LMFDB - Embedded newform 5929.2.a.bk.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5929,2,Mod(1,5929)] 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("5929.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5929, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5929 = 7^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5929.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-2,4,-4,0,0,0,8,0,0,-14,0,0,14,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(18)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.3433033584$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.58383808.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 15x^{2} - 7 $ x^6 - 8x^4 + 15x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 847)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.34670$ of defining polynomial
Character	$\chi$	$=$	5929.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.34670 q^{2} -2.28514 q^{3} +3.50702 q^{4} -3.50702 q^{5} -5.36255 q^{6} +3.53653 q^{8} +2.22188 q^{9} -8.22993 q^{10} -8.01404 q^{12} +4.05720 q^{13} +8.01404 q^{15} +1.28514 q^{16} +2.49517 q^{17} +5.21409 q^{18} -3.53653 q^{19} -12.2992 q^{20} +2.34841 q^{23} -8.08147 q^{24} +7.29918 q^{25} +9.52106 q^{26} +1.77812 q^{27} +9.01460 q^{29} +18.8066 q^{30} -6.95077 q^{31} -4.05720 q^{32} +5.85543 q^{34} +7.79216 q^{36} -2.44375 q^{37} -8.29918 q^{38} -9.27129 q^{39} -12.4027 q^{40} +6.37097 q^{41} -7.85772 q^{43} -7.79216 q^{45} +5.51102 q^{46} -2.77812 q^{47} -2.93673 q^{48} +17.1290 q^{50} -5.70182 q^{51} +14.2287 q^{52} +3.17265 q^{53} +4.17273 q^{54} +8.08147 q^{57} +21.1546 q^{58} -0.714858 q^{59} +28.1054 q^{60} -1.33829 q^{61} -16.3114 q^{62} -12.0913 q^{64} -14.2287 q^{65} +0.126533 q^{67} +8.75061 q^{68} -5.36645 q^{69} -5.22188 q^{71} +7.85772 q^{72} -4.13979 q^{73} -5.73476 q^{74} -16.6797 q^{75} -12.4027 q^{76} -21.7570 q^{78} -4.02426 q^{79} -4.50702 q^{80} -10.7289 q^{81} +14.9508 q^{82} -1.15688 q^{83} -8.75061 q^{85} -18.4397 q^{86} -20.5997 q^{87} -11.6476 q^{89} -18.2859 q^{90} +8.23591 q^{92} +15.8835 q^{93} -6.51943 q^{94} +12.4027 q^{95} +9.27129 q^{96} +5.61640 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} + 4 q^{4} - 4 q^{5} + 8 q^{9} - 14 q^{12} + 14 q^{15} - 4 q^{16} - 28 q^{20} - 4 q^{23} - 2 q^{25} + 6 q^{26} + 16 q^{27} - 14 q^{31} + 18 q^{36} - 4 q^{37} - 4 q^{38} - 18 q^{45} - 22 q^{47}+ \cdots - 10 q^{97}+O(q^{100}) $ 6 * q - 2 * q^3 + 4 * q^4 - 4 * q^5 + 8 * q^9 - 14 * q^12 + 14 * q^15 - 4 * q^16 - 28 * q^20 - 4 * q^23 - 2 * q^25 + 6 * q^26 + 16 * q^27 - 14 * q^31 + 18 * q^36 - 4 * q^37 - 4 * q^38 - 18 * q^45 - 22 * q^47 - 24 * q^48 - 14 * q^53 + 46 * q^58 - 16 * q^59 + 60 * q^60 + 2 * q^64 - 12 * q^67 - 62 * q^69 - 26 * q^71 - 50 * q^75 - 40 * q^78 - 10 * q^80 - 42 * q^81 + 62 * q^82 - 18 * q^86 - 6 * q^89 + 10 * q^92 - 8 * q^93 - 10 * 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 $

$2$	2.34670	1.65937	0.829685	−	0.558232i	$-0.188520\pi$
$2$	2.34670	1.65937	0.829685	+	0.558232i	$0.188520\pi$
$3$	−2.28514	−1.31933	−0.659664	−	0.751561i	$-0.729301\pi$
$3$	−2.28514	−1.31933	−0.659664	+	0.751561i	$0.729301\pi$
$4$	3.50702	1.75351
$5$	−3.50702	−1.56839	−0.784193	−	0.620517i	$-0.786923\pi$
$5$	−3.50702	−1.56839	−0.784193	+	0.620517i	$0.786923\pi$
$6$	−5.36255	−2.18925
$7$	0	0
$7$	0	0
$8$	3.53653	1.25035
$9$	2.22188	0.740625
$10$	−8.22993	−2.60253
$11$	0	0
$11$	0	0
$12$	−8.01404	−2.31345
$13$	4.05720	1.12527	0.562633	−	0.826707i	$-0.309788\pi$
$13$	4.05720	1.12527	0.562633	+	0.826707i	$0.309788\pi$
$14$	0	0
$15$	8.01404	2.06922
$16$	1.28514	0.321286
$17$	2.49517	0.605168	0.302584	−	0.953123i	$-0.402151\pi$
$17$	2.49517	0.605168	0.302584	+	0.953123i	$0.402151\pi$
$18$	5.21409	1.22897
$19$	−3.53653	−0.811335	−0.405667	−	0.914021i	$-0.632961\pi$
$19$	−3.53653	−0.811335	−0.405667	+	0.914021i	$0.632961\pi$
$20$	−12.2992	−2.75018
$21$	0	0
$22$	0	0
$23$	2.34841	0.489677	0.244839	−	0.969564i	$-0.421265\pi$
$23$	2.34841	0.489677	0.244839	+	0.969564i	$0.421265\pi$
$24$	−8.08147	−1.64962
$25$	7.29918	1.45984
$26$	9.52106	1.86723
$27$	1.77812	0.342200
$28$	0	0
$29$	9.01460	1.67397	0.836985	−	0.547226i	$-0.184316\pi$
$29$	9.01460	1.67397	0.836985	+	0.547226i	$0.184316\pi$
$30$	18.8066	3.43359
$31$	−6.95077	−1.24840	−0.624198	−	0.781266i	$-0.714574\pi$
$31$	−6.95077	−1.24840	−0.624198	+	0.781266i	$0.714574\pi$
$32$	−4.05720	−0.717219
$33$	0	0
$34$	5.85543	1.00420
$35$	0	0
$36$	7.79216	1.29869
$37$	−2.44375	−0.401750	−0.200875	−	0.979617i	$-0.564379\pi$
$37$	−2.44375	−0.401750	−0.200875	+	0.979617i	$0.564379\pi$
$38$	−8.29918	−1.34630
$39$	−9.27129	−1.48459
$40$	−12.4027	−1.96103
$41$	6.37097	0.994978	0.497489	−	0.867470i	$-0.334256\pi$
$41$	6.37097	0.994978	0.497489	+	0.867470i	$0.334256\pi$
$42$	0	0
$43$	−7.85772	−1.19829	−0.599146	−	0.800640i	$-0.704493\pi$
$43$	−7.85772	−1.19829	−0.599146	+	0.800640i	$0.704493\pi$
$44$	0	0
$45$	−7.79216	−1.16159
$46$	5.51102	0.812556
$47$	−2.77812	−0.405231	−0.202616	−	0.979258i	$-0.564944\pi$
$47$	−2.77812	−0.405231	−0.202616	+	0.979258i	$0.564944\pi$
$48$	−2.93673	−0.423881
$49$	0	0
$50$	17.1290	2.42241
$51$	−5.70182	−0.798415
$52$	14.2287	1.97316
$53$	3.17265	0.435797	0.217898	−	0.975971i	$-0.430080\pi$
$53$	3.17265	0.435797	0.217898	+	0.975971i	$0.430080\pi$
$54$	4.17273	0.567837
$55$	0	0
$56$	0	0
$57$	8.08147	1.07042
$58$	21.1546	2.77774
$59$	−0.714858	−0.0930665	−0.0465333	−	0.998917i	$-0.514817\pi$
$59$	−0.714858	−0.0930665	−0.0465333	+	0.998917i	$0.514817\pi$
$60$	28.1054	3.62839
$61$	−1.33829	−0.171350	−0.0856752	−	0.996323i	$-0.527305\pi$
$61$	−1.33829	−0.171350	−0.0856752	+	0.996323i	$0.527305\pi$
$62$	−16.3114	−2.07155
$63$	0	0
$64$	−12.0913	−1.51142
$65$	−14.2287	−1.76485
$66$	0	0
$67$	0.126533	0.0154584	0.00772921	−	0.999970i	$-0.497540\pi$
$67$	0.126533	0.0154584	0.00772921	+	0.999970i	$0.497540\pi$
$68$	8.75061	1.06117
$69$	−5.36645	−0.646045
$70$	0	0
$71$	−5.22188	−0.619723	−0.309861	−	0.950782i	$-0.600283\pi$
$71$	−5.22188	−0.619723	−0.309861	+	0.950782i	$0.600283\pi$
$72$	7.85772	0.926042
$73$	−4.13979	−0.484526	−0.242263	−	0.970211i	$-0.577890\pi$
$73$	−4.13979	−0.484526	−0.242263	+	0.970211i	$0.577890\pi$
$74$	−5.73476	−0.666653
$75$	−16.6797	−1.92600
$76$	−12.4027	−1.42268
$77$	0	0
$78$	−21.7570	−2.46349
$79$	−4.02426	−0.452765	−0.226382	−	0.974038i	$-0.572690\pi$
$79$	−4.02426	−0.452765	−0.226382	+	0.974038i	$0.572690\pi$
$80$	−4.50702	−0.503900
$81$	−10.7289	−1.19210
$82$	14.9508	1.65104
$83$	−1.15688	−0.126984	−0.0634921	−	0.997982i	$-0.520224\pi$
$83$	−1.15688	−0.126984	−0.0634921	+	0.997982i	$0.520224\pi$
$84$	0	0
$85$	−8.75061	−0.949137
$86$	−18.4397	−1.98841
$87$	−20.5997	−2.20851
$88$	0	0
$89$	−11.6476	−1.23464	−0.617321	−	0.786711i	$-0.711782\pi$
$89$	−11.6476	−1.23464	−0.617321	+	0.786711i	$0.711782\pi$
$90$	−18.2859	−1.92750
$91$	0	0
$92$	8.23591	0.858653
$93$	15.8835	1.64704
$94$	−6.51943	−0.672428
$95$	12.4027	1.27249
$96$	9.27129	0.946247
$97$	5.61640	0.570259	0.285129	−	0.958489i	$-0.407963\pi$
$97$	5.61640	0.570259	0.285129	+	0.958489i	$0.407963\pi$
$98$	0	0
$99$	0	0
$100$	25.5984	2.55984
$101$	5.76770	0.573908	0.286954	−	0.957944i	$-0.407357\pi$
$101$	5.76770	0.573908	0.286954	+	0.957944i	$0.407357\pi$
$102$	−13.3805	−1.32487
$103$	3.23591	0.318844	0.159422	−	0.987211i	$-0.449037\pi$
$103$	3.23591	0.318844	0.159422	+	0.987211i	$0.449037\pi$
$104$	14.3484	1.40698
$105$	0	0
$106$	7.44526	0.723148
$107$	−15.7081	−1.51856	−0.759282	−	0.650762i	$-0.774449\pi$
$107$	−15.7081	−1.51856	−0.759282	+	0.650762i	$0.774449\pi$
$108$	6.23591	0.600051
$109$	6.25544	0.599163	0.299581	−	0.954071i	$-0.403153\pi$
$109$	6.25544	0.599163	0.299581	+	0.954071i	$0.403153\pi$
$110$	0	0
$111$	5.58432	0.530040
$112$	0	0
$113$	−17.8202	−1.67639	−0.838193	−	0.545373i	$-0.816388\pi$
$113$	−17.8202	−1.67639	−0.838193	+	0.545373i	$0.816388\pi$
$114$	18.9648	1.77622
$115$	−8.23591	−0.768003
$116$	31.6144	2.93532
$117$	9.01460	0.833401
$118$	−1.67756	−0.154432
$119$	0	0
$120$	28.3419	2.58725
$121$	0	0
$122$	−3.14057	−0.284334
$123$	−14.5586	−1.31270
$124$	−24.3765	−2.18907
$125$	−8.06327	−0.721200
$126$	0	0
$127$	−15.9721	−1.41730	−0.708649	−	0.705561i	$-0.750695\pi$
$127$	−15.9721	−1.41730	−0.708649	+	0.705561i	$0.750695\pi$
$128$	−20.2604	−1.79078
$129$	17.9560	1.58094
$130$	−33.3905	−2.92854
$131$	−12.3697	−1.08075	−0.540374	−	0.841425i	$-0.681717\pi$
$131$	−12.3697	−1.08075	−0.540374	+	0.841425i	$0.681717\pi$
$132$	0	0
$133$	0	0
$134$	0.296935	0.0256512
$135$	−6.23591	−0.536702
$136$	8.82424	0.756672
$137$	15.2851	1.30590	0.652949	−	0.757402i	$-0.273532\pi$
$137$	15.2851	1.30590	0.652949	+	0.757402i	$0.273532\pi$
$138$	−12.5935	−1.07203
$139$	−6.11428	−0.518607	−0.259303	−	0.965796i	$-0.583493\pi$
$139$	−6.11428	−0.518607	−0.259303	+	0.965796i	$0.583493\pi$
$140$	0	0
$141$	6.34841	0.534632
$142$	−12.2542	−1.02835
$143$	0	0
$144$	2.85543	0.237952
$145$	−31.6144	−2.62543
$146$	−9.71486	−0.804007
$147$	0	0
$148$	−8.57028	−0.704473
$149$	−7.45257	−0.610538	−0.305269	−	0.952266i	$-0.598746\pi$
$149$	−7.45257	−0.610538	−0.305269	+	0.952266i	$0.598746\pi$
$150$	−39.1422	−3.19595
$151$	−19.7068	−1.60371	−0.801857	−	0.597516i	$-0.796154\pi$
$151$	−19.7068	−1.60371	−0.801857	+	0.597516i	$0.796154\pi$
$152$	−12.5070	−1.01445
$153$	5.54396	0.448203
$154$	0	0
$155$	24.3765	1.95797
$156$	−32.5146	−2.60325
$157$	−20.5843	−1.64281	−0.821404	−	0.570347i	$-0.806809\pi$
$157$	−20.5843	−1.64281	−0.821404	+	0.570347i	$0.806809\pi$
$158$	−9.44375	−0.751305
$159$	−7.24995	−0.574958
$160$	14.2287	1.12488
$161$	0	0
$162$	−25.1775	−1.97813
$163$	22.4578	1.75903	0.879515	−	0.475871i	$-0.157867\pi$
$163$	22.4578	1.75903	0.879515	+	0.475871i	$0.157867\pi$
$164$	22.3431	1.74470
$165$	0	0
$166$	−2.71486	−0.210714
$167$	−9.52797	−0.737297	−0.368648	−	0.929569i	$-0.620179\pi$
$167$	−9.52797	−0.737297	−0.368648	+	0.929569i	$0.620179\pi$
$168$	0	0
$169$	3.46090	0.266223
$170$	−20.5351	−1.57497
$171$	−7.85772	−0.600895
$172$	−27.5572	−2.10122
$173$	17.3930	1.32237	0.661183	−	0.750225i	$-0.270055\pi$
$173$	17.3930	1.32237	0.661183	+	0.750225i	$0.270055\pi$
$174$	−48.3413	−3.66474
$175$	0	0
$176$	0	0
$177$	1.63355	0.122785
$178$	−27.3334	−2.04873
$179$	−8.84139	−0.660837	−0.330418	−	0.943835i	$-0.607190\pi$
$179$	−8.84139	−0.660837	−0.330418	+	0.943835i	$0.607190\pi$
$180$	−27.3273	−2.03685
$181$	10.7149	0.796429	0.398215	−	0.917292i	$-0.369630\pi$
$181$	10.7149	0.796429	0.398215	+	0.917292i	$0.369630\pi$
$182$	0	0
$183$	3.05818	0.226067
$184$	8.30521	0.612268
$185$	8.57028	0.630100
$186$	37.2739	2.73305
$187$	0	0
$188$	−9.74293	−0.710576
$189$	0	0
$190$	29.1054	2.11153
$191$	−11.4718	−0.830072	−0.415036	−	0.909805i	$-0.636231\pi$
$191$	−11.4718	−0.830072	−0.415036	+	0.909805i	$0.636231\pi$
$192$	27.6304	1.99406
$193$	−8.00619	−0.576298	−0.288149	−	0.957586i	$-0.593040\pi$
$193$	−8.00619	−0.576298	−0.288149	+	0.957586i	$0.593040\pi$
$194$	13.1800	0.946271
$195$	32.5146	2.32842
$196$	0	0
$197$	−15.8989	−1.13275	−0.566376	−	0.824147i	$-0.691655\pi$
$197$	−15.8989	−1.13275	−0.566376	+	0.824147i	$0.691655\pi$
$198$	0	0
$199$	13.1054	0.929016	0.464508	−	0.885569i	$-0.346231\pi$
$199$	13.1054	0.929016	0.464508	+	0.885569i	$0.346231\pi$
$200$	25.8137	1.82531
$201$	−0.289145	−0.0203947
$202$	13.5351	0.952326
$203$	0	0
$204$	−19.9964	−1.40003
$205$	−22.3431	−1.56051
$206$	7.59373	0.529080
$207$	5.21787	0.362667
$208$	5.21409	0.361532
$209$	0	0
$210$	0	0
$211$	−5.80065	−0.399333	−0.199666	−	0.979864i	$-0.563986\pi$
$211$	−5.80065	−0.399333	−0.199666	+	0.979864i	$0.563986\pi$
$212$	11.1265	0.764173
$213$	11.9327	0.817617
$214$	−36.8623	−2.51986
$215$	27.5572	1.87938
$216$	6.28838	0.427870
$217$	0	0
$218$	14.6797	0.994232
$219$	9.46001	0.639248
$220$	0	0
$221$	10.1234	0.680975
$222$	13.1048	0.879533
$223$	−2.97193	−0.199015	−0.0995074	−	0.995037i	$-0.531727\pi$
$223$	−2.97193	−0.199015	−0.0995074	+	0.995037i	$0.531727\pi$
$224$	0	0
$225$	16.2179	1.08119
$226$	−41.8188	−2.78175
$227$	−23.0122	−1.52738	−0.763688	−	0.645585i	$-0.776613\pi$
$227$	−23.0122	−1.52738	−0.763688	+	0.645585i	$0.776613\pi$
$228$	28.3419	1.87699
$229$	29.3734	1.94105	0.970523	−	0.241007i	$-0.0774776\pi$
$229$	29.3734	1.94105	0.970523	+	0.241007i	$0.0774776\pi$
$230$	−19.3272	−1.27440
$231$	0	0
$232$	31.8804	2.09305
$233$	−9.23835	−0.605224	−0.302612	−	0.953114i	$-0.597859\pi$
$233$	−9.23835	−0.605224	−0.302612	+	0.953114i	$0.597859\pi$
$234$	21.1546	1.38292
$235$	9.74293	0.635559
$236$	−2.50702	−0.163193
$237$	9.19601	0.597345
$238$	0	0
$239$	11.8417	0.765978	0.382989	−	0.923753i	$-0.374895\pi$
$239$	11.8417	0.765978	0.382989	+	0.923753i	$0.374895\pi$
$240$	10.2992	0.664809
$241$	−1.52909	−0.0984974	−0.0492487	−	0.998787i	$-0.515683\pi$
$241$	−1.52909	−0.0984974	−0.0492487	+	0.998787i	$0.515683\pi$
$242$	0	0
$243$	19.1827	1.23057
$244$	−4.69341	−0.300465
$245$	0	0
$246$	−34.1646	−2.17826
$247$	−14.3484	−0.912967
$248$	−24.5816	−1.56093
$249$	2.64364	0.167534
$250$	−18.9221	−1.19674
$251$	2.93273	0.185112	0.0925562	−	0.995707i	$-0.470496\pi$
$251$	2.93273	0.185112	0.0925562	+	0.995707i	$0.470496\pi$
$252$	0	0
$253$	0	0
$254$	−37.4819	−2.35182
$255$	19.9964	1.25222
$256$	−23.3624	−1.46015
$257$	−12.9508	−0.807847	−0.403923	−	0.914793i	$-0.632354\pi$
$257$	−12.9508	−0.807847	−0.403923	+	0.914793i	$0.632354\pi$
$258$	42.1375	2.62336
$259$	0	0
$260$	−49.9003	−3.09468
$261$	20.0293	1.23978
$262$	−29.0281	−1.79336
$263$	−17.8311	−1.09951	−0.549756	−	0.835325i	$-0.685279\pi$
$263$	−17.8311	−1.09951	−0.549756	+	0.835325i	$0.685279\pi$
$264$	0	0
$265$	−11.1265	−0.683497
$266$	0	0
$267$	26.6164	1.62890
$268$	0.443752	0.0271065
$269$	1.07730	0.0656844	0.0328422	−	0.999461i	$-0.489544\pi$
$269$	1.07730	0.0656844	0.0328422	+	0.999461i	$0.489544\pi$
$270$	−14.6338	−0.890587
$271$	21.5987	1.31203	0.656013	−	0.754749i	$-0.272242\pi$
$271$	21.5987	1.31203	0.656013	+	0.754749i	$0.272242\pi$
$272$	3.20665	0.194432
$273$	0	0
$274$	35.8697	2.16697
$275$	0	0
$276$	−18.8202	−1.13285
$277$	−1.76014	−0.105757	−0.0528784	−	0.998601i	$-0.516840\pi$
$277$	−1.76014	−0.105757	−0.0528784	+	0.998601i	$0.516840\pi$
$278$	−14.3484	−0.860560
$279$	−15.4438	−0.924593
$280$	0	0
$281$	5.21409	0.311046	0.155523	−	0.987832i	$-0.450294\pi$
$281$	5.21409	0.311046	0.155523	+	0.987832i	$0.450294\pi$
$282$	14.8978	0.887153
$283$	−8.37840	−0.498044	−0.249022	−	0.968498i	$-0.580109\pi$
$283$	−8.37840	−0.498044	−0.249022	+	0.968498i	$0.580109\pi$
$284$	−18.3132	−1.08669
$285$	−28.3419	−1.67883
$286$	0	0
$287$	0	0
$288$	−9.01460	−0.531191
$289$	−10.7741	−0.633772
$290$	−74.1896	−4.35656
$291$	−12.8343	−0.752358
$292$	−14.5183	−0.849620
$293$	−4.31389	−0.252020	−0.126010	−	0.992029i	$-0.540217\pi$
$293$	−4.31389	−0.252020	−0.126010	+	0.992029i	$0.540217\pi$
$294$	0	0
$295$	2.50702	0.145964
$296$	−8.64239	−0.502329
$297$	0	0
$298$	−17.4890	−1.01311
$299$	9.52797	0.551017
$300$	−58.4959	−3.37726
$301$	0	0
$302$	−46.2459	−2.66116
$303$	−13.1800	−0.757173
$304$	−4.54494	−0.260670
$305$	4.69341	0.268744
$306$	13.0100	0.743734
$307$	−10.5014	−0.599344	−0.299672	−	0.954042i	$-0.596877\pi$
$307$	−10.5014	−0.599344	−0.299672	+	0.954042i	$0.596877\pi$
$308$	0	0
$309$	−7.39452	−0.420660
$310$	57.2044	3.24899
$311$	12.4226	0.704421	0.352210	−	0.935921i	$-0.385430\pi$
$311$	12.4226	0.704421	0.352210	+	0.935921i	$0.385430\pi$
$312$	−32.7882	−1.85626
$313$	5.76409	0.325805	0.162903	−	0.986642i	$-0.447914\pi$
$313$	5.76409	0.325805	0.162903	+	0.986642i	$0.447914\pi$
$314$	−48.3053	−2.72603
$315$	0	0
$316$	−14.1132	−0.793928
$317$	4.08042	0.229179	0.114590	−	0.993413i	$-0.463445\pi$
$317$	4.08042	0.229179	0.114590	+	0.993413i	$0.463445\pi$
$318$	−17.0135	−0.954069
$319$	0	0
$320$	42.4046	2.37049
$321$	35.8953	2.00348
$322$	0	0
$323$	−8.82424	−0.490994
$324$	−37.6264	−2.09036
$325$	29.6143	1.64270
$326$	52.7018	2.91888
$327$	−14.2946	−0.790492
$328$	22.5311	1.24407
$329$	0	0
$330$	0	0
$331$	11.3453	0.623594	0.311797	−	0.950149i	$-0.399069\pi$
$331$	11.3453	0.623594	0.311797	+	0.950149i	$0.399069\pi$
$332$	−4.05720	−0.222668
$333$	−5.42972	−0.297547
$334$	−22.3593	−1.22345
$335$	−0.443752	−0.0242448
$336$	0	0
$337$	35.4149	1.92917	0.964586	−	0.263767i	$-0.0849650\pi$
$337$	35.4149	1.92917	0.964586	+	0.263767i	$0.0849650\pi$
$338$	8.12172	0.441763
$339$	40.7218	2.21170
$340$	−30.6886	−1.66432
$341$	0	0
$342$	−18.4397	−0.997107
$343$	0	0
$344$	−27.7890	−1.49828
$345$	18.8202	1.01325
$346$	40.8162	2.19429
$347$	−26.2189	−1.40750	−0.703752	−	0.710446i	$-0.748493\pi$
$347$	−26.2189	−1.40750	−0.703752	+	0.710446i	$0.748493\pi$
$348$	−72.2434	−3.87265
$349$	−35.0850	−1.87806	−0.939029	−	0.343837i	$-0.888273\pi$
$349$	−35.0850	−1.87806	−0.939029	+	0.343837i	$0.888273\pi$
$350$	0	0
$351$	7.21421	0.385066
$352$	0	0
$353$	−4.91958	−0.261843	−0.130921	−	0.991393i	$-0.541794\pi$
$353$	−4.91958	−0.261843	−0.130921	+	0.991393i	$0.541794\pi$
$354$	3.83346	0.203746
$355$	18.3132	0.971965
$356$	−40.8483	−2.16496
$357$	0	0
$358$	−20.7481	−1.09657
$359$	23.2689	1.22809	0.614044	−	0.789272i	$-0.289542\pi$
$359$	23.2689	1.22809	0.614044	+	0.789272i	$0.289542\pi$
$360$	−27.5572	−1.45239
$361$	−6.49298	−0.341736
$362$	25.1446	1.32157
$363$	0	0
$364$	0	0
$365$	14.5183	0.759923
$366$	7.17665	0.375129
$367$	32.8162	1.71299	0.856497	−	0.516152i	$-0.172636\pi$
$367$	32.8162	1.71299	0.856497	+	0.516152i	$0.172636\pi$
$368$	3.01804	0.157326
$369$	14.1555	0.736906
$370$	20.1119	1.04557
$371$	0	0
$372$	55.7037	2.88810
$373$	26.4426	1.36915	0.684574	−	0.728943i	$-0.259988\pi$
$373$	26.4426	1.36915	0.684574	+	0.728943i	$0.259988\pi$
$374$	0	0
$375$	18.4257	0.951500
$376$	−9.82491	−0.506681
$377$	36.5741	1.88366
$378$	0	0
$379$	12.5531	0.644811	0.322406	−	0.946602i	$-0.395509\pi$
$379$	12.5531	0.644811	0.322406	+	0.946602i	$0.395509\pi$
$380$	43.4964	2.23132
$381$	36.4986	1.86988
$382$	−26.9210	−1.37740
$383$	−27.9960	−1.43053	−0.715264	−	0.698854i	$-0.753694\pi$
$383$	−27.9960	−1.43053	−0.715264	+	0.698854i	$0.753694\pi$
$384$	46.2979	2.36263
$385$	0	0
$386$	−18.7882	−0.956292
$387$	−17.4589	−0.887485
$388$	19.6968	0.999954
$389$	18.2179	0.923683	0.461841	−	0.886963i	$-0.347189\pi$
$389$	18.2179	0.923683	0.461841	+	0.886963i	$0.347189\pi$
$390$	76.3021	3.86371
$391$	5.85968	0.296337
$392$	0	0
$393$	28.2666	1.42586
$394$	−37.3101	−1.87966
$395$	14.1132	0.710110
$396$	0	0
$397$	−13.2671	−0.665857	−0.332928	−	0.942952i	$-0.608037\pi$
$397$	−13.2671	−0.665857	−0.332928	+	0.942952i	$0.608037\pi$
$398$	30.7544	1.54158
$399$	0	0
$400$	9.38049	0.469024
$401$	−2.27111	−0.113414	−0.0567068	−	0.998391i	$-0.518060\pi$
$401$	−2.27111	−0.113414	−0.0567068	+	0.998391i	$0.518060\pi$
$402$	−0.678538	−0.0338424
$403$	−28.2007	−1.40478
$404$	20.2274	1.00635
$405$	37.6264	1.86967
$406$	0	0
$407$	0	0
$408$	−20.1646	−0.998299
$409$	−35.3747	−1.74916	−0.874582	−	0.484877i	$-0.838864\pi$
$409$	−35.3747	−1.74916	−0.874582	+	0.484877i	$0.838864\pi$
$410$	−52.4326	−2.58946
$411$	−34.9287	−1.72291
$412$	11.3484	0.559096
$413$	0	0
$414$	12.2448	0.601799
$415$	4.05720	0.199160
$416$	−16.4609	−0.807062
$417$	13.9720	0.684212
$418$	0	0
$419$	−35.7249	−1.74528	−0.872638	−	0.488368i	$-0.837592\pi$
$419$	−35.7249	−1.74528	−0.872638	+	0.488368i	$0.837592\pi$
$420$	0	0
$421$	−30.0873	−1.46637	−0.733184	−	0.680031i	$-0.761966\pi$
$421$	−30.0873	−1.46637	−0.733184	+	0.680031i	$0.761966\pi$
$422$	−13.6124	−0.662641
$423$	−6.17265	−0.300124
$424$	11.2202	0.544899
$425$	18.2127	0.883446
$426$	28.0026	1.35673
$427$	0	0
$428$	−55.0887	−2.66281
$429$	0	0
$430$	64.6685	3.11859
$431$	6.41122	0.308817	0.154409	−	0.988007i	$-0.450653\pi$
$431$	6.41122	0.308817	0.154409	+	0.988007i	$0.450653\pi$
$432$	2.28514	0.109944
$433$	−35.1686	−1.69010	−0.845049	−	0.534690i	$-0.820429\pi$
$433$	−35.1686	−1.69010	−0.845049	+	0.534690i	$0.820429\pi$
$434$	0	0
$435$	72.2434	3.46380
$436$	21.9379	1.05064
$437$	−8.30521	−0.397292
$438$	22.1998	1.06075
$439$	−14.6265	−0.698086	−0.349043	−	0.937107i	$-0.613493\pi$
$439$	−14.6265	−0.698086	−0.349043	+	0.937107i	$0.613493\pi$
$440$	0	0
$441$	0	0
$442$	23.7567	1.12999
$443$	6.71174	0.318885	0.159442	−	0.987207i	$-0.449030\pi$
$443$	6.71174	0.318885	0.159442	+	0.987207i	$0.449030\pi$
$444$	19.5843	0.929431
$445$	40.8483	1.93640
$446$	−6.97423	−0.330239
$447$	17.0302	0.805500
$448$	0	0
$449$	6.14057	0.289791	0.144896	−	0.989447i	$-0.453715\pi$
$449$	6.14057	0.289791	0.144896	+	0.989447i	$0.453715\pi$
$450$	38.0585	1.79410
$451$	0	0
$452$	−62.4959	−2.93956
$453$	45.0328	2.11582
$454$	−54.0029	−2.53448
$455$	0	0
$456$	28.5803	1.33840
$457$	29.4491	1.37757	0.688785	−	0.724965i	$-0.258144\pi$
$457$	29.4491	1.37757	0.688785	+	0.724965i	$0.258144\pi$
$458$	68.9306	3.22092
$459$	4.43672	0.207089
$460$	−28.8835	−1.34670
$461$	−27.8139	−1.29542	−0.647710	−	0.761887i	$-0.724273\pi$
$461$	−27.8139	−1.29542	−0.647710	+	0.761887i	$0.724273\pi$
$462$	0	0
$463$	17.2319	0.800835	0.400417	−	0.916333i	$-0.368865\pi$
$463$	17.2319	0.800835	0.400417	+	0.916333i	$0.368865\pi$
$464$	11.5851	0.537823
$465$	−55.7037	−2.58320
$466$	−21.6797	−1.00429
$467$	14.0250	0.648998	0.324499	−	0.945886i	$-0.394804\pi$
$467$	14.0250	0.648998	0.324499	+	0.945886i	$0.394804\pi$
$468$	31.6144	1.46138
$469$	0	0
$470$	22.8638	1.05463
$471$	47.0381	2.16740
$472$	−2.52811	−0.116366
$473$	0	0
$474$	21.5803	0.991217
$475$	−25.8137	−1.18442
$476$	0	0
$477$	7.04923	0.322762
$478$	27.7890	1.27104
$479$	28.3821	1.29681	0.648406	−	0.761295i	$-0.275436\pi$
$479$	28.3821	1.29681	0.648406	+	0.761295i	$0.275436\pi$
$480$	−32.5146	−1.48408
$481$	−9.91480	−0.452076
$482$	−3.58832	−0.163444
$483$	0	0
$484$	0	0
$485$	−19.6968	−0.894386
$486$	45.0161	2.04197
$487$	−19.9820	−0.905469	−0.452735	−	0.891645i	$-0.649551\pi$
$487$	−19.9820	−0.905469	−0.452735	+	0.891645i	$0.649551\pi$
$488$	−4.73290	−0.214248
$489$	−51.3192	−2.32074
$490$	0	0
$491$	27.6304	1.24694	0.623471	−	0.781847i	$-0.285722\pi$
$491$	27.6304	1.24694	0.623471	+	0.781847i	$0.285722\pi$
$492$	−51.0572	−2.30183
$493$	22.4930	1.01303
$494$	−33.6715	−1.51495
$495$	0	0
$496$	−8.93273	−0.401091
$497$	0	0
$498$	6.20384	0.278001
$499$	30.5632	1.36820	0.684098	−	0.729390i	$-0.260196\pi$
$499$	30.5632	1.36820	0.684098	+	0.729390i	$0.260196\pi$
$500$	−28.2780	−1.26463
$501$	21.7728	0.972736
$502$	6.88225	0.307170
$503$	−27.3737	−1.22053	−0.610266	−	0.792196i	$-0.708938\pi$
$503$	−27.3737	−1.22053	−0.610266	+	0.792196i	$0.708938\pi$
$504$	0	0
$505$	−20.2274	−0.900110
$506$	0	0
$507$	−7.90866	−0.351236
$508$	−56.0146	−2.48524
$509$	−3.62352	−0.160610	−0.0803048	−	0.996770i	$-0.525589\pi$
$509$	−3.62352	−0.160610	−0.0803048	+	0.996770i	$0.525589\pi$
$510$	46.9256	2.07790
$511$	0	0
$512$	−14.3040	−0.632152
$513$	−6.28838	−0.277639
$514$	−30.3916	−1.34052
$515$	−11.3484	−0.500071
$516$	62.9721	2.77219
$517$	0	0
$518$	0	0
$519$	−39.7455	−1.74463
$520$	−50.3201	−2.20668
$521$	−25.9608	−1.13736	−0.568682	−	0.822558i	$-0.692546\pi$
$521$	−25.9608	−1.13736	−0.568682	+	0.822558i	$0.692546\pi$
$522$	47.0029	2.05726
$523$	−0.694777	−0.0303805	−0.0151902	−	0.999885i	$-0.504835\pi$
$523$	−0.694777	−0.0303805	−0.0151902	+	0.999885i	$0.504835\pi$
$524$	−43.3808	−1.89510
$525$	0	0
$526$	−41.8443	−1.82450
$527$	−17.3434	−0.755489
$528$	0	0
$529$	−17.4850	−0.760216
$530$	−26.1107	−1.13418
$531$	−1.58832	−0.0689274
$532$	0	0
$533$	25.8483	1.11961
$534$	62.4608	2.70294
$535$	55.0887	2.38169
$536$	0.447486	0.0193285
$537$	20.2038	0.871860
$538$	2.52811	0.108995
$539$	0	0
$540$	−21.8695	−0.941112
$541$	24.4331	1.05046	0.525231	−	0.850960i	$-0.323979\pi$
$541$	24.4331	1.05046	0.525231	+	0.850960i	$0.323979\pi$
$542$	50.6857	2.17714
$543$	−24.4850	−1.05075
$544$	−10.1234	−0.434038
$545$	−21.9379	−0.939718
$546$	0	0
$547$	26.9706	1.15318	0.576590	−	0.817033i	$-0.304383\pi$
$547$	26.9706	1.15318	0.576590	+	0.817033i	$0.304383\pi$
$548$	53.6053	2.28990
$549$	−2.97351	−0.126907
$550$	0	0
$551$	−31.8804	−1.35815
$552$	−18.9786	−0.807782
$553$	0	0
$554$	−4.13054	−0.175490
$555$	−19.5843	−0.831308
$556$	−21.4429	−0.909382
$557$	11.2458	0.476499	0.238250	−	0.971204i	$-0.423426\pi$
$557$	11.2458	0.476499	0.238250	+	0.971204i	$0.423426\pi$
$558$	−36.2419	−1.53424
$559$	−31.8804	−1.34840
$560$	0	0
$561$	0	0
$562$	12.2359	0.516141
$563$	22.8805	0.964297	0.482149	−	0.876089i	$-0.339857\pi$
$563$	22.8805	0.964297	0.482149	+	0.876089i	$0.339857\pi$
$564$	22.2640	0.937483
$565$	62.4959	2.62922
$566$	−19.6616	−0.826440
$567$	0	0
$568$	−18.4673	−0.774871
$569$	−38.1082	−1.59758	−0.798789	−	0.601611i	$-0.794526\pi$
$569$	−38.1082	−1.59758	−0.798789	+	0.601611i	$0.794526\pi$
$570$	−66.5099	−2.78579
$571$	2.85799	0.119603	0.0598015	−	0.998210i	$-0.480953\pi$
$571$	2.85799	0.119603	0.0598015	+	0.998210i	$0.480953\pi$
$572$	0	0
$573$	26.2148	1.09514
$574$	0	0
$575$	17.1415	0.714848
$576$	−26.8655	−1.11939
$577$	10.4266	0.434065	0.217033	−	0.976164i	$-0.430362\pi$
$577$	10.4266	0.434065	0.217033	+	0.976164i	$0.430362\pi$
$578$	−25.2837	−1.05166
$579$	18.2953	0.760326
$580$	−110.872	−4.60372
$581$	0	0
$582$	−30.1182	−1.24844
$583$	0	0
$584$	−14.6405	−0.605827
$585$	−31.6144	−1.30709
$586$	−10.1234	−0.418195
$587$	16.4899	0.680610	0.340305	−	0.940315i	$-0.389470\pi$
$587$	16.4899	0.680610	0.340305	+	0.940315i	$0.389470\pi$
$588$	0	0
$589$	24.5816	1.01287
$590$	5.88323	0.242209
$591$	36.3313	1.49447
$592$	−3.14057	−0.129077
$593$	−11.8564	−0.486882	−0.243441	−	0.969916i	$-0.578276\pi$
$593$	−11.8564	−0.486882	−0.243441	+	0.969916i	$0.578276\pi$
$594$	0	0
$595$	0	0
$596$	−26.1363	−1.07058
$597$	−29.9477	−1.22568
$598$	22.3593	0.914341
$599$	−42.6084	−1.74093	−0.870466	−	0.492229i	$-0.836182\pi$
$599$	−42.6084	−1.74093	−0.870466	+	0.492229i	$0.836182\pi$
$600$	−58.9881	−2.40818
$601$	−20.9295	−0.853733	−0.426867	−	0.904315i	$-0.640383\pi$
$601$	−20.9295	−0.853733	−0.426867	+	0.904315i	$0.640383\pi$
$602$	0	0
$603$	0.281140	0.0114489
$604$	−69.1120	−2.81213
$605$	0	0
$606$	−30.9296	−1.25643
$607$	11.8747	0.481978	0.240989	−	0.970528i	$-0.422528\pi$
$607$	11.8747	0.481978	0.240989	+	0.970528i	$0.422528\pi$
$608$	14.3484	0.581905
$609$	0	0
$610$	11.0140	0.445945
$611$	−11.2714	−0.455993
$612$	19.4428	0.785928
$613$	18.0951	0.730854	0.365427	−	0.930840i	$-0.380923\pi$
$613$	18.0951	0.730854	0.365427	+	0.930840i	$0.380923\pi$
$614$	−24.6436	−0.994534
$615$	51.0572	2.05882
$616$	0	0
$617$	−5.19672	−0.209212	−0.104606	−	0.994514i	$-0.533358\pi$
$617$	−5.19672	−0.209212	−0.104606	+	0.994514i	$0.533358\pi$
$618$	−17.3528	−0.698030
$619$	16.1546	0.649308	0.324654	−	0.945833i	$-0.394752\pi$
$619$	16.1546	0.649308	0.324654	+	0.945833i	$0.394752\pi$
$620$	85.4888	3.43331
$621$	4.17576	0.167568
$622$	29.1522	1.16890
$623$	0	0
$624$	−11.9149	−0.476979
$625$	−8.21787	−0.328715
$626$	13.5266	0.540632
$627$	0	0
$628$	−72.1896	−2.88068
$629$	−6.09758	−0.243126
$630$	0	0
$631$	21.2148	0.844546	0.422273	−	0.906469i	$-0.361232\pi$
$631$	21.2148	0.844546	0.422273	+	0.906469i	$0.361232\pi$
$632$	−14.2319	−0.566115
$633$	13.2553	0.526851
$634$	9.57553	0.380293
$635$	56.0146	2.22287
$636$	−25.4257	−1.00820
$637$	0	0
$638$	0	0
$639$	−11.6024	−0.458982
$640$	71.0536	2.80864
$641$	40.9788	1.61857	0.809284	−	0.587418i	$-0.199856\pi$
$641$	40.9788	1.61857	0.809284	+	0.587418i	$0.199856\pi$
$642$	84.2357	3.32452
$643$	24.8514	0.980045	0.490022	−	0.871710i	$-0.336989\pi$
$643$	24.8514	0.980045	0.490022	+	0.871710i	$0.336989\pi$
$644$	0	0
$645$	−62.9721	−2.47952
$646$	−20.7079	−0.814740
$647$	−14.2188	−0.558997	−0.279499	−	0.960146i	$-0.590168\pi$
$647$	−14.2188	−0.558997	−0.279499	+	0.960146i	$0.590168\pi$
$648$	−37.9430	−1.49054
$649$	0	0
$650$	69.4959	2.72585
$651$	0	0
$652$	78.7599	3.08448
$653$	−49.7991	−1.94879	−0.974394	−	0.224846i	$-0.927812\pi$
$653$	−49.7991	−1.94879	−0.974394	+	0.224846i	$0.927812\pi$
$654$	−33.5451	−1.31172
$655$	43.3808	1.69503
$656$	8.18760	0.319672
$657$	−9.19810	−0.358852
$658$	0	0
$659$	7.27116	0.283244	0.141622	−	0.989921i	$-0.454768\pi$
$659$	7.27116	0.283244	0.141622	+	0.989921i	$0.454768\pi$
$660$	0	0
$661$	−4.59144	−0.178586	−0.0892931	−	0.996005i	$-0.528461\pi$
$661$	−4.59144	−0.178586	−0.0892931	+	0.996005i	$0.528461\pi$
$662$	26.6240	1.03477
$663$	−23.1335	−0.898429
$664$	−4.09134	−0.158775
$665$	0	0
$666$	−12.7419	−0.493740
$667$	21.1700	0.819705
$668$	−33.4148	−1.29286
$669$	6.79127	0.262566
$670$	−1.04136	−0.0402311
$671$	0	0
$672$	0	0
$673$	12.7419	0.491165	0.245583	−	0.969376i	$-0.421021\pi$
$673$	12.7419	0.491165	0.245583	+	0.969376i	$0.421021\pi$
$674$	83.1083	3.20121
$675$	12.9788	0.499556
$676$	12.1375	0.466825
$677$	10.9065	0.419171	0.209586	−	0.977790i	$-0.432788\pi$
$677$	10.9065	0.419171	0.209586	+	0.977790i	$0.432788\pi$
$678$	95.5619	3.67003
$679$	0	0
$680$	−30.9468	−1.18675
$681$	52.5863	2.01511
$682$	0	0
$683$	23.0100	0.880455	0.440227	−	0.897886i	$-0.354898\pi$
$683$	23.0100	0.880455	0.440227	+	0.897886i	$0.354898\pi$
$684$	−27.5572	−1.05368
$685$	−53.6053	−2.04815
$686$	0	0
$687$	−67.1223	−2.56088
$688$	−10.0983	−0.384994
$689$	12.8721	0.490387
$690$	44.1655	1.68135
$691$	−7.58432	−0.288521	−0.144261	−	0.989540i	$-0.546080\pi$
$691$	−7.58432	−0.288521	−0.144261	+	0.989540i	$0.546080\pi$
$692$	60.9976	2.31878
$693$	0	0
$694$	−61.5280	−2.33557
$695$	21.4429	0.813376
$696$	−72.8512	−2.76142
$697$	15.8967	0.602129
$698$	−82.3342	−3.11639
$699$	21.1109	0.798489
$700$	0	0
$701$	−24.9867	−0.943736	−0.471868	−	0.881669i	$-0.656420\pi$
$701$	−24.9867	−0.943736	−0.471868	+	0.881669i	$0.656420\pi$
$702$	16.9296	0.638967
$703$	8.64239	0.325954
$704$	0	0
$705$	−22.2640	−0.838510
$706$	−11.5448	−0.434494
$707$	0	0
$708$	5.72889	0.215305
$709$	6.78816	0.254935	0.127467	−	0.991843i	$-0.459315\pi$
$709$	6.78816	0.254935	0.127467	+	0.991843i	$0.459315\pi$
$710$	42.9757	1.61285
$711$	−8.94141	−0.335329
$712$	−41.1920	−1.54374
$713$	−16.3233	−0.611311
$714$	0	0
$715$	0	0
$716$	−31.0069	−1.15878
$717$	−27.0601	−1.01058
$718$	54.6053	2.03785
$719$	−27.1898	−1.01401	−0.507004	−	0.861944i	$-0.669247\pi$
$719$	−27.1898	−1.01401	−0.507004	+	0.861944i	$0.669247\pi$
$720$	−10.0140	−0.373201
$721$	0	0
$722$	−15.2371	−0.567066
$723$	3.49419	0.129950
$724$	37.5772	1.39655
$725$	65.7992	2.44372
$726$	0	0
$727$	−43.1335	−1.59973	−0.799866	−	0.600179i	$-0.795096\pi$
$727$	−43.1335	−1.59973	−0.799866	+	0.600179i	$0.795096\pi$
$728$	0	0
$729$	−11.6485	−0.431425
$730$	34.0702	1.26099
$731$	−19.6064	−0.725168
$732$	10.7251	0.396411
$733$	−33.3322	−1.23115	−0.615576	−	0.788077i	$-0.711077\pi$
$733$	−33.3322	−1.23115	−0.615576	+	0.788077i	$0.711077\pi$
$734$	77.0100	2.84249
$735$	0	0
$736$	−9.52797	−0.351206
$737$	0	0
$738$	33.2188	1.22280
$739$	39.4392	1.45079	0.725397	−	0.688331i	$-0.241656\pi$
$739$	39.4392	1.45079	0.725397	+	0.688331i	$0.241656\pi$
$740$	30.0561	1.10489
$741$	32.7882	1.20450
$742$	0	0
$743$	0.240445	0.00882107	0.00441053	−	0.999990i	$-0.498596\pi$
$743$	0.240445	0.00882107	0.00441053	+	0.999990i	$0.498596\pi$
$744$	56.1724	2.05938
$745$	26.1363	0.957560
$746$	62.0530	2.27192
$747$	−2.57045	−0.0940477
$748$	0	0
$749$	0	0
$750$	43.2397	1.57889
$751$	−24.1758	−0.882186	−0.441093	−	0.897461i	$-0.645409\pi$
$751$	−24.1758	−0.882186	−0.441093	+	0.897461i	$0.645409\pi$
$752$	−3.57028	−0.130195
$753$	−6.70171	−0.244224
$754$	85.8286	3.12569
$755$	69.1120	2.51524
$756$	0	0
$757$	−41.4647	−1.50706	−0.753530	−	0.657413i	$-0.771651\pi$
$757$	−41.4647	−1.50706	−0.753530	+	0.657413i	$0.771651\pi$
$758$	29.4585	1.06998
$759$	0	0
$760$	43.8623	1.59105
$761$	48.8333	1.77021	0.885103	−	0.465396i	$-0.154088\pi$
$761$	48.8333	1.77021	0.885103	+	0.465396i	$0.154088\pi$
$762$	85.6514	3.10282
$763$	0	0
$764$	−40.2319	−1.45554
$765$	−19.4428	−0.702955
$766$	−65.6983	−2.37378
$767$	−2.90032	−0.104725
$768$	53.3865	1.92642
$769$	−11.9149	−0.429663	−0.214832	−	0.976651i	$-0.568920\pi$
$769$	−11.9149	−0.429663	−0.214832	+	0.976651i	$0.568920\pi$
$770$	0	0
$771$	29.5944	1.06581
$772$	−28.0779	−1.01054
$773$	−34.9327	−1.25644	−0.628222	−	0.778035i	$-0.716217\pi$
$773$	−34.9327	−1.25644	−0.628222	+	0.778035i	$0.716217\pi$
$774$	−40.9708	−1.47267
$775$	−50.7349	−1.82245
$776$	19.8625	0.713024
$777$	0	0
$778$	42.7520	1.53273
$779$	−22.5311	−0.807260
$780$	114.029	4.08290
$781$	0	0
$782$	13.7509	0.491733
$783$	16.0291	0.572833
$784$	0	0
$785$	72.1896	2.57656
$786$	66.3333	2.36603
$787$	13.3635	0.476358	0.238179	−	0.971221i	$-0.423450\pi$
$787$	13.3635	0.476358	0.238179	+	0.971221i	$0.423450\pi$
$788$	−55.7579	−1.98629
$789$	40.7466	1.45062
$790$	33.1194	1.17834
$791$	0	0
$792$	0	0
$793$	−5.42972	−0.192815
$794$	−31.1340	−1.10490
$795$	25.4257	0.901757
$796$	45.9608	1.62904
$797$	16.4326	0.582074	0.291037	−	0.956712i	$-0.406000\pi$
$797$	16.4326	0.582074	0.291037	+	0.956712i	$0.406000\pi$
$798$	0	0
$799$	−6.93189	−0.245233
$800$	−29.6143	−1.04702
$801$	−25.8795	−0.914407
$802$	−5.32961	−0.188195
$803$	0	0
$804$	−1.01404	−0.0357623
$805$	0	0
$806$	−66.1787	−2.33104
$807$	−2.46179	−0.0866592
$808$	20.3976	0.717586
$809$	−39.5474	−1.39041	−0.695206	−	0.718811i	$-0.744687\pi$
$809$	−39.5474	−1.39041	−0.695206	+	0.718811i	$0.744687\pi$
$810$	88.2981	3.10248
$811$	22.1027	0.776129	0.388065	−	0.921632i	$-0.373144\pi$
$811$	22.1027	0.776129	0.388065	+	0.921632i	$0.373144\pi$
$812$	0	0
$813$	−49.3561	−1.73099
$814$	0	0
$815$	−78.7599	−2.75884
$816$	−7.32765	−0.256519
$817$	27.7890	0.972216
$818$	−83.0138	−2.90251
$819$	0	0
$820$	−78.3577	−2.73637
$821$	43.8686	1.53102	0.765512	−	0.643422i	$-0.222486\pi$
$821$	43.8686	1.53102	0.765512	+	0.643422i	$0.222486\pi$
$822$	−81.9674	−2.85894
$823$	13.0149	0.453672	0.226836	−	0.973933i	$-0.427162\pi$
$823$	13.0149	0.453672	0.226836	+	0.973933i	$0.427162\pi$
$824$	11.4439	0.398667
$825$	0	0
$826$	0	0
$827$	−11.5119	−0.400307	−0.200153	−	0.979765i	$-0.564144\pi$
$827$	−11.5119	−0.400307	−0.200153	+	0.979765i	$0.564144\pi$
$828$	18.2992	0.635940
$829$	−23.2148	−0.806282	−0.403141	−	0.915138i	$-0.632082\pi$
$829$	−23.2148	−0.806282	−0.403141	+	0.915138i	$0.632082\pi$
$830$	9.52106	0.330481
$831$	4.02218	0.139528
$832$	−49.0570	−1.70075
$833$	0	0
$834$	32.7882	1.13536
$835$	33.4148	1.15637
$836$	0	0
$837$	−12.3593	−0.427201
$838$	−83.8357	−2.89606
$839$	−36.1976	−1.24968	−0.624840	−	0.780753i	$-0.714836\pi$
$839$	−36.1976	−1.24968	−0.624840	+	0.780753i	$0.714836\pi$
$840$	0	0
$841$	52.2631	1.80218
$842$	−70.6061	−2.43325
$843$	−11.9149	−0.410372
$844$	−20.3430	−0.700234
$845$	−12.1375	−0.417541
$846$	−14.4854	−0.498017
$847$	0	0
$848$	4.07730	0.140015
$849$	19.1458	0.657084
$850$	42.7398	1.46596
$851$	−5.73893	−0.196728
$852$	41.8483	1.43370
$853$	24.9705	0.854974	0.427487	−	0.904022i	$-0.359399\pi$
$853$	24.9705	0.854974	0.427487	+	0.904022i	$0.359399\pi$
$854$	0	0
$855$	27.5572	0.942436
$856$	−55.5522	−1.89874
$857$	8.31252	0.283950	0.141975	−	0.989870i	$-0.454655\pi$
$857$	8.31252	0.283950	0.141975	+	0.989870i	$0.454655\pi$
$858$	0	0
$859$	20.0521	0.684170	0.342085	−	0.939669i	$-0.388867\pi$
$859$	20.0521	0.684170	0.342085	+	0.939669i	$0.388867\pi$
$860$	96.6436	3.29552
$861$	0	0
$862$	15.0452	0.512442
$863$	15.0352	0.511804	0.255902	−	0.966703i	$-0.417628\pi$
$863$	15.0352	0.511804	0.255902	+	0.966703i	$0.417628\pi$
$864$	−7.21421	−0.245432
$865$	−60.9976	−2.07398
$866$	−82.5304	−2.80450
$867$	24.6204	0.836153
$868$	0	0
$869$	0	0
$870$	169.534	5.74773
$871$	0.513369	0.0173948
$872$	22.1225	0.749163
$873$	12.4789	0.422348
$874$	−19.4899	−0.659255
$875$	0	0
$876$	33.1764	1.12093
$877$	23.3609	0.788842	0.394421	−	0.918930i	$-0.370945\pi$
$877$	23.3609	0.788842	0.394421	+	0.918930i	$0.370945\pi$
$878$	−34.3241	−1.15838
$879$	9.85785	0.332497
$880$	0	0
$881$	−6.22276	−0.209650	−0.104825	−	0.994491i	$-0.533428\pi$
$881$	−6.22276	−0.209650	−0.104825	+	0.994491i	$0.533428\pi$
$882$	0	0
$883$	7.42571	0.249895	0.124948	−	0.992163i	$-0.460124\pi$
$883$	7.42571	0.249895	0.124948	+	0.992163i	$0.460124\pi$
$884$	35.5030	1.19410
$885$	−5.72889	−0.192575
$886$	15.7505	0.529147
$887$	−18.7899	−0.630902	−0.315451	−	0.948942i	$-0.602156\pi$
$887$	−18.7899	−0.630902	−0.315451	+	0.948942i	$0.602156\pi$
$888$	19.7491	0.662736
$889$	0	0
$890$	95.8589	3.21320
$891$	0	0
$892$	−10.4226	−0.348974
$893$	9.82491	0.328778
$894$	39.9648	1.33662
$895$	31.0069	1.03645
$896$	0	0
$897$	−21.7728	−0.726972
$898$	14.4101	0.480871
$899$	−62.6585	−2.08978
$900$	56.8764	1.89588
$901$	7.91630	0.263730
$902$	0	0
$903$	0	0
$904$	−63.0217	−2.09607
$905$	−37.5772	−1.24911
$906$	105.679	3.51094
$907$	16.1063	0.534800	0.267400	−	0.963586i	$-0.413836\pi$
$907$	16.1063	0.534800	0.267400	+	0.963586i	$0.413836\pi$
$908$	−80.7044	−2.67827
$909$	12.8151	0.425051
$910$	0	0
$911$	16.8040	0.556740	0.278370	−	0.960474i	$-0.410206\pi$
$911$	16.8040	0.556740	0.278370	+	0.960474i	$0.410206\pi$
$912$	10.3858	0.343909
$913$	0	0
$914$	69.1083	2.28590
$915$	−10.7251	−0.354561
$916$	103.013	3.40364
$917$	0	0
$918$	10.4117	0.343636
$919$	−46.3381	−1.52855	−0.764277	−	0.644888i	$-0.776904\pi$
$919$	−46.3381	−1.52855	−0.764277	+	0.644888i	$0.776904\pi$
$920$	−29.1265	−0.960273
$921$	23.9971	0.790732
$922$	−65.2709	−2.14958
$923$	−21.1862	−0.697353
$924$	0	0
$925$	−17.8374	−0.586490
$926$	40.4382	1.32888
$927$	7.18980	0.236144
$928$	−36.5741	−1.20060
$929$	10.3865	0.340771	0.170385	−	0.985378i	$-0.445499\pi$
$929$	10.3865	0.340771	0.170385	+	0.985378i	$0.445499\pi$
$930$	−130.720	−4.28648
$931$	0	0
$932$	−32.3991	−1.06127
$933$	−28.3874	−0.929362
$934$	32.9124	1.07693
$935$	0	0
$936$	31.8804	1.04204
$937$	−13.6421	−0.445669	−0.222834	−	0.974856i	$-0.571531\pi$
$937$	−13.6421	−0.445669	−0.222834	+	0.974856i	$0.571531\pi$
$938$	0	0
$939$	−13.1718	−0.429844
$940$	34.1686	1.11446
$941$	−46.0647	−1.50167	−0.750834	−	0.660491i	$-0.770348\pi$
$941$	−46.0647	−1.50167	−0.750834	+	0.660491i	$0.770348\pi$
$942$	110.385	3.59652
$943$	14.9616	0.487218
$944$	−0.918694	−0.0299009
$945$	0	0
$946$	0	0
$947$	−18.1415	−0.589518	−0.294759	−	0.955572i	$-0.595239\pi$
$947$	−18.1415	−0.589518	−0.294759	+	0.955572i	$0.595239\pi$
$948$	32.2506	1.04745
$949$	−16.7960	−0.545220
$950$	−60.5772	−1.96538
$951$	−9.32434	−0.302362
$952$	0	0
$953$	−28.2739	−0.915881	−0.457941	−	0.888983i	$-0.651413\pi$
$953$	−28.2739	−0.915881	−0.457941	+	0.888983i	$0.651413\pi$
$954$	16.5425	0.535582
$955$	40.2319	1.30187
$956$	41.5292	1.34315
$957$	0	0
$958$	66.6044	2.15189
$959$	0	0
$960$	−96.9005	−3.12745
$961$	17.3132	0.558491
$962$	−23.2671	−0.750161
$963$	−34.9015	−1.12469
$964$	−5.36255	−0.172716
$965$	28.0779	0.903858
$966$	0	0
$967$	45.6026	1.46648	0.733241	−	0.679969i	$-0.238007\pi$
$967$	45.6026	1.46648	0.733241	+	0.679969i	$0.238007\pi$
$968$	0	0
$969$	20.1646	0.647782
$970$	−46.2226	−1.48412
$971$	−6.37648	−0.204631	−0.102316	−	0.994752i	$-0.532625\pi$
$971$	−6.37648	−0.204631	−0.102316	+	0.994752i	$0.532625\pi$
$972$	67.2740	2.15782
$973$	0	0
$974$	−46.8917	−1.50251
$975$	−67.6728	−2.16726
$976$	−1.71989	−0.0550524
$977$	−42.2419	−1.35144	−0.675720	−	0.737159i	$-0.736167\pi$
$977$	−42.2419	−1.35144	−0.675720	+	0.737159i	$0.736167\pi$
$978$	−120.431	−3.85096
$979$	0	0
$980$	0	0
$981$	13.8988	0.443755
$982$	64.8403	2.06914
$983$	−33.5743	−1.07085	−0.535427	−	0.844582i	$-0.679849\pi$
$983$	−33.5743	−1.07085	−0.535427	+	0.844582i	$0.679849\pi$
$984$	−51.4868	−1.64134
$985$	55.7579	1.77659
$986$	52.7844	1.68100
$987$	0	0
$988$	−50.3201	−1.60090
$989$	−18.4531	−0.586776
$990$	0	0
$991$	36.6155	1.16313	0.581565	−	0.813500i	$-0.302441\pi$
$991$	36.6155	1.16313	0.581565	+	0.813500i	$0.302441\pi$
$992$	28.2007	0.895373
$993$	−25.9256	−0.822724
$994$	0	0
$995$	−45.9608	−1.45706
$996$	9.27129	0.293772
$997$	15.2298	0.482332	0.241166	−	0.970484i	$-0.422470\pi$
$997$	15.2298	0.482332	0.241166	+	0.970484i	$0.422470\pi$
$998$	71.7227	2.27034
$999$	−4.34529	−0.137479

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5929.2.a.bk.1.6		6
7.2	even	3		847.2.e.e.606.1	yes	12
7.4	even	3		847.2.e.e.485.1	✓	12
7.6	odd	2		5929.2.a.bl.1.6		6
11.10	odd	2	inner	5929.2.a.bk.1.1		6
77.2	odd	30		847.2.n.k.81.6		48
77.4	even	15		847.2.n.k.632.1		48
77.9	even	15		847.2.n.k.81.1		48
77.16	even	15		847.2.n.k.487.6		48
77.18	odd	30		847.2.n.k.632.6		48
77.25	even	15		847.2.n.k.9.6		48
77.30	odd	30		847.2.n.k.130.6		48
77.32	odd	6		847.2.e.e.485.6	yes	12
77.37	even	15		847.2.n.k.753.6		48
77.39	odd	30		847.2.n.k.366.6		48
77.46	odd	30		847.2.n.k.807.1		48
77.51	odd	30		847.2.n.k.753.1		48
77.53	even	15		847.2.n.k.807.6		48
77.58	even	15		847.2.n.k.130.1		48
77.60	even	15		847.2.n.k.366.1		48
77.65	odd	6		847.2.e.e.606.6	yes	12
77.72	odd	30		847.2.n.k.487.1		48
77.74	odd	30		847.2.n.k.9.1		48
77.76	even	2		5929.2.a.bl.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.e.e.485.1	✓	12	7.4	even	3
847.2.e.e.485.6	yes	12	77.32	odd	6
847.2.e.e.606.1	yes	12	7.2	even	3
847.2.e.e.606.6	yes	12	77.65	odd	6
847.2.n.k.9.1		48	77.74	odd	30
847.2.n.k.9.6		48	77.25	even	15
847.2.n.k.81.1		48	77.9	even	15
847.2.n.k.81.6		48	77.2	odd	30
847.2.n.k.130.1		48	77.58	even	15
847.2.n.k.130.6		48	77.30	odd	30
847.2.n.k.366.1		48	77.60	even	15
847.2.n.k.366.6		48	77.39	odd	30
847.2.n.k.487.1		48	77.72	odd	30
847.2.n.k.487.6		48	77.16	even	15
847.2.n.k.632.1		48	77.4	even	15
847.2.n.k.632.6		48	77.18	odd	30
847.2.n.k.753.1		48	77.51	odd	30
847.2.n.k.753.6		48	77.37	even	15
847.2.n.k.807.1		48	77.46	odd	30
847.2.n.k.807.6		48	77.53	even	15
5929.2.a.bk.1.1		6	11.10	odd	2	inner
5929.2.a.bk.1.6		6	1.1	even	1	trivial
5929.2.a.bl.1.1		6	77.76	even	2
5929.2.a.bl.1.6		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5929 = 7^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5929.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34670 q^{2} -2.28514 q^{3} +3.50702 q^{4} -3.50702 q^{5} -5.36255 q^{6} +3.53653 q^{8} +2.22188 q^{9} -8.22993 q^{10} -8.01404 q^{12} +4.05720 q^{13} +8.01404 q^{15} +1.28514 q^{16} +2.49517 q^{17} +5.21409 q^{18} -3.53653 q^{19} -12.2992 q^{20} +2.34841 q^{23} -8.08147 q^{24} +7.29918 q^{25} +9.52106 q^{26} +1.77812 q^{27} +9.01460 q^{29} +18.8066 q^{30} -6.95077 q^{31} -4.05720 q^{32} +5.85543 q^{34} +7.79216 q^{36} -2.44375 q^{37} -8.29918 q^{38} -9.27129 q^{39} -12.4027 q^{40} +6.37097 q^{41} -7.85772 q^{43} -7.79216 q^{45} +5.51102 q^{46} -2.77812 q^{47} -2.93673 q^{48} +17.1290 q^{50} -5.70182 q^{51} +14.2287 q^{52} +3.17265 q^{53} +4.17273 q^{54} +8.08147 q^{57} +21.1546 q^{58} -0.714858 q^{59} +28.1054 q^{60} -1.33829 q^{61} -16.3114 q^{62} -12.0913 q^{64} -14.2287 q^{65} +0.126533 q^{67} +8.75061 q^{68} -5.36645 q^{69} -5.22188 q^{71} +7.85772 q^{72} -4.13979 q^{73} -5.73476 q^{74} -16.6797 q^{75} -12.4027 q^{76} -21.7570 q^{78} -4.02426 q^{79} -4.50702 q^{80} -10.7289 q^{81} +14.9508 q^{82} -1.15688 q^{83} -8.75061 q^{85} -18.4397 q^{86} -20.5997 q^{87} -11.6476 q^{89} -18.2859 q^{90} +8.23591 q^{92} +15.8835 q^{93} -6.51943 q^{94} +12.4027 q^{95} +9.27129 q^{96} +5.61640 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 4 q^{4} - 4 q^{5} + 8 q^{9} - 14 q^{12} + 14 q^{15} - 4 q^{16} - 28 q^{20} - 4 q^{23} - 2 q^{25} + 6 q^{26} + 16 q^{27} - 14 q^{31} + 18 q^{36} - 4 q^{37} - 4 q^{38} - 18 q^{45} - 22 q^{47}+ \cdots - 10 q^{97}+O(q^{100}) \) 6 * q - 2 * q^3 + 4 * q^4 - 4 * q^5 + 8 * q^9 - 14 * q^12 + 14 * q^15 - 4 * q^16 - 28 * q^20 - 4 * q^23 - 2 * q^25 + 6 * q^26 + 16 * q^27 - 14 * q^31 + 18 * q^36 - 4 * q^37 - 4 * q^38 - 18 * q^45 - 22 * q^47 - 24 * q^48 - 14 * q^53 + 46 * q^58 - 16 * q^59 + 60 * q^60 + 2 * q^64 - 12 * q^67 - 62 * q^69 - 26 * q^71 - 50 * q^75 - 40 * q^78 - 10 * q^80 - 42 * q^81 + 62 * q^82 - 18 * q^86 - 6 * q^89 + 10 * q^92 - 8 * q^93 - 10 * q^97

Introduction

L-functions

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