	9.08197	0.477998
$362$	−5.75450	−0.302450
$363$	0	0
$364$	0	0
$365$	14.3188	0.749481
$366$	0	0
$367$	22.9938	1.20027	0.600133	−	0.799900i	$-0.295114\pi$
$367$	22.9938	1.20027	0.600133	+	0.799900i	$0.295114\pi$
$368$	9.45981	0.493127
$369$	0	0
$370$	1.04003	0.0540685
$371$	0	0
$372$	0	0
$373$	−18.0386	−0.934002	−0.467001	−	0.884257i	$-0.654665\pi$
$373$	−18.0386	−0.934002	−0.467001	+	0.884257i	$0.654665\pi$
$374$	−4.22964	−0.218709
$375$	0	0
$376$	−26.9257	−1.38859
$377$	−6.48608	−0.334050
$378$	0	0
$379$	−9.99684	−0.513503	−0.256752	−	0.966477i	$-0.582652\pi$
$379$	−9.99684	−0.513503	−0.256752	+	0.966477i	$0.582652\pi$
$380$	−13.2418	−0.679289
$381$	0	0
$382$	20.1414	1.03053
$383$	−10.1526	−0.518773	−0.259386	−	0.965774i	$-0.583520\pi$
$383$	−10.1526	−0.518773	−0.259386	+	0.965774i	$0.583520\pi$
$384$	0	0
$385$	0	0
$386$	−0.693286	−0.0352873
$387$	0	0
$388$	−7.92083	−0.402119
$389$	−31.3169	−1.58783	−0.793914	−	0.608030i	$-0.791960\pi$
$389$	−31.3169	−1.58783	−0.793914	+	0.608030i	$0.791960\pi$
$390$	0	0
$391$	−2.84159	−0.143705
$392$	0	0
$393$	0	0
$394$	2.26637	0.114178
$395$	28.8647	1.45234
$396$	0	0
$397$	22.9229	1.15047	0.575235	−	0.817988i	$-0.304911\pi$
$397$	22.9229	1.15047	0.575235	+	0.817988i	$0.304911\pi$
$398$	−20.3297	−1.01904
$399$	0	0
$400$	−13.4586	−0.672930
$401$	−18.3226	−0.914988	−0.457494	−	0.889213i	$-0.651253\pi$
$401$	−18.3226	−0.914988	−0.457494	+	0.889213i	$0.651253\pi$
$402$	0	0
$403$	10.7147	0.533740
$404$	−3.26441	−0.162411
$405$	0	0
$406$	0	0
$407$	−1.68010	−0.0832794
$408$	0	0
$409$	4.92084	0.243320	0.121660	−	0.992572i	$-0.461178\pi$
$409$	4.92084	0.243320	0.121660	+	0.992572i	$0.461178\pi$
$410$	−1.02877	−0.0508075
$411$	0	0
$412$	−8.56398	−0.421917
$413$	0	0
$414$	0	0
$415$	0.400145	0.0196423
$416$	3.89245	0.190843
$417$	0	0
$418$	−37.1936	−1.81920
$419$	19.6349	0.959230	0.479615	−	0.877479i	$-0.340776\pi$
$419$	19.6349	0.959230	0.479615	+	0.877479i	$0.340776\pi$
$420$	0	0
$421$	−4.93028	−0.240287	−0.120144	−	0.992757i	$-0.538335\pi$
$421$	−4.93028	−0.240287	−0.120144	+	0.992757i	$0.538335\pi$
$422$	21.3629	1.03993
$423$	0	0
$424$	−11.1920	−0.543531
$425$	4.04277	0.196103
$426$	0	0
$427$	0	0
$428$	−2.56328	−0.123901
$429$	0	0
$430$	46.3409	2.23476
$431$	33.0547	1.59219	0.796094	−	0.605172i	$-0.206896\pi$
$431$	33.0547	1.59219	0.796094	+	0.605172i	$0.206896\pi$
$432$	0	0
$433$	6.62761	0.318503	0.159251	−	0.987238i	$-0.449092\pi$
$433$	6.62761	0.318503	0.159251	+	0.987238i	$0.449092\pi$
$434$	0	0
$435$	0	0
$436$	9.07231	0.434485
$437$	−24.9877	−1.19532
$438$	0	0
$439$	−21.5644	−1.02921	−0.514607	−	0.857426i	$-0.672062\pi$
$439$	−21.5644	−1.02921	−0.514607	+	0.857426i	$0.672062\pi$
$440$	65.5708	3.12597
$441$	0	0
$442$	0.679056	0.0322994
$443$	13.1473	0.624645	0.312323	−	0.949976i	$-0.398893\pi$
$443$	13.1473	0.624645	0.312323	+	0.949976i	$0.398893\pi$
$444$	0	0
$445$	33.7935	1.60196
$446$	−20.0262	−0.948269
$447$	0	0
$448$	0	0
$449$	−18.5922	−0.877422	−0.438711	−	0.898628i	$-0.644565\pi$
$449$	−18.5922	−0.877422	−0.438711	+	0.898628i	$0.644565\pi$
$450$	0	0
$451$	1.66191	0.0782565
$452$	3.88148	0.182569
$453$	0	0
$454$	−25.4696	−1.19535
$455$	0	0
$456$	0	0
$457$	−5.81527	−0.272027	−0.136013	−	0.990707i	$-0.543429\pi$
$457$	−5.81527	−0.272027	−0.136013	+	0.990707i	$0.543429\pi$
$458$	16.3063	0.761943
$459$	0	0
$460$	11.7827	0.549370
$461$	−3.87560	−0.180505	−0.0902523	−	0.995919i	$-0.528767\pi$
$461$	−3.87560	−0.180505	−0.0902523	+	0.995919i	$0.528767\pi$
$462$	0	0
$463$	−12.3411	−0.573540	−0.286770	−	0.957999i	$-0.592582\pi$
$463$	−12.3411	−0.573540	−0.286770	+	0.957999i	$0.592582\pi$
$464$	13.0123	0.604079
$465$	0	0
$466$	−25.7454	−1.19263
$467$	30.0566	1.39085	0.695427	−	0.718597i	$-0.255215\pi$
$467$	30.0566	1.39085	0.695427	+	0.718597i	$0.255215\pi$
$468$	0	0
$469$	0	0
$470$	33.7453	1.55656
$471$	0	0
$472$	−13.1731	−0.606339
$473$	−74.8606	−3.44209
$474$	0	0
$475$	35.5503	1.63116
$476$	0	0
$477$	0	0
$478$	−6.05117	−0.276774
$479$	11.7233	0.535653	0.267827	−	0.963467i	$-0.413695\pi$
$479$	11.7233	0.535653	0.267827	+	0.963467i	$0.413695\pi$
$480$	0	0
$481$	0.269735	0.0122989
$482$	6.69658	0.305021
$483$	0	0
$484$	−20.2990	−0.922681
$485$	37.1144	1.68528
$486$	0	0
$487$	21.4819	0.973436	0.486718	−	0.873559i	$-0.338194\pi$
$487$	21.4819	0.973436	0.486718	+	0.873559i	$0.338194\pi$
$488$	11.0135	0.498556
$489$	0	0
$490$	0	0
$491$	−4.33221	−0.195510	−0.0977549	−	0.995211i	$-0.531166\pi$
$491$	−4.33221	−0.195510	−0.0977549	+	0.995211i	$0.531166\pi$
$492$	0	0
$493$	−3.90870	−0.176039
$494$	5.97132	0.268662
$495$	0	0
$496$	−21.4957	−0.965187
$497$	0	0
$498$	0	0
$499$	37.8237	1.69322	0.846610	−	0.532214i	$-0.178640\pi$
$499$	37.8237	1.69322	0.846610	+	0.532214i	$0.178640\pi$
$500$	−4.26937	−0.190932
$501$	0	0
$502$	−9.60152	−0.428537
$503$	37.2173	1.65944	0.829718	−	0.558182i	$-0.188501\pi$
$503$	37.2173	1.65944	0.829718	+	0.558182i	$0.188501\pi$
$504$	0	0
$505$	15.2959	0.680660
$506$	33.0953	1.47126
$507$	0	0
$508$	4.69072	0.208117
$509$	−8.45232	−0.374643	−0.187321	−	0.982299i	$-0.559981\pi$
$509$	−8.45232	−0.374643	−0.187321	+	0.982299i	$0.559981\pi$
$510$	0	0
$511$	0	0
$512$	−20.1531	−0.890651
$513$	0	0
$514$	18.7865	0.828635
$515$	40.1279	1.76825
$516$	0	0
$517$	−54.5133	−2.39749
$518$	0	0
$519$	0	0
$520$	−10.5272	−0.461649
$521$	−27.5641	−1.20761	−0.603803	−	0.797134i	$-0.706349\pi$
$521$	−27.5641	−1.20761	−0.603803	+	0.797134i	$0.706349\pi$
$522$	0	0
$523$	−17.5097	−0.765645	−0.382823	−	0.923822i	$-0.625048\pi$
$523$	−17.5097	−0.765645	−0.382823	+	0.923822i	$0.625048\pi$
$524$	−12.8950	−0.563321
$525$	0	0
$526$	−31.3085	−1.36512
$527$	6.45701	0.281272
$528$	0	0
$529$	−0.765693	−0.0332910
$530$	14.0266	0.609279
$531$	0	0
$532$	0	0
$533$	−0.266816	−0.0115571
$534$	0	0
$535$	12.0107	0.519267
$536$	−4.23068	−0.182738
$537$	0	0
$538$	2.64803	0.114165
$539$	0	0
$540$	0	0
$541$	−31.5505	−1.35646	−0.678230	−	0.734849i	$-0.737253\pi$
$541$	−31.5505	−1.35646	−0.678230	+	0.734849i	$0.737253\pi$
$542$	−7.93722	−0.340933
$543$	0	0
$544$	2.34570	0.100571
$545$	−42.5098	−1.82092
$546$	0	0
$547$	10.3525	0.442639	0.221319	−	0.975201i	$-0.428964\pi$
$547$	10.3525	0.442639	0.221319	+	0.975201i	$0.428964\pi$
$548$	4.51225	0.192754
$549$	0	0
$550$	−47.0851	−2.00772
$551$	−34.3713	−1.46427
$552$	0	0
$553$	0	0
$554$	−18.3766	−0.780747
$555$	0	0
$556$	5.71633	0.242427
$557$	−27.8350	−1.17941	−0.589704	−	0.807619i	$-0.700756\pi$
$557$	−27.8350	−1.17941	−0.589704	+	0.807619i	$0.700756\pi$
$558$	0	0
$559$	12.0187	0.508335
$560$	0	0
$561$	0	0
$562$	25.4120	1.07194
$563$	27.7834	1.17093	0.585466	−	0.810697i	$-0.300912\pi$
$563$	27.7834	1.17093	0.585466	+	0.810697i	$0.300912\pi$
$564$	0	0
$565$	−18.1873	−0.765146
$566$	28.0094	1.17732
$567$	0	0
$568$	−2.27748	−0.0955611
$569$	38.4902	1.61359	0.806797	−	0.590828i	$-0.201199\pi$
$569$	38.4902	1.61359	0.806797	+	0.590828i	$0.201199\pi$
$570$	0	0
$571$	−33.3413	−1.39529	−0.697646	−	0.716443i	$-0.745769\pi$
$571$	−33.3413	−1.39529	−0.697646	+	0.716443i	$0.745769\pi$
$572$	4.54860	0.190187
$573$	0	0
$574$	0	0
$575$	−31.6331	−1.31919
$576$	0	0
$577$	−2.35508	−0.0980431	−0.0490216	−	0.998798i	$-0.515610\pi$
$577$	−2.35508	−0.0980431	−0.0490216	+	0.998798i	$0.515610\pi$
$578$	−18.7468	−0.779764
$579$	0	0
$580$	16.2075	0.672978
$581$	0	0
$582$	0	0
$583$	−22.6591	−0.938445
$584$	−12.8741	−0.532734
$585$	0	0
$586$	27.3061	1.12801
$587$	20.8410	0.860202	0.430101	−	0.902781i	$-0.358478\pi$
$587$	20.8410	0.860202	0.430101	+	0.902781i	$0.358478\pi$
$588$	0	0
$589$	56.7800	2.33958
$590$	16.5095	0.679685
$591$	0	0
$592$	−0.541137	−0.0222406
$593$	8.42497	0.345972	0.172986	−	0.984924i	$-0.444658\pi$
$593$	8.42497	0.345972	0.172986	+	0.984924i	$0.444658\pi$
$594$	0	0
$595$	0	0
$596$	6.07881	0.248998
$597$	0	0
$598$	−5.31335	−0.217279
$599$	22.6744	0.926449	0.463225	−	0.886241i	$-0.346692\pi$
$599$	22.6744	0.926449	0.463225	+	0.886241i	$0.346692\pi$
$600$	0	0
$601$	12.9905	0.529895	0.264948	−	0.964263i	$-0.414645\pi$
$601$	12.9905	0.529895	0.264948	+	0.964263i	$0.414645\pi$
$602$	0	0
$603$	0	0
$604$	14.8746	0.605238
$605$	95.1142	3.86694
$606$	0	0
$607$	32.3286	1.31218	0.656089	−	0.754683i	$-0.272209\pi$
$607$	32.3286	1.31218	0.656089	+	0.754683i	$0.272209\pi$
$608$	20.6270	0.836536
$609$	0	0
$610$	−13.8029	−0.558863
$611$	8.75196	0.354066
$612$	0	0
$613$	31.0323	1.25338	0.626691	−	0.779268i	$-0.284409\pi$
$613$	31.0323	1.25338	0.626691	+	0.779268i	$0.284409\pi$
$614$	−16.7920	−0.677669
$615$	0	0
$616$	0	0
$617$	−27.2587	−1.09739	−0.548697	−	0.836022i	$-0.684876\pi$
$617$	−27.2587	−1.09739	−0.548697	+	0.836022i	$0.684876\pi$
$618$	0	0
$619$	1.44083	0.0579119	0.0289560	−	0.999581i	$-0.490782\pi$
$619$	1.44083	0.0579119	0.0289560	+	0.999581i	$0.490782\pi$
$620$	−26.7741	−1.07527
$621$	0	0
$622$	19.9424	0.799618
$623$	0	0
$624$	0	0
$625$	−13.5380	−0.541520
$626$	21.0345	0.840708
$627$	0	0
$628$	−3.34227	−0.133371
$629$	0.162550	0.00648129
$630$	0	0
$631$	−19.3826	−0.771610	−0.385805	−	0.922580i	$-0.626076\pi$
$631$	−19.3826	−0.771610	−0.385805	+	0.922580i	$0.626076\pi$
$632$	−25.9523	−1.03233
$633$	0	0
$634$	0.00314695	0.000124981 0
$635$	−21.9791	−0.872216
$636$	0	0
$637$	0	0
$638$	45.5236	1.80230
$639$	0	0
$640$	5.74420	0.227059
$641$	−0.962996	−0.0380361	−0.0190180	−	0.999819i	$-0.506054\pi$
$641$	−0.962996	−0.0380361	−0.0190180	+	0.999819i	$0.506054\pi$
$642$	0	0
$643$	5.51311	0.217416	0.108708	−	0.994074i	$-0.465329\pi$
$643$	5.51311	0.217416	0.108708	+	0.994074i	$0.465329\pi$
$644$	0	0
$645$	0	0
$646$	3.59848	0.141580
$647$	5.74549	0.225879	0.112939	−	0.993602i	$-0.463973\pi$
$647$	5.74549	0.225879	0.112939	+	0.993602i	$0.463973\pi$
$648$	0	0
$649$	−26.6700	−1.04689
$650$	7.55938	0.296503
$651$	0	0
$652$	−6.65610	−0.260673
$653$	−6.35614	−0.248735	−0.124368	−	0.992236i	$-0.539690\pi$
$653$	−6.35614	−0.248735	−0.124368	+	0.992236i	$0.539690\pi$
$654$	0	0
$655$	60.4217	2.36087
$656$	0.535281	0.0208992
$657$	0	0
$658$	0	0
$659$	−18.7092	−0.728806	−0.364403	−	0.931241i	$-0.618727\pi$
$659$	−18.7092	−0.728806	−0.364403	+	0.931241i	$0.618727\pi$
$660$	0	0
$661$	−36.8249	−1.43232	−0.716162	−	0.697934i	$-0.754103\pi$
$661$	−36.8249	−1.43232	−0.716162	+	0.697934i	$0.754103\pi$
$662$	37.8561	1.47132
$663$	0	0
$664$	−0.359772	−0.0139618
$665$	0	0
$666$	0	0
$667$	30.5840	1.18422
$668$	10.2086	0.394981
$669$	0	0
$670$	5.30221	0.204842
$671$	22.2977	0.860792
$672$	0	0
$673$	40.4024	1.55740	0.778699	−	0.627397i	$-0.215880\pi$
$673$	40.4024	1.55740	0.778699	+	0.627397i	$0.215880\pi$
$674$	4.94414	0.190441
$675$	0	0
$676$	−0.730265	−0.0280871
$677$	−2.01655	−0.0775022	−0.0387511	−	0.999249i	$-0.512338\pi$
$677$	−2.01655	−0.0775022	−0.0387511	+	0.999249i	$0.512338\pi$
$678$	0	0
$679$	0	0
$680$	−6.34399	−0.243281
$681$	0	0
$682$	−75.2031	−2.87968
$683$	−13.9015	−0.531925	−0.265962	−	0.963983i	$-0.585690\pi$
$683$	−13.9015	−0.531925	−0.265962	+	0.963983i	$0.585690\pi$
$684$	0	0
$685$	−21.1429	−0.807828
$686$	0	0
$687$	0	0
$688$	−24.1116	−0.919247
$689$	3.63786	0.138591
$690$	0	0
$691$	23.8046	0.905571	0.452785	−	0.891620i	$-0.350430\pi$
$691$	23.8046	0.905571	0.452785	+	0.891620i	$0.350430\pi$
$692$	9.68920	0.368328
$693$	0	0
$694$	0.0803782	0.00305111
$695$	−26.7848	−1.01601
$696$	0	0
$697$	−0.160791	−0.00609038
$698$	4.25367	0.161004
$699$	0	0
$700$	0	0
$701$	15.6607	0.591498	0.295749	−	0.955266i	$-0.404431\pi$
$701$	15.6607	0.591498	0.295749	+	0.955266i	$0.404431\pi$
$702$	0	0
$703$	1.42939	0.0539105
$704$	−52.3116	−1.97157
$705$	0	0
$706$	7.49762	0.282177
$707$	0	0
$708$	0	0
$709$	−23.6825	−0.889413	−0.444707	−	0.895676i	$-0.646692\pi$
$709$	−23.6825	−0.889413	−0.444707	+	0.895676i	$0.646692\pi$
$710$	2.85432	0.107121
$711$	0	0
$712$	−30.3838	−1.13868
$713$	−50.5235	−1.89212
$714$	0	0
$715$	−21.3132	−0.797070
$716$	−14.3853	−0.537605
$717$	0	0
$718$	−21.2027	−0.791278
$719$	38.1586	1.42308	0.711538	−	0.702647i	$-0.247999\pi$
$719$	38.1586	1.42308	0.711538	+	0.702647i	$0.247999\pi$
$720$	0	0
$721$	0	0
$722$	10.2338	0.380862
$723$	0	0
$724$	3.72934	0.138600
$725$	−43.5123	−1.61601
$726$	0	0
$727$	−41.6980	−1.54649	−0.773246	−	0.634106i	$-0.781368\pi$
$727$	−41.6980	−1.54649	−0.773246	+	0.634106i	$0.781368\pi$
$728$	0	0
$729$	0	0
$730$	16.1348	0.597176
$731$	7.24278	0.267884
$732$	0	0
$733$	−20.2046	−0.746275	−0.373137	−	0.927776i	$-0.621718\pi$
$733$	−20.2046	−0.746275	−0.373137	+	0.927776i	$0.621718\pi$
$734$	25.9100	0.956355
$735$	0	0
$736$	−18.3542	−0.676543
$737$	−8.56537	−0.315510
$738$	0	0
$739$	−14.0261	−0.515958	−0.257979	−	0.966151i	$-0.583057\pi$
$739$	−14.0261	−0.515958	−0.257979	+	0.966151i	$0.583057\pi$
$740$	−0.674015	−0.0247773
$741$	0	0
$742$	0	0
$743$	−37.7853	−1.38621	−0.693103	−	0.720838i	$-0.743757\pi$
$743$	−37.7853	−1.38621	−0.693103	+	0.720838i	$0.743757\pi$
$744$	0	0
$745$	−28.4832	−1.04355
$746$	−20.3263	−0.744199
$747$	0	0
$748$	2.74112	0.100225
$749$	0	0
$750$	0	0
$751$	−7.14110	−0.260582	−0.130291	−	0.991476i	$-0.541591\pi$
$751$	−7.14110	−0.260582	−0.130291	+	0.991476i	$0.541591\pi$
$752$	−17.5580	−0.640275
$753$	0	0
$754$	−7.30868	−0.266166
$755$	−69.6973	−2.53654
$756$	0	0
$757$	−17.9270	−0.651568	−0.325784	−	0.945444i	$-0.605628\pi$
$757$	−17.9270	−0.651568	−0.325784	+	0.945444i	$0.605628\pi$
$758$	−11.2647	−0.409152
$759$	0	0
$760$	−55.7862	−2.02358
$761$	−12.0253	−0.435916	−0.217958	−	0.975958i	$-0.569940\pi$
$761$	−12.0253	−0.435916	−0.217958	+	0.975958i	$0.569940\pi$
$762$	0	0
$763$	0	0
$764$	−13.0531	−0.472245
$765$	0	0
$766$	−11.4402	−0.413351
$767$	4.28179	0.154606
$768$	0	0
$769$	−0.871602	−0.0314308	−0.0157154	−	0.999877i	$-0.505003\pi$
$769$	−0.871602	−0.0314308	−0.0157154	+	0.999877i	$0.505003\pi$
$770$	0	0
$771$	0	0
$772$	0.449300	0.0161707
$773$	14.2385	0.512122	0.256061	−	0.966661i	$-0.417575\pi$
$773$	14.2385	0.512122	0.256061	+	0.966661i	$0.417575\pi$
$774$	0	0
$775$	71.8806	2.58203
$776$	−33.3697	−1.19790
$777$	0	0
$778$	−35.2886	−1.26516
$779$	−1.41392	−0.0506590
$780$	0	0
$781$	−4.61096	−0.164993
$782$	−3.20197	−0.114502
$783$	0	0
$784$	0	0
$785$	15.6608	0.558957
$786$	0	0
$787$	−35.7765	−1.27530	−0.637648	−	0.770328i	$-0.720092\pi$
$787$	−35.7765	−1.27530	−0.637648	+	0.770328i	$0.720092\pi$
$788$	−1.46877	−0.0523229
$789$	0	0
$790$	32.5254	1.15720
$791$	0	0
$792$	0	0
$793$	−3.57983	−0.127123
$794$	25.8302	0.916678
$795$	0	0
$796$	13.1752	0.466981
$797$	18.8290	0.666958	0.333479	−	0.942758i	$-0.391777\pi$
$797$	18.8290	0.666958	0.333479	+	0.942758i	$0.391777\pi$
$798$	0	0
$799$	5.27418	0.186587
$800$	26.1127	0.923224
$801$	0	0
$802$	−20.6464	−0.729050
$803$	−26.0647	−0.919803
$804$	0	0
$805$	0	0
$806$	12.0736	0.425276
$807$	0	0
$808$	−13.7526	−0.483816
$809$	−46.0538	−1.61916	−0.809582	−	0.587006i	$-0.800306\pi$
$809$	−46.0538	−1.61916	−0.809582	+	0.587006i	$0.800306\pi$
$810$	0	0
$811$	45.8404	1.60968	0.804838	−	0.593495i	$-0.202252\pi$
$811$	45.8404	1.60968	0.804838	+	0.593495i	$0.202252\pi$
$812$	0	0
$813$	0	0
$814$	−1.89318	−0.0663558
$815$	31.1882	1.09248
$816$	0	0
$817$	63.6898	2.22822
$818$	5.54493	0.193874
$819$	0	0
$820$	0.666720	0.0232829
$821$	26.0675	0.909761	0.454880	−	0.890553i	$-0.349682\pi$
$821$	26.0675	0.909761	0.454880	+	0.890553i	$0.349682\pi$
$822$	0	0
$823$	−25.0080	−0.871722	−0.435861	−	0.900014i	$-0.643556\pi$
$823$	−25.0080	−0.871722	−0.435861	+	0.900014i	$0.643556\pi$
$824$	−36.0792	−1.25688
$825$	0	0
$826$	0	0
$827$	32.1098	1.11657	0.558284	−	0.829650i	$-0.311460\pi$
$827$	32.1098	1.11657	0.558284	+	0.829650i	$0.311460\pi$
$828$	0	0
$829$	11.8754	0.412451	0.206225	−	0.978505i	$-0.433882\pi$
$829$	11.8754	0.412451	0.206225	+	0.978505i	$0.433882\pi$
$830$	0.450893	0.0156507
$831$	0	0
$832$	8.39847	0.291165
$833$	0	0
$834$	0	0
$835$	−47.8339	−1.65536
$836$	24.1041	0.833659
$837$	0	0
$838$	22.1251	0.764300
$839$	11.7957	0.407232	0.203616	−	0.979051i	$-0.434731\pi$
$839$	11.7957	0.407232	0.203616	+	0.979051i	$0.434731\pi$
$840$	0	0
$841$	13.0693	0.450664
$842$	−5.55556	−0.191457
$843$	0	0
$844$	−13.8447	−0.476555
$845$	3.42178	0.117713
$846$	0	0
$847$	0	0
$848$	−7.29821	−0.250621
$849$	0	0
$850$	4.55549	0.156252
$851$	−1.27189	−0.0435998
$852$	0	0
$853$	30.7053	1.05133	0.525665	−	0.850692i	$-0.323817\pi$
$853$	30.7053	1.05133	0.525665	+	0.850692i	$0.323817\pi$
$854$	0	0
$855$	0	0
$856$	−10.7988	−0.369097
$857$	−36.7982	−1.25700	−0.628502	−	0.777808i	$-0.716331\pi$
$857$	−36.7982	−1.25700	−0.628502	+	0.777808i	$0.716331\pi$
$858$	0	0
$859$	−5.82807	−0.198851	−0.0994256	−	0.995045i	$-0.531701\pi$
$859$	−5.82807	−0.198851	−0.0994256	+	0.995045i	$0.531701\pi$
$860$	−30.0323	−1.02409
$861$	0	0
$862$	37.2469	1.26863
$863$	−41.0881	−1.39865	−0.699327	−	0.714802i	$-0.746517\pi$
$863$	−41.0881	−1.39865	−0.699327	+	0.714802i	$0.746517\pi$
$864$	0	0
$865$	−45.4003	−1.54366
$866$	7.46816	0.253778
$867$	0	0
$868$	0	0
$869$	−52.5427	−1.78239
$870$	0	0
$871$	1.37515	0.0465950
$872$	38.2207	1.29432
$873$	0	0
$874$	−28.1567	−0.952415
$875$	0	0
$876$	0	0
$877$	−11.5614	−0.390400	−0.195200	−	0.980763i	$-0.562536\pi$
$877$	−11.5614	−0.390400	−0.195200	+	0.980763i	$0.562536\pi$
$878$	−24.2993	−0.820062
$879$	0	0
$880$	42.7582	1.44138
$881$	18.8516	0.635125	0.317563	−	0.948237i	$-0.397136\pi$
$881$	18.8516	0.635125	0.317563	+	0.948237i	$0.397136\pi$
$882$	0	0
$883$	−0.0238069	−0.000801165 0	−0.000400582	−	1.00000i	$-0.500128\pi$
$883$	−0.0238069	−0.000801165 0	−0.000400582		1.00000i	$0.500128\pi$
$884$	−0.440078	−0.0148014
$885$	0	0
$886$	14.8147	0.497708
$887$	−15.1878	−0.509957	−0.254979	−	0.966947i	$-0.582068\pi$
$887$	−15.1878	−0.509957	−0.254979	+	0.966947i	$0.582068\pi$
$888$	0	0
$889$	0	0
$890$	38.0793	1.27642
$891$	0	0
$892$	12.9785	0.434551
$893$	46.3788	1.55201
$894$	0	0
$895$	67.4048	2.25310
$896$	0	0
$897$	0	0
$898$	−20.9502	−0.699117
$899$	−69.4967	−2.31785
$900$	0	0
$901$	2.19227	0.0730353
$902$	1.87269	0.0623536
$903$	0	0
$904$	16.3523	0.543868
$905$	−17.4744	−0.580869
$906$	0	0
$907$	8.62323	0.286330	0.143165	−	0.989699i	$-0.454272\pi$
$907$	8.62323	0.286330	0.143165	+	0.989699i	$0.454272\pi$
$908$	16.5062	0.547777
$909$	0	0
$910$	0	0
$911$	10.5379	0.349135	0.174568	−	0.984645i	$-0.444147\pi$
$911$	10.5379	0.349135	0.174568	+	0.984645i	$0.444147\pi$
$912$	0	0
$913$	−0.728389	−0.0241061
$914$	−6.55279	−0.216747
$915$	0	0
$916$	−10.5677	−0.349166
$917$	0	0
$918$	0	0
$919$	2.83822	0.0936244	0.0468122	−	0.998904i	$-0.485094\pi$
$919$	2.83822	0.0936244	0.0468122	+	0.998904i	$0.485094\pi$
$920$	49.6392	1.63656
$921$	0	0
$922$	−4.36712	−0.143823
$923$	0.740276	0.0243665
$924$	0	0
$925$	1.80953	0.0594971
$926$	−13.9063	−0.456989
$927$	0	0
$928$	−25.2467	−0.828764
$929$	−57.2054	−1.87685	−0.938425	−	0.345484i	$-0.887715\pi$
$929$	−57.2054	−1.87685	−0.938425	+	0.345484i	$0.887715\pi$
$930$	0	0
$931$	0	0
$932$	16.6849	0.546532
$933$	0	0
$934$	33.8685	1.10821
$935$	−12.8440	−0.420042
$936$	0	0
$937$	−3.71475	−0.121355	−0.0606777	−	0.998157i	$-0.519326\pi$
$937$	−3.71475	−0.121355	−0.0606777	+	0.998157i	$0.519326\pi$
$938$	0	0
$939$	0	0
$940$	−21.8694	−0.713302
$941$	−41.7767	−1.36188	−0.680941	−	0.732338i	$-0.738429\pi$
$941$	−41.7767	−1.36188	−0.680941	+	0.732338i	$0.738429\pi$
$942$	0	0
$943$	1.25812	0.0409701
$944$	−8.59005	−0.279582
$945$	0	0
$946$	−84.3548	−2.74261
$947$	−11.8074	−0.383688	−0.191844	−	0.981425i	$-0.561447\pi$
$947$	−11.8074	−0.383688	−0.191844	+	0.981425i	$0.561447\pi$
$948$	0	0
$949$	4.18461	0.135838
$950$	40.0590	1.29968
$951$	0	0
$952$	0	0
$953$	−17.1436	−0.555334	−0.277667	−	0.960677i	$-0.589561\pi$
$953$	−17.1436	−0.555334	−0.277667	+	0.960677i	$0.589561\pi$
$954$	0	0
$955$	61.1626	1.97917
$956$	3.92160	0.126834
$957$	0	0
$958$	13.2102	0.426801
$959$	0	0
$960$	0	0
$961$	83.8058	2.70341
$962$	0.303944	0.00979955
$963$	0	0
$964$	−4.33987	−0.139778
$965$	−2.10527	−0.0677710
$966$	0	0
$967$	14.7591	0.474622	0.237311	−	0.971434i	$-0.423734\pi$
$967$	14.7591	0.474622	0.237311	+	0.971434i	$0.423734\pi$
$968$	−85.5175	−2.74864
$969$	0	0
$970$	41.8214	1.34280
$971$	30.6869	0.984790	0.492395	−	0.870372i	$-0.336122\pi$
$971$	30.6869	0.984790	0.492395	+	0.870372i	$0.336122\pi$
$972$	0	0
$973$	0	0
$974$	24.2063	0.775620
$975$	0	0
$976$	7.18179	0.229883
$977$	36.1440	1.15635	0.578175	−	0.815913i	$-0.303765\pi$
$977$	36.1440	1.15635	0.578175	+	0.815913i	$0.303765\pi$
$978$	0	0
$979$	−61.5146	−1.96602
$980$	0	0
$981$	0	0
$982$	−4.88164	−0.155779
$983$	−46.1138	−1.47080	−0.735401	−	0.677633i	$-0.763006\pi$
$983$	−46.1138	−1.47080	−0.735401	+	0.677633i	$0.763006\pi$
$984$	0	0
$985$	6.88218	0.219285
$986$	−4.40442	−0.140265
$987$	0	0
$988$	−3.86985	−0.123116
$989$	−56.6719	−1.80206
$990$	0	0
$991$	−28.2237	−0.896556	−0.448278	−	0.893894i	$-0.647962\pi$
$991$	−28.2237	−0.896556	−0.448278	+	0.893894i	$0.647962\pi$
$992$	41.7066	1.32418
$993$	0	0
$994$	0	0
$995$	−61.7344	−1.95711
$996$	0	0
$997$	−6.36012	−0.201427	−0.100713	−	0.994915i	$-0.532113\pi$
$997$	−6.36012	−0.201427	−0.100713	+	0.994915i	$0.532113\pi$
$998$	42.6207	1.34913
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.by.1.7		10
3.2	odd	2	inner	5733.2.a.by.1.4		10
7.2	even	3		819.2.j.j.235.4	✓	20
7.4	even	3		819.2.j.j.352.4	yes	20
7.6	odd	2		5733.2.a.bz.1.7		10
21.2	odd	6		819.2.j.j.235.7	yes	20
21.11	odd	6		819.2.j.j.352.7	yes	20
21.20	even	2		5733.2.a.bz.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.j.j.235.4	✓	20	7.2	even	3
819.2.j.j.235.7	yes	20	21.2	odd	6
819.2.j.j.352.4	yes	20	7.4	even	3
819.2.j.j.352.7	yes	20	21.11	odd	6
5733.2.a.by.1.4		10	3.2	odd	2	inner
5733.2.a.by.1.7		10	1.1	even	1	trivial
5733.2.a.bz.1.4		10	21.20	even	2
5733.2.a.bz.1.7		10	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q+1.12683 q^{2} -0.730265 q^{4} +3.42178 q^{5} -3.07653 q^{8} +3.85575 q^{10} -6.22870 q^{11} +1.00000 q^{13} -2.00618 q^{16} +0.602628 q^{17} +5.29924 q^{19} -2.49881 q^{20} -7.01866 q^{22} -4.71533 q^{23} +6.70856 q^{25} +1.12683 q^{26} -6.48608 q^{29} +10.7147 q^{31} +3.89245 q^{32} +0.679056 q^{34} +0.269735 q^{37} +5.97132 q^{38} -10.5272 q^{40} -0.266816 q^{41} +12.0187 q^{43} +4.54860 q^{44} -5.31335 q^{46} +8.75196 q^{47} +7.55938 q^{50} -0.730265 q^{52} +3.63786 q^{53} -21.3132 q^{55} -7.30868 q^{58} +4.28179 q^{59} -3.57983 q^{61} +12.0736 q^{62} +8.39847 q^{64} +3.42178 q^{65} +1.37515 q^{67} -0.440078 q^{68} +0.740276 q^{71} +4.18461 q^{73} +0.303944 q^{74} -3.86985 q^{76} +8.43557 q^{79} -6.86471 q^{80} -0.300654 q^{82} +0.116941 q^{83} +2.06206 q^{85} +13.5429 q^{86} +19.1628 q^{88} +9.87599 q^{89} +3.44344 q^{92} +9.86193 q^{94} +18.1328 q^{95} +10.8465 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 16 q^{4} - 4 q^{10} + 10 q^{13} + 32 q^{16} + 6 q^{19} + 10 q^{22} + 24 q^{25} + 12 q^{31} + 34 q^{34} + 26 q^{37} - 70 q^{40} + 40 q^{43} - 6 q^{46} + 16 q^{52} - 24 q^{55} + 36 q^{58} - 22 q^{61}+ \cdots + 18 q^{97}+O(q^{100}) \) 10 * q + 16 * q^4 - 4 * q^10 + 10 * q^13 + 32 * q^16 + 6 * q^19 + 10 * q^22 + 24 * q^25 + 12 * q^31 + 34 * q^34 + 26 * q^37 - 70 * q^40 + 40 * q^43 - 6 * q^46 + 16 * q^52 - 24 * q^55 + 36 * q^58 - 22 * q^61 + 78 * q^64 + 26 * q^67 + 24 * q^73 + 4 * q^76 + 28 * q^79 - 28 * q^82 - 20 * q^85 + 12 * q^88 + 4 * q^94 + 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more