LMFDB - Embedded newform 6039.2.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43689 q^{2} +3.93843 q^{4} -3.08470 q^{5} +4.69684 q^{7} -4.72373 q^{8} +O(q^{10})$	\(q-2.43689 q^{2} +3.93843 q^{4} -3.08470 q^{5} +4.69684 q^{7} -4.72373 q^{8} +7.51707 q^{10} +1.00000 q^{11} +0.516905 q^{13} -11.4457 q^{14} +3.63436 q^{16} +7.14031 q^{17} -5.09505 q^{19} -12.1489 q^{20} -2.43689 q^{22} -8.35055 q^{23} +4.51538 q^{25} -1.25964 q^{26} +18.4982 q^{28} +7.36230 q^{29} +5.08121 q^{31} +0.590936 q^{32} -17.4001 q^{34} -14.4883 q^{35} -5.37159 q^{37} +12.4161 q^{38} +14.5713 q^{40} -9.33468 q^{41} +4.02303 q^{43} +3.93843 q^{44} +20.3494 q^{46} -2.85519 q^{47} +15.0603 q^{49} -11.0035 q^{50} +2.03579 q^{52} -5.63336 q^{53} -3.08470 q^{55} -22.1866 q^{56} -17.9411 q^{58} -2.20064 q^{59} +1.00000 q^{61} -12.3823 q^{62} -8.70877 q^{64} -1.59450 q^{65} -3.62908 q^{67} +28.1216 q^{68} +35.3065 q^{70} -16.6484 q^{71} -4.86700 q^{73} +13.0900 q^{74} -20.0665 q^{76} +4.69684 q^{77} -1.20008 q^{79} -11.2109 q^{80} +22.7476 q^{82} +0.616496 q^{83} -22.0257 q^{85} -9.80367 q^{86} -4.72373 q^{88} -17.2281 q^{89} +2.42782 q^{91} -32.8880 q^{92} +6.95779 q^{94} +15.7167 q^{95} -15.8083 q^{97} -36.7003 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.43689	−1.72314	−0.861570	−	0.507638i	$-0.830519\pi$
$2$	−2.43689	−1.72314	−0.861570	+	0.507638i	$0.830519\pi$
$3$	0	0
$3$	0	0
$4$	3.93843	1.96921
$5$	−3.08470	−1.37952	−0.689760	−	0.724038i	$-0.742284\pi$
$5$	−3.08470	−1.37952	−0.689760	+	0.724038i	$0.742284\pi$
$6$	0	0
$7$	4.69684	1.77524	0.887619	−	0.460578i	$-0.152358\pi$
$7$	4.69684	1.77524	0.887619	+	0.460578i	$0.152358\pi$
$8$	−4.72373	−1.67009
$9$	0	0
$10$	7.51707	2.37711
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.516905	0.143364	0.0716818	−	0.997428i	$-0.477163\pi$
$13$	0.516905	0.143364	0.0716818	+	0.997428i	$0.477163\pi$
$14$	−11.4457	−3.05899
$15$	0	0
$16$	3.63436	0.908590
$17$	7.14031	1.73178	0.865890	−	0.500234i	$-0.166753\pi$
$17$	7.14031	1.73178	0.865890	+	0.500234i	$0.166753\pi$
$18$	0	0
$19$	−5.09505	−1.16889	−0.584443	−	0.811435i	$-0.698687\pi$
$19$	−5.09505	−1.16889	−0.584443	+	0.811435i	$0.698687\pi$
$20$	−12.1489	−2.71657
$21$	0	0
$22$	−2.43689	−0.519546
$23$	−8.35055	−1.74121	−0.870605	−	0.491983i	$-0.836272\pi$
$23$	−8.35055	−1.74121	−0.870605	+	0.491983i	$0.836272\pi$
$24$	0	0
$25$	4.51538	0.903076
$26$	−1.25964	−0.247036
$27$	0	0
$28$	18.4982	3.49582
$29$	7.36230	1.36714	0.683572	−	0.729883i	$-0.260425\pi$
$29$	7.36230	1.36714	0.683572	+	0.729883i	$0.260425\pi$
$30$	0	0
$31$	5.08121	0.912612	0.456306	−	0.889823i	$-0.349172\pi$
$31$	5.08121	0.912612	0.456306	+	0.889823i	$0.349172\pi$
$32$	0.590936	0.104464
$33$	0	0
$34$	−17.4001	−2.98410
$35$	−14.4883	−2.44898
$36$	0	0
$37$	−5.37159	−0.883083	−0.441542	−	0.897241i	$-0.645568\pi$
$37$	−5.37159	−0.883083	−0.441542	+	0.897241i	$0.645568\pi$
$38$	12.4161	2.01415
$39$	0	0
$40$	14.5713	2.30393
$41$	−9.33468	−1.45783	−0.728916	−	0.684603i	$-0.759976\pi$
$41$	−9.33468	−1.45783	−0.728916	+	0.684603i	$0.759976\pi$
$42$	0	0
$43$	4.02303	0.613506	0.306753	−	0.951789i	$-0.400757\pi$
$43$	4.02303	0.613506	0.306753	+	0.951789i	$0.400757\pi$
$44$	3.93843	0.593740
$45$	0	0
$46$	20.3494	3.00035
$47$	−2.85519	−0.416473	−0.208236	−	0.978079i	$-0.566772\pi$
$47$	−2.85519	−0.416473	−0.208236	+	0.978079i	$0.566772\pi$
$48$	0	0
$49$	15.0603	2.15147
$50$	−11.0035	−1.55613
$51$	0	0
$52$	2.03579	0.282314
$53$	−5.63336	−0.773801	−0.386901	−	0.922121i	$-0.626454\pi$
$53$	−5.63336	−0.773801	−0.386901	+	0.922121i	$0.626454\pi$
$54$	0	0
$55$	−3.08470	−0.415941
$56$	−22.1866	−2.96481
$57$	0	0
$58$	−17.9411	−2.35578
$59$	−2.20064	−0.286499	−0.143249	−	0.989687i	$-0.545755\pi$
$59$	−2.20064	−0.286499	−0.143249	+	0.989687i	$0.545755\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−12.3823	−1.57256
$63$	0	0
$64$	−8.70877	−1.08860
$65$	−1.59450	−0.197773
$66$	0	0
$67$	−3.62908	−0.443363	−0.221681	−	0.975119i	$-0.571155\pi$
$67$	−3.62908	−0.443363	−0.221681	+	0.975119i	$0.571155\pi$
$68$	28.1216	3.41025
$69$	0	0
$70$	35.3065	4.21993
$71$	−16.6484	−1.97580	−0.987902	−	0.155081i	$-0.950436\pi$
$71$	−16.6484	−1.97580	−0.987902	+	0.155081i	$0.950436\pi$
$72$	0	0
$73$	−4.86700	−0.569640	−0.284820	−	0.958581i	$-0.591934\pi$
$73$	−4.86700	−0.569640	−0.284820	+	0.958581i	$0.591934\pi$
$74$	13.0900	1.52168
$75$	0	0
$76$	−20.0665	−2.30179
$77$	4.69684	0.535254
$78$	0	0
$79$	−1.20008	−0.135020	−0.0675099	−	0.997719i	$-0.521505\pi$
$79$	−1.20008	−0.135020	−0.0675099	+	0.997719i	$0.521505\pi$
$80$	−11.2109	−1.25342
$81$	0	0
$82$	22.7476	2.51205
$83$	0.616496	0.0676692	0.0338346	−	0.999427i	$-0.489228\pi$
$83$	0.616496	0.0676692	0.0338346	+	0.999427i	$0.489228\pi$
$84$	0	0
$85$	−22.0257	−2.38903
$86$	−9.80367	−1.05716
$87$	0	0
$88$	−4.72373	−0.503552
$89$	−17.2281	−1.82618	−0.913089	−	0.407761i	$-0.866310\pi$
$89$	−17.2281	−1.82618	−0.913089	+	0.407761i	$0.866310\pi$
$90$	0	0
$91$	2.42782	0.254505
$92$	−32.8880	−3.42881
$93$	0	0
$94$	6.95779	0.717641
$95$	15.7167	1.61250
$96$	0	0
$97$	−15.8083	−1.60509	−0.802546	−	0.596590i	$-0.796522\pi$
$97$	−15.8083	−1.60509	−0.802546	+	0.596590i	$0.796522\pi$
$98$	−36.7003	−3.70729
$99$	0	0
$100$	17.7835	1.77835
$101$	9.80037	0.975173	0.487587	−	0.873075i	$-0.337877\pi$
$101$	9.80037	0.975173	0.487587	+	0.873075i	$0.337877\pi$
$102$	0	0
$103$	−18.2536	−1.79858	−0.899289	−	0.437356i	$-0.855915\pi$
$103$	−18.2536	−1.79858	−0.899289	+	0.437356i	$0.855915\pi$
$104$	−2.44172	−0.239430
$105$	0	0
$106$	13.7279	1.33337
$107$	12.8295	1.24027	0.620136	−	0.784494i	$-0.287077\pi$
$107$	12.8295	1.24027	0.620136	+	0.784494i	$0.287077\pi$
$108$	0	0
$109$	−8.64156	−0.827711	−0.413855	−	0.910343i	$-0.635818\pi$
$109$	−8.64156	−0.827711	−0.413855	+	0.910343i	$0.635818\pi$
$110$	7.51707	0.716725
$111$	0	0
$112$	17.0700	1.61296
$113$	9.82453	0.924214	0.462107	−	0.886824i	$-0.347094\pi$
$113$	9.82453	0.924214	0.462107	+	0.886824i	$0.347094\pi$
$114$	0	0
$115$	25.7589	2.40203
$116$	28.9959	2.69220
$117$	0	0
$118$	5.36271	0.493677
$119$	33.5369	3.07432
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.43689	−0.220626
$123$	0	0
$124$	20.0120	1.79713
$125$	1.49491	0.133709
$126$	0	0
$127$	−11.6833	−1.03673	−0.518363	−	0.855161i	$-0.673458\pi$
$127$	−11.6833	−1.03673	−0.518363	+	0.855161i	$0.673458\pi$
$128$	20.0404	1.77134
$129$	0	0
$130$	3.88561	0.340791
$131$	16.8721	1.47412	0.737059	−	0.675828i	$-0.236214\pi$
$131$	16.8721	1.47412	0.737059	+	0.675828i	$0.236214\pi$
$132$	0	0
$133$	−23.9306	−2.07505
$134$	8.84367	0.763977
$135$	0	0
$136$	−33.7289	−2.89223
$137$	−16.0875	−1.37445	−0.687224	−	0.726446i	$-0.741171\pi$
$137$	−16.0875	−1.37445	−0.687224	+	0.726446i	$0.741171\pi$
$138$	0	0
$139$	10.4951	0.890180	0.445090	−	0.895486i	$-0.353172\pi$
$139$	10.4951	0.890180	0.445090	+	0.895486i	$0.353172\pi$
$140$	−57.0613	−4.82256
$141$	0	0
$142$	40.5703	3.40459
$143$	0.516905	0.0432258
$144$	0	0
$145$	−22.7105	−1.88600
$146$	11.8604	0.981570
$147$	0	0
$148$	−21.1556	−1.73898
$149$	11.5051	0.942532	0.471266	−	0.881991i	$-0.343797\pi$
$149$	11.5051	0.942532	0.471266	+	0.881991i	$0.343797\pi$
$150$	0	0
$151$	12.2524	0.997082	0.498541	−	0.866866i	$-0.333869\pi$
$151$	12.2524	0.997082	0.498541	+	0.866866i	$0.333869\pi$
$152$	24.0677	1.95215
$153$	0	0
$154$	−11.4457	−0.922319
$155$	−15.6740	−1.25897
$156$	0	0
$157$	−19.6074	−1.56484	−0.782422	−	0.622748i	$-0.786016\pi$
$157$	−19.6074	−1.56484	−0.782422	+	0.622748i	$0.786016\pi$
$158$	2.92447	0.232658
$159$	0	0
$160$	−1.82286	−0.144110
$161$	−39.2212	−3.09106
$162$	0	0
$163$	10.5893	0.829419	0.414709	−	0.909954i	$-0.363883\pi$
$163$	10.5893	0.829419	0.414709	+	0.909954i	$0.363883\pi$
$164$	−36.7640	−2.87078
$165$	0	0
$166$	−1.50233	−0.116604
$167$	2.59047	0.200456	0.100228	−	0.994964i	$-0.468043\pi$
$167$	2.59047	0.200456	0.100228	+	0.994964i	$0.468043\pi$
$168$	0	0
$169$	−12.7328	−0.979447
$170$	53.6742	4.11663
$171$	0	0
$172$	15.8444	1.20812
$173$	2.99310	0.227561	0.113781	−	0.993506i	$-0.463704\pi$
$173$	2.99310	0.227561	0.113781	+	0.993506i	$0.463704\pi$
$174$	0	0
$175$	21.2080	1.60317
$176$	3.63436	0.273950
$177$	0	0
$178$	41.9830	3.14676
$179$	−19.6746	−1.47055	−0.735274	−	0.677770i	$-0.762946\pi$
$179$	−19.6746	−1.47055	−0.735274	+	0.677770i	$0.762946\pi$
$180$	0	0
$181$	14.2699	1.06068	0.530338	−	0.847786i	$-0.322065\pi$
$181$	14.2699	1.06068	0.530338	+	0.847786i	$0.322065\pi$
$182$	−5.91633	−0.438547
$183$	0	0
$184$	39.4458	2.90798
$185$	16.5697	1.21823
$186$	0	0
$187$	7.14031	0.522151
$188$	−11.2450	−0.820124
$189$	0	0
$190$	−38.2999	−2.77857
$191$	8.13745	0.588805	0.294403	−	0.955681i	$-0.404879\pi$
$191$	8.13745	0.588805	0.294403	+	0.955681i	$0.404879\pi$
$192$	0	0
$193$	−1.47706	−0.106321	−0.0531607	−	0.998586i	$-0.516930\pi$
$193$	−1.47706	−0.106321	−0.0531607	+	0.998586i	$0.516930\pi$
$194$	38.5231	2.76580
$195$	0	0
$196$	59.3139	4.23671
$197$	−1.49825	−0.106746	−0.0533730	−	0.998575i	$-0.516997\pi$
$197$	−1.49825	−0.106746	−0.0533730	+	0.998575i	$0.516997\pi$
$198$	0	0
$199$	11.9244	0.845301	0.422651	−	0.906293i	$-0.361100\pi$
$199$	11.9244	0.845301	0.422651	+	0.906293i	$0.361100\pi$
$200$	−21.3294	−1.50822
$201$	0	0
$202$	−23.8824	−1.68036
$203$	34.5795	2.42701
$204$	0	0
$205$	28.7947	2.01111
$206$	44.4819	3.09920
$207$	0	0
$208$	1.87862	0.130259
$209$	−5.09505	−0.352432
$210$	0	0
$211$	−5.91620	−0.407288	−0.203644	−	0.979045i	$-0.565278\pi$
$211$	−5.91620	−0.407288	−0.203644	+	0.979045i	$0.565278\pi$
$212$	−22.1866	−1.52378
$213$	0	0
$214$	−31.2640	−2.13716
$215$	−12.4098	−0.846344
$216$	0	0
$217$	23.8656	1.62010
$218$	21.0585	1.42626
$219$	0	0
$220$	−12.1489	−0.819077
$221$	3.69086	0.248274
$222$	0	0
$223$	8.04047	0.538430	0.269215	−	0.963080i	$-0.413236\pi$
$223$	8.04047	0.538430	0.269215	+	0.963080i	$0.413236\pi$
$224$	2.77553	0.185448
$225$	0	0
$226$	−23.9413	−1.59255
$227$	−19.2046	−1.27465	−0.637327	−	0.770593i	$-0.719960\pi$
$227$	−19.2046	−1.27465	−0.637327	+	0.770593i	$0.719960\pi$
$228$	0	0
$229$	−0.953052	−0.0629795	−0.0314897	−	0.999504i	$-0.510025\pi$
$229$	−0.953052	−0.0629795	−0.0314897	+	0.999504i	$0.510025\pi$
$230$	−62.7717	−4.13904
$231$	0	0
$232$	−34.7776	−2.28326
$233$	21.5512	1.41187	0.705933	−	0.708279i	$-0.250528\pi$
$233$	21.5512	1.41187	0.705933	+	0.708279i	$0.250528\pi$
$234$	0	0
$235$	8.80741	0.574532
$236$	−8.66705	−0.564177
$237$	0	0
$238$	−81.7257	−5.29749
$239$	−4.72703	−0.305766	−0.152883	−	0.988244i	$-0.548856\pi$
$239$	−4.72703	−0.305766	−0.152883	+	0.988244i	$0.548856\pi$
$240$	0	0
$241$	−10.6935	−0.688827	−0.344414	−	0.938818i	$-0.611922\pi$
$241$	−10.6935	−0.688827	−0.344414	+	0.938818i	$0.611922\pi$
$242$	−2.43689	−0.156649
$243$	0	0
$244$	3.93843	0.252132
$245$	−46.4565	−2.96800
$246$	0	0
$247$	−2.63366	−0.167576
$248$	−24.0023	−1.52415
$249$	0	0
$250$	−3.64293	−0.230399
$251$	20.6863	1.30571	0.652854	−	0.757484i	$-0.273571\pi$
$251$	20.6863	1.30571	0.652854	+	0.757484i	$0.273571\pi$
$252$	0	0
$253$	−8.35055	−0.524994
$254$	28.4709	1.78642
$255$	0	0
$256$	−31.4188	−1.96367
$257$	−3.72781	−0.232534	−0.116267	−	0.993218i	$-0.537093\pi$
$257$	−3.72781	−0.232534	−0.116267	+	0.993218i	$0.537093\pi$
$258$	0	0
$259$	−25.2295	−1.56768
$260$	−6.27981	−0.389457
$261$	0	0
$262$	−41.1153	−2.54011
$263$	30.3705	1.87272	0.936361	−	0.351038i	$-0.114171\pi$
$263$	30.3705	1.87272	0.936361	+	0.351038i	$0.114171\pi$
$264$	0	0
$265$	17.3772	1.06747
$266$	58.3163	3.57560
$267$	0	0
$268$	−14.2929	−0.873077
$269$	4.46725	0.272373	0.136186	−	0.990683i	$-0.456515\pi$
$269$	4.46725	0.272373	0.136186	+	0.990683i	$0.456515\pi$
$270$	0	0
$271$	1.23735	0.0751638	0.0375819	−	0.999294i	$-0.488034\pi$
$271$	1.23735	0.0751638	0.0375819	+	0.999294i	$0.488034\pi$
$272$	25.9505	1.57348
$273$	0	0
$274$	39.2034	2.36837
$275$	4.51538	0.272288
$276$	0	0
$277$	−22.6687	−1.36203	−0.681015	−	0.732269i	$-0.738461\pi$
$277$	−22.6687	−1.36203	−0.681015	+	0.732269i	$0.738461\pi$
$278$	−25.5753	−1.53390
$279$	0	0
$280$	68.4391	4.09002
$281$	−29.3898	−1.75325	−0.876623	−	0.481177i	$-0.840209\pi$
$281$	−29.3898	−1.75325	−0.876623	+	0.481177i	$0.840209\pi$
$282$	0	0
$283$	−9.39140	−0.558261	−0.279130	−	0.960253i	$-0.590046\pi$
$283$	−9.39140	−0.558261	−0.279130	+	0.960253i	$0.590046\pi$
$284$	−65.5686	−3.89078
$285$	0	0
$286$	−1.25964	−0.0744841
$287$	−43.8435	−2.58800
$288$	0	0
$289$	33.9841	1.99906
$290$	55.3430	3.24985
$291$	0	0
$292$	−19.1683	−1.12174
$293$	23.4035	1.36725	0.683625	−	0.729834i	$-0.260402\pi$
$293$	23.4035	1.36725	0.683625	+	0.729834i	$0.260402\pi$
$294$	0	0
$295$	6.78831	0.395230
$296$	25.3739	1.47483
$297$	0	0
$298$	−28.0366	−1.62411
$299$	−4.31644	−0.249626
$300$	0	0
$301$	18.8955	1.08912
$302$	−29.8576	−1.71811
$303$	0	0
$304$	−18.5173	−1.06204
$305$	−3.08470	−0.176629
$306$	0	0
$307$	−24.0970	−1.37529	−0.687645	−	0.726047i	$-0.741355\pi$
$307$	−24.0970	−1.37529	−0.687645	+	0.726047i	$0.741355\pi$
$308$	18.4982	1.05403
$309$	0	0
$310$	38.1958	2.16938
$311$	−32.2368	−1.82798	−0.913989	−	0.405738i	$-0.867015\pi$
$311$	−32.2368	−1.82798	−0.913989	+	0.405738i	$0.867015\pi$
$312$	0	0
$313$	6.21401	0.351236	0.175618	−	0.984458i	$-0.443808\pi$
$313$	6.21401	0.351236	0.175618	+	0.984458i	$0.443808\pi$
$314$	47.7812	2.69645
$315$	0	0
$316$	−4.72644	−0.265883
$317$	19.4663	1.09334	0.546668	−	0.837349i	$-0.315896\pi$
$317$	19.4663	1.09334	0.546668	+	0.837349i	$0.315896\pi$
$318$	0	0
$319$	7.36230	0.412210
$320$	26.8639	1.50174
$321$	0	0
$322$	95.5777	5.32633
$323$	−36.3803	−2.02425
$324$	0	0
$325$	2.33402	0.129468
$326$	−25.8050	−1.42921
$327$	0	0
$328$	44.0946	2.43472
$329$	−13.4104	−0.739338
$330$	0	0
$331$	−22.6233	−1.24349	−0.621746	−	0.783219i	$-0.713576\pi$
$331$	−22.6233	−1.24349	−0.621746	+	0.783219i	$0.713576\pi$
$332$	2.42803	0.133255
$333$	0	0
$334$	−6.31268	−0.345415
$335$	11.1946	0.611628
$336$	0	0
$337$	−13.4954	−0.735142	−0.367571	−	0.929995i	$-0.619811\pi$
$337$	−13.4954	−0.735142	−0.367571	+	0.929995i	$0.619811\pi$
$338$	31.0284	1.68772
$339$	0	0
$340$	−86.7467	−4.70450
$341$	5.08121	0.275163
$342$	0	0
$343$	37.8579	2.04414
$344$	−19.0037	−1.02461
$345$	0	0
$346$	−7.29386	−0.392120
$347$	−21.3747	−1.14745	−0.573726	−	0.819047i	$-0.694503\pi$
$347$	−21.3747	−1.14745	−0.573726	+	0.819047i	$0.694503\pi$
$348$	0	0
$349$	−31.5400	−1.68829	−0.844147	−	0.536111i	$-0.819893\pi$
$349$	−31.5400	−1.68829	−0.844147	+	0.536111i	$0.819893\pi$
$350$	−51.6816	−2.76250
$351$	0	0
$352$	0.590936	0.0314970
$353$	32.0139	1.70393	0.851964	−	0.523601i	$-0.175412\pi$
$353$	32.0139	1.70393	0.851964	+	0.523601i	$0.175412\pi$
$354$	0	0
$355$	51.3554	2.72566
$356$	−67.8517	−3.59613
$357$	0	0
$358$	47.9448	2.53396
$359$	−17.8611	−0.942670	−0.471335	−	0.881954i	$-0.656228\pi$
$359$	−17.8611	−0.942670	−0.471335	+	0.881954i	$0.656228\pi$
$360$	0	0
$361$	6.95958	0.366294
$362$	−34.7743	−1.82770
$363$	0	0
$364$	9.56179	0.501174
$365$	15.0133	0.785830
$366$	0	0
$367$	5.79220	0.302351	0.151175	−	0.988507i	$-0.451694\pi$
$367$	5.79220	0.302351	0.151175	+	0.988507i	$0.451694\pi$
$368$	−30.3489	−1.58205
$369$	0	0
$370$	−40.3786	−2.09918
$371$	−26.4590	−1.37368
$372$	0	0
$373$	23.9502	1.24009	0.620046	−	0.784565i	$-0.287114\pi$
$373$	23.9502	1.24009	0.620046	+	0.784565i	$0.287114\pi$
$374$	−17.4001	−0.899740
$375$	0	0
$376$	13.4872	0.695548
$377$	3.80561	0.195999
$378$	0	0
$379$	14.8797	0.764316	0.382158	−	0.924097i	$-0.375181\pi$
$379$	14.8797	0.764316	0.382158	+	0.924097i	$0.375181\pi$
$380$	61.8992	3.17536
$381$	0	0
$382$	−19.8301	−1.01459
$383$	−0.133239	−0.00680821	−0.00340411	−	0.999994i	$-0.501084\pi$
$383$	−0.133239	−0.00680821	−0.00340411	+	0.999994i	$0.501084\pi$
$384$	0	0
$385$	−14.4883	−0.738394
$386$	3.59944	0.183207
$387$	0	0
$388$	−62.2599	−3.16077
$389$	−20.2102	−1.02470	−0.512350	−	0.858777i	$-0.671225\pi$
$389$	−20.2102	−1.02470	−0.512350	+	0.858777i	$0.671225\pi$
$390$	0	0
$391$	−59.6255	−3.01539
$392$	−71.1408	−3.59315
$393$	0	0
$394$	3.65107	0.183938
$395$	3.70190	0.186263
$396$	0	0
$397$	3.82899	0.192172	0.0960858	−	0.995373i	$-0.469368\pi$
$397$	3.82899	0.192172	0.0960858	+	0.995373i	$0.469368\pi$
$398$	−29.0585	−1.45657
$399$	0	0
$400$	16.4105	0.820526
$401$	−2.17503	−0.108616	−0.0543079	−	0.998524i	$-0.517295\pi$
$401$	−2.17503	−0.108616	−0.0543079	+	0.998524i	$0.517295\pi$
$402$	0	0
$403$	2.62650	0.130835
$404$	38.5981	1.92033
$405$	0	0
$406$	−84.2665	−4.18208
$407$	−5.37159	−0.266260
$408$	0	0
$409$	18.8552	0.932332	0.466166	−	0.884697i	$-0.345635\pi$
$409$	18.8552	0.932332	0.466166	+	0.884697i	$0.345635\pi$
$410$	−70.1695	−3.46542
$411$	0	0
$412$	−71.8903	−3.54178
$413$	−10.3360	−0.508603
$414$	0	0
$415$	−1.90171	−0.0933511
$416$	0.305458	0.0149763
$417$	0	0
$418$	12.4161	0.607290
$419$	−7.00090	−0.342016	−0.171008	−	0.985270i	$-0.554702\pi$
$419$	−7.00090	−0.342016	−0.171008	+	0.985270i	$0.554702\pi$
$420$	0	0
$421$	4.72997	0.230525	0.115262	−	0.993335i	$-0.463229\pi$
$421$	4.72997	0.230525	0.115262	+	0.993335i	$0.463229\pi$
$422$	14.4171	0.701814
$423$	0	0
$424$	26.6105	1.29232
$425$	32.2412	1.56393
$426$	0	0
$427$	4.69684	0.227296
$428$	50.5279	2.44236
$429$	0	0
$430$	30.2414	1.45837
$431$	11.6039	0.558939	0.279470	−	0.960155i	$-0.409841\pi$
$431$	11.6039	0.558939	0.279470	+	0.960155i	$0.409841\pi$
$432$	0	0
$433$	−14.1807	−0.681481	−0.340740	−	0.940157i	$-0.610678\pi$
$433$	−14.1807	−0.681481	−0.340740	+	0.940157i	$0.610678\pi$
$434$	−58.1579	−2.79167
$435$	0	0
$436$	−34.0341	−1.62994
$437$	42.5465	2.03527
$438$	0	0
$439$	−29.9385	−1.42889	−0.714443	−	0.699694i	$-0.753320\pi$
$439$	−29.9385	−1.42889	−0.714443	+	0.699694i	$0.753320\pi$
$440$	14.5713	0.694660
$441$	0	0
$442$	−8.99422	−0.427811
$443$	−2.20160	−0.104601	−0.0523005	−	0.998631i	$-0.516655\pi$
$443$	−2.20160	−0.104601	−0.0523005	+	0.998631i	$0.516655\pi$
$444$	0	0
$445$	53.1436	2.51925
$446$	−19.5937	−0.927790
$447$	0	0
$448$	−40.9037	−1.93252
$449$	−4.84586	−0.228690	−0.114345	−	0.993441i	$-0.536477\pi$
$449$	−4.84586	−0.228690	−0.114345	+	0.993441i	$0.536477\pi$
$450$	0	0
$451$	−9.33468	−0.439553
$452$	38.6932	1.81997
$453$	0	0
$454$	46.7995	2.19641
$455$	−7.48910	−0.351094
$456$	0	0
$457$	31.4841	1.47276	0.736381	−	0.676567i	$-0.236533\pi$
$457$	31.4841	1.47276	0.736381	+	0.676567i	$0.236533\pi$
$458$	2.32248	0.108522
$459$	0	0
$460$	101.450	4.73012
$461$	10.1872	0.474467	0.237233	−	0.971453i	$-0.423759\pi$
$461$	10.1872	0.474467	0.237233	+	0.971453i	$0.423759\pi$
$462$	0	0
$463$	40.1188	1.86448	0.932239	−	0.361843i	$-0.117852\pi$
$463$	40.1188	1.86448	0.932239	+	0.361843i	$0.117852\pi$
$464$	26.7573	1.24217
$465$	0	0
$466$	−52.5179	−2.43284
$467$	−3.47807	−0.160946	−0.0804729	−	0.996757i	$-0.525643\pi$
$467$	−3.47807	−0.160946	−0.0804729	+	0.996757i	$0.525643\pi$
$468$	0	0
$469$	−17.0452	−0.787075
$470$	−21.4627	−0.990000
$471$	0	0
$472$	10.3952	0.478479
$473$	4.02303	0.184979
$474$	0	0
$475$	−23.0061	−1.05559
$476$	132.083	6.05400
$477$	0	0
$478$	11.5192	0.526878
$479$	−8.52107	−0.389338	−0.194669	−	0.980869i	$-0.562363\pi$
$479$	−8.52107	−0.389338	−0.194669	+	0.980869i	$0.562363\pi$
$480$	0	0
$481$	−2.77660	−0.126602
$482$	26.0588	1.18695
$483$	0	0
$484$	3.93843	0.179019
$485$	48.7639	2.21426
$486$	0	0
$487$	5.98983	0.271425	0.135713	−	0.990748i	$-0.456668\pi$
$487$	5.98983	0.271425	0.135713	+	0.990748i	$0.456668\pi$
$488$	−4.72373	−0.213833
$489$	0	0
$490$	113.209	5.11428
$491$	−7.12577	−0.321581	−0.160791	−	0.986989i	$-0.551404\pi$
$491$	−7.12577	−0.321581	−0.160791	+	0.986989i	$0.551404\pi$
$492$	0	0
$493$	52.5691	2.36759
$494$	6.41793	0.288756
$495$	0	0
$496$	18.4669	0.829190
$497$	−78.1949	−3.50752
$498$	0	0
$499$	−36.6701	−1.64158	−0.820790	−	0.571230i	$-0.806467\pi$
$499$	−36.6701	−1.64158	−0.820790	+	0.571230i	$0.806467\pi$
$500$	5.88760	0.263302
$501$	0	0
$502$	−50.4102	−2.24992
$503$	−21.2520	−0.947581	−0.473791	−	0.880638i	$-0.657115\pi$
$503$	−21.2520	−0.947581	−0.473791	+	0.880638i	$0.657115\pi$
$504$	0	0
$505$	−30.2312	−1.34527
$506$	20.3494	0.904639
$507$	0	0
$508$	−46.0138	−2.04153
$509$	−2.22831	−0.0987682	−0.0493841	−	0.998780i	$-0.515726\pi$
$509$	−2.22831	−0.0987682	−0.0493841	+	0.998780i	$0.515726\pi$
$510$	0	0
$511$	−22.8595	−1.01125
$512$	36.4832	1.61234
$513$	0	0
$514$	9.08426	0.400690
$515$	56.3068	2.48117
$516$	0	0
$517$	−2.85519	−0.125571
$518$	61.4814	2.70134
$519$	0	0
$520$	7.53198	0.330299
$521$	−13.6331	−0.597277	−0.298638	−	0.954366i	$-0.596532\pi$
$521$	−13.6331	−0.597277	−0.298638	+	0.954366i	$0.596532\pi$
$522$	0	0
$523$	18.4157	0.805262	0.402631	−	0.915362i	$-0.368096\pi$
$523$	18.4157	0.805262	0.402631	+	0.915362i	$0.368096\pi$
$524$	66.4494	2.90286
$525$	0	0
$526$	−74.0094	−3.22696
$527$	36.2814	1.58044
$528$	0	0
$529$	46.7316	2.03181
$530$	−42.3464	−1.83941
$531$	0	0
$532$	−94.2491	−4.08622
$533$	−4.82514	−0.209000
$534$	0	0
$535$	−39.5751	−1.71098
$536$	17.1428	0.740457
$537$	0	0
$538$	−10.8862	−0.469337
$539$	15.0603	0.648693
$540$	0	0
$541$	16.8155	0.722953	0.361477	−	0.932381i	$-0.382273\pi$
$541$	16.8155	0.722953	0.361477	+	0.932381i	$0.382273\pi$
$542$	−3.01529	−0.129518
$543$	0	0
$544$	4.21947	0.180908
$545$	26.6566	1.14184
$546$	0	0
$547$	−7.74048	−0.330959	−0.165480	−	0.986213i	$-0.552917\pi$
$547$	−7.74048	−0.330959	−0.165480	+	0.986213i	$0.552917\pi$
$548$	−63.3595	−2.70658
$549$	0	0
$550$	−11.0035	−0.469190
$551$	−37.5113	−1.59804
$552$	0	0
$553$	−5.63660	−0.239692
$554$	55.2411	2.34697
$555$	0	0
$556$	41.3340	1.75295
$557$	14.9601	0.633879	0.316940	−	0.948446i	$-0.397345\pi$
$557$	14.9601	0.633879	0.316940	+	0.948446i	$0.397345\pi$
$558$	0	0
$559$	2.07952	0.0879544
$560$	−52.6559	−2.22512
$561$	0	0
$562$	71.6196	3.02109
$563$	24.6634	1.03944	0.519719	−	0.854337i	$-0.326037\pi$
$563$	24.6634	1.03944	0.519719	+	0.854337i	$0.326037\pi$
$564$	0	0
$565$	−30.3057	−1.27497
$566$	22.8858	0.961962
$567$	0	0
$568$	78.6427	3.29977
$569$	−39.1435	−1.64098	−0.820490	−	0.571661i	$-0.806299\pi$
$569$	−39.1435	−1.64098	−0.820490	+	0.571661i	$0.806299\pi$
$570$	0	0
$571$	6.74022	0.282069	0.141035	−	0.990005i	$-0.454957\pi$
$571$	6.74022	0.282069	0.141035	+	0.990005i	$0.454957\pi$
$572$	2.03579	0.0851208
$573$	0	0
$574$	106.842	4.45949
$575$	−37.7059	−1.57244
$576$	0	0
$577$	−4.83163	−0.201143	−0.100572	−	0.994930i	$-0.532067\pi$
$577$	−4.83163	−0.201143	−0.100572	+	0.994930i	$0.532067\pi$
$578$	−82.8154	−3.44467
$579$	0	0
$580$	−89.4436	−3.71395
$581$	2.89558	0.120129
$582$	0	0
$583$	−5.63336	−0.233310
$584$	22.9904	0.951351
$585$	0	0
$586$	−57.0318	−2.35596
$587$	9.81000	0.404902	0.202451	−	0.979292i	$-0.435109\pi$
$587$	9.81000	0.404902	0.202451	+	0.979292i	$0.435109\pi$
$588$	0	0
$589$	−25.8890	−1.06674
$590$	−16.5423	−0.681038
$591$	0	0
$592$	−19.5223	−0.802361
$593$	−47.4367	−1.94799	−0.973997	−	0.226562i	$-0.927251\pi$
$593$	−47.4367	−1.94799	−0.973997	+	0.226562i	$0.927251\pi$
$594$	0	0
$595$	−103.451	−4.24109
$596$	45.3119	1.85605
$597$	0	0
$598$	10.5187	0.430141
$599$	23.4046	0.956288	0.478144	−	0.878282i	$-0.341310\pi$
$599$	23.4046	0.956288	0.478144	+	0.878282i	$0.341310\pi$
$600$	0	0
$601$	−15.1003	−0.615954	−0.307977	−	0.951394i	$-0.599652\pi$
$601$	−15.1003	−0.615954	−0.307977	+	0.951394i	$0.599652\pi$
$602$	−46.0463	−1.87671
$603$	0	0
$604$	48.2550	1.96347
$605$	−3.08470	−0.125411
$606$	0	0
$607$	−9.10047	−0.369377	−0.184688	−	0.982797i	$-0.559128\pi$
$607$	−9.10047	−0.369377	−0.184688	+	0.982797i	$0.559128\pi$
$608$	−3.01085	−0.122106
$609$	0	0
$610$	7.51707	0.304357
$611$	−1.47586	−0.0597070
$612$	0	0
$613$	1.22644	0.0495354	0.0247677	−	0.999693i	$-0.492115\pi$
$613$	1.22644	0.0495354	0.0247677	+	0.999693i	$0.492115\pi$
$614$	58.7218	2.36982
$615$	0	0
$616$	−22.1866	−0.893924
$617$	−18.1079	−0.728995	−0.364497	−	0.931204i	$-0.618759\pi$
$617$	−18.1079	−0.728995	−0.364497	+	0.931204i	$0.618759\pi$
$618$	0	0
$619$	−29.0561	−1.16787	−0.583933	−	0.811802i	$-0.698487\pi$
$619$	−29.0561	−1.16787	−0.583933	+	0.811802i	$0.698487\pi$
$620$	−61.7310	−2.47917
$621$	0	0
$622$	78.5574	3.14987
$623$	−80.9177	−3.24190
$624$	0	0
$625$	−27.1882	−1.08753
$626$	−15.1428	−0.605230
$627$	0	0
$628$	−77.2225	−3.08151
$629$	−38.3548	−1.52931
$630$	0	0
$631$	0.530754	0.0211290	0.0105645	−	0.999944i	$-0.496637\pi$
$631$	0.530754	0.0211290	0.0105645	+	0.999944i	$0.496637\pi$
$632$	5.66887	0.225496
$633$	0	0
$634$	−47.4372	−1.88397
$635$	36.0395	1.43018
$636$	0	0
$637$	7.78474	0.308443
$638$	−17.9411	−0.710295
$639$	0	0
$640$	−61.8187	−2.44360
$641$	−30.0454	−1.18672	−0.593361	−	0.804936i	$-0.702200\pi$
$641$	−30.0454	−1.18672	−0.593361	+	0.804936i	$0.702200\pi$
$642$	0	0
$643$	4.68213	0.184645	0.0923226	−	0.995729i	$-0.470571\pi$
$643$	4.68213	0.184645	0.0923226	+	0.995729i	$0.470571\pi$
$644$	−154.470	−6.08696
$645$	0	0
$646$	88.6547	3.48807
$647$	−40.9451	−1.60972	−0.804860	−	0.593465i	$-0.797760\pi$
$647$	−40.9451	−1.60972	−0.804860	+	0.593465i	$0.797760\pi$
$648$	0	0
$649$	−2.20064	−0.0863826
$650$	−5.68775	−0.223092
$651$	0	0
$652$	41.7052	1.63330
$653$	−22.7116	−0.888773	−0.444386	−	0.895835i	$-0.646578\pi$
$653$	−22.7116	−0.888773	−0.444386	+	0.895835i	$0.646578\pi$
$654$	0	0
$655$	−52.0453	−2.03358
$656$	−33.9256	−1.32457
$657$	0	0
$658$	32.6796	1.27398
$659$	−11.0384	−0.429997	−0.214998	−	0.976614i	$-0.568975\pi$
$659$	−11.0384	−0.429997	−0.214998	+	0.976614i	$0.568975\pi$
$660$	0	0
$661$	10.7029	0.416293	0.208146	−	0.978098i	$-0.433257\pi$
$661$	10.7029	0.416293	0.208146	+	0.978098i	$0.433257\pi$
$662$	55.1306	2.14271
$663$	0	0
$664$	−2.91216	−0.113014
$665$	73.8189	2.86257
$666$	0	0
$667$	−61.4792	−2.38049
$668$	10.2024	0.394742
$669$	0	0
$670$	−27.2801	−1.05392
$671$	1.00000	0.0386046
$672$	0	0
$673$	8.26589	0.318627	0.159313	−	0.987228i	$-0.449072\pi$
$673$	8.26589	0.318627	0.159313	+	0.987228i	$0.449072\pi$
$674$	32.8868	1.26675
$675$	0	0
$676$	−50.1473	−1.92874
$677$	−23.6498	−0.908937	−0.454469	−	0.890763i	$-0.650171\pi$
$677$	−23.6498	−0.908937	−0.454469	+	0.890763i	$0.650171\pi$
$678$	0	0
$679$	−74.2491	−2.84942
$680$	104.044	3.98989
$681$	0	0
$682$	−12.3823	−0.474144
$683$	13.1886	0.504647	0.252323	−	0.967643i	$-0.418805\pi$
$683$	13.1886	0.504647	0.252323	+	0.967643i	$0.418805\pi$
$684$	0	0
$685$	49.6251	1.89608
$686$	−92.2555	−3.52233
$687$	0	0
$688$	14.6211	0.557425
$689$	−2.91191	−0.110935
$690$	0	0
$691$	35.5040	1.35064	0.675318	−	0.737526i	$-0.264006\pi$
$691$	35.5040	1.35064	0.675318	+	0.737526i	$0.264006\pi$
$692$	11.7881	0.448117
$693$	0	0
$694$	52.0877	1.97722
$695$	−32.3741	−1.22802
$696$	0	0
$697$	−66.6526	−2.52465
$698$	76.8594	2.90917
$699$	0	0
$700$	83.5262	3.15699
$701$	−50.7441	−1.91658	−0.958290	−	0.285798i	$-0.907741\pi$
$701$	−50.7441	−1.91658	−0.958290	+	0.285798i	$0.907741\pi$
$702$	0	0
$703$	27.3685	1.03222
$704$	−8.70877	−0.328224
$705$	0	0
$706$	−78.0143	−2.93611
$707$	46.0308	1.73117
$708$	0	0
$709$	−39.4580	−1.48188	−0.740939	−	0.671573i	$-0.765619\pi$
$709$	−39.4580	−1.48188	−0.740939	+	0.671573i	$0.765619\pi$
$710$	−125.147	−4.69670
$711$	0	0
$712$	81.3811	3.04988
$713$	−42.4309	−1.58905
$714$	0	0
$715$	−1.59450	−0.0596308
$716$	−77.4869	−2.89582
$717$	0	0
$718$	43.5254	1.62435
$719$	8.90606	0.332140	0.166070	−	0.986114i	$-0.446892\pi$
$719$	8.90606	0.332140	0.166070	+	0.986114i	$0.446892\pi$
$720$	0	0
$721$	−85.7340	−3.19290
$722$	−16.9597	−0.631175
$723$	0	0
$724$	56.2012	2.08870
$725$	33.2436	1.23464
$726$	0	0
$727$	−18.9221	−0.701780	−0.350890	−	0.936417i	$-0.614121\pi$
$727$	−18.9221	−0.701780	−0.350890	+	0.936417i	$0.614121\pi$
$728$	−11.4684	−0.425046
$729$	0	0
$730$	−36.5856	−1.35410
$731$	28.7257	1.06246
$732$	0	0
$733$	5.56864	0.205682	0.102841	−	0.994698i	$-0.467207\pi$
$733$	5.56864	0.205682	0.102841	+	0.994698i	$0.467207\pi$
$734$	−14.1150	−0.520993
$735$	0	0
$736$	−4.93464	−0.181893
$737$	−3.62908	−0.133679
$738$	0	0
$739$	3.10008	0.114038	0.0570191	−	0.998373i	$-0.481840\pi$
$739$	3.10008	0.114038	0.0570191	+	0.998373i	$0.481840\pi$
$740$	65.2587	2.39896
$741$	0	0
$742$	64.4776	2.36705
$743$	−24.0693	−0.883018	−0.441509	−	0.897257i	$-0.645557\pi$
$743$	−24.0693	−0.883018	−0.441509	+	0.897257i	$0.645557\pi$
$744$	0	0
$745$	−35.4897	−1.30024
$746$	−58.3639	−2.13685
$747$	0	0
$748$	28.1216	1.02823
$749$	60.2580	2.20178
$750$	0	0
$751$	−26.7291	−0.975358	−0.487679	−	0.873023i	$-0.662156\pi$
$751$	−26.7291	−0.975358	−0.487679	+	0.873023i	$0.662156\pi$
$752$	−10.3768	−0.378403
$753$	0	0
$754$	−9.27385	−0.337734
$755$	−37.7948	−1.37549
$756$	0	0
$757$	−35.9366	−1.30614	−0.653069	−	0.757299i	$-0.726519\pi$
$757$	−35.9366	−1.30614	−0.653069	+	0.757299i	$0.726519\pi$
$758$	−36.2601	−1.31702
$759$	0	0
$760$	−74.2416	−2.69303
$761$	16.4162	0.595087	0.297544	−	0.954708i	$-0.403833\pi$
$761$	16.4162	0.595087	0.297544	+	0.954708i	$0.403833\pi$
$762$	0	0
$763$	−40.5880	−1.46938
$764$	32.0488	1.15948
$765$	0	0
$766$	0.324690	0.0117315
$767$	−1.13752	−0.0410735
$768$	0	0
$769$	26.4352	0.953277	0.476638	−	0.879099i	$-0.341855\pi$
$769$	26.4352	0.953277	0.476638	+	0.879099i	$0.341855\pi$
$770$	35.3065	1.27236
$771$	0	0
$772$	−5.81731	−0.209370
$773$	−45.9815	−1.65384	−0.826920	−	0.562320i	$-0.809909\pi$
$773$	−45.9815	−1.65384	−0.826920	+	0.562320i	$0.809909\pi$
$774$	0	0
$775$	22.9436	0.824158
$776$	74.6743	2.68065
$777$	0	0
$778$	49.2501	1.76570
$779$	47.5607	1.70404
$780$	0	0
$781$	−16.6484	−0.595727
$782$	145.301	5.19594
$783$	0	0
$784$	54.7345	1.95481
$785$	60.4831	2.15873
$786$	0	0
$787$	35.3287	1.25933	0.629667	−	0.776865i	$-0.283191\pi$
$787$	35.3287	1.25933	0.629667	+	0.776865i	$0.283191\pi$
$788$	−5.90076	−0.210206
$789$	0	0
$790$	−9.02111	−0.320957
$791$	46.1442	1.64070
$792$	0	0
$793$	0.516905	0.0183558
$794$	−9.33083	−0.331139
$795$	0	0
$796$	46.9635	1.66458
$797$	9.52814	0.337504	0.168752	−	0.985659i	$-0.446026\pi$
$797$	9.52814	0.337504	0.168752	+	0.985659i	$0.446026\pi$
$798$	0	0
$799$	−20.3870	−0.721239
$800$	2.66830	0.0943386
$801$	0	0
$802$	5.30030	0.187160
$803$	−4.86700	−0.171753
$804$	0	0
$805$	120.986	4.26418
$806$	−6.40049	−0.225448
$807$	0	0
$808$	−46.2944	−1.62863
$809$	−12.6739	−0.445592	−0.222796	−	0.974865i	$-0.571518\pi$
$809$	−12.6739	−0.445592	−0.222796	+	0.974865i	$0.571518\pi$
$810$	0	0
$811$	−41.1799	−1.44602	−0.723011	−	0.690836i	$-0.757243\pi$
$811$	−41.1799	−1.44602	−0.723011	+	0.690836i	$0.757243\pi$
$812$	136.189	4.77930
$813$	0	0
$814$	13.0900	0.458803
$815$	−32.6649	−1.14420
$816$	0	0
$817$	−20.4975	−0.717118
$818$	−45.9481	−1.60654
$819$	0	0
$820$	113.406	3.96031
$821$	4.33526	0.151302	0.0756508	−	0.997134i	$-0.475897\pi$
$821$	4.33526	0.151302	0.0756508	+	0.997134i	$0.475897\pi$
$822$	0	0
$823$	−14.4880	−0.505020	−0.252510	−	0.967594i	$-0.581256\pi$
$823$	−14.4880	−0.505020	−0.252510	+	0.967594i	$0.581256\pi$
$824$	86.2250	3.00379
$825$	0	0
$826$	25.1878	0.876395
$827$	−36.2463	−1.26041	−0.630203	−	0.776430i	$-0.717029\pi$
$827$	−36.2463	−1.26041	−0.630203	+	0.776430i	$0.717029\pi$
$828$	0	0
$829$	−13.6514	−0.474132	−0.237066	−	0.971494i	$-0.576186\pi$
$829$	−13.6514	−0.474132	−0.237066	+	0.971494i	$0.576186\pi$
$830$	4.63425	0.160857
$831$	0	0
$832$	−4.50160	−0.156065
$833$	107.535	3.72587
$834$	0	0
$835$	−7.99082	−0.276534
$836$	−20.0665	−0.694015
$837$	0	0
$838$	17.0604	0.589342
$839$	−43.0401	−1.48591	−0.742955	−	0.669341i	$-0.766576\pi$
$839$	−43.0401	−1.48591	−0.742955	+	0.669341i	$0.766576\pi$
$840$	0	0
$841$	25.2035	0.869085
$842$	−11.5264	−0.397227
$843$	0	0
$844$	−23.3005	−0.802037
$845$	39.2769	1.35117
$846$	0	0
$847$	4.69684	0.161385
$848$	−20.4737	−0.703068
$849$	0	0
$850$	−78.5683	−2.69487
$851$	44.8557	1.53763
$852$	0	0
$853$	45.8832	1.57101	0.785506	−	0.618855i	$-0.212403\pi$
$853$	45.8832	1.57101	0.785506	+	0.618855i	$0.212403\pi$
$854$	−11.4457	−0.391663
$855$	0	0
$856$	−60.6030	−2.07137
$857$	−38.2011	−1.30492	−0.652462	−	0.757822i	$-0.726264\pi$
$857$	−38.2011	−1.30492	−0.652462	+	0.757822i	$0.726264\pi$
$858$	0	0
$859$	−4.71415	−0.160845	−0.0804224	−	0.996761i	$-0.525627\pi$
$859$	−4.71415	−0.160845	−0.0804224	+	0.996761i	$0.525627\pi$
$860$	−48.8753	−1.66663
$861$	0	0
$862$	−28.2774	−0.963131
$863$	−10.7165	−0.364795	−0.182397	−	0.983225i	$-0.558386\pi$
$863$	−10.7165	−0.364795	−0.182397	+	0.983225i	$0.558386\pi$
$864$	0	0
$865$	−9.23283	−0.313926
$866$	34.5568	1.17429
$867$	0	0
$868$	93.9930	3.19033
$869$	−1.20008	−0.0407100
$870$	0	0
$871$	−1.87589	−0.0635621
$872$	40.8204	1.38235
$873$	0	0
$874$	−103.681	−3.50706
$875$	7.02136	0.237365
$876$	0	0
$877$	6.09552	0.205831	0.102915	−	0.994690i	$-0.467183\pi$
$877$	6.09552	0.205831	0.102915	+	0.994690i	$0.467183\pi$
$878$	72.9567	2.46217
$879$	0	0
$880$	−11.2109	−0.377920
$881$	−35.4378	−1.19393	−0.596965	−	0.802267i	$-0.703627\pi$
$881$	−35.4378	−1.19393	−0.596965	+	0.802267i	$0.703627\pi$
$882$	0	0
$883$	−16.2215	−0.545897	−0.272948	−	0.962029i	$-0.587999\pi$
$883$	−16.2215	−0.545897	−0.272948	+	0.962029i	$0.587999\pi$
$884$	14.5362	0.488905
$885$	0	0
$886$	5.36505	0.180242
$887$	33.5907	1.12787	0.563933	−	0.825820i	$-0.309288\pi$
$887$	33.5907	1.12787	0.563933	+	0.825820i	$0.309288\pi$
$888$	0	0
$889$	−54.8746	−1.84043
$890$	−129.505	−4.34102
$891$	0	0
$892$	31.6668	1.06028
$893$	14.5474	0.486809
$894$	0	0
$895$	60.6902	2.02865
$896$	94.1266	3.14455
$897$	0	0
$898$	11.8088	0.394065
$899$	37.4094	1.24767
$900$	0	0
$901$	−40.2239	−1.34005
$902$	22.7476	0.757412
$903$	0	0
$904$	−46.4084	−1.54352
$905$	−44.0185	−1.46322
$906$	0	0
$907$	−0.236217	−0.00784347	−0.00392174	−	0.999992i	$-0.501248\pi$
$907$	−0.236217	−0.00784347	−0.00392174	+	0.999992i	$0.501248\pi$
$908$	−75.6360	−2.51007
$909$	0	0
$910$	18.2501	0.604985
$911$	−20.6660	−0.684696	−0.342348	−	0.939573i	$-0.611222\pi$
$911$	−20.6660	−0.684696	−0.342348	+	0.939573i	$0.611222\pi$
$912$	0	0
$913$	0.616496	0.0204030
$914$	−76.7232	−2.53778
$915$	0	0
$916$	−3.75353	−0.124020
$917$	79.2454	2.61691
$918$	0	0
$919$	16.6639	0.549692	0.274846	−	0.961488i	$-0.411373\pi$
$919$	16.6639	0.549692	0.274846	+	0.961488i	$0.411373\pi$
$920$	−121.678	−4.01162
$921$	0	0
$922$	−24.8251	−0.817573
$923$	−8.60565	−0.283258
$924$	0	0
$925$	−24.2547	−0.797491
$926$	−97.7650	−3.21276
$927$	0	0
$928$	4.35065	0.142817
$929$	9.44702	0.309947	0.154973	−	0.987919i	$-0.450471\pi$
$929$	9.44702	0.309947	0.154973	+	0.987919i	$0.450471\pi$
$930$	0	0
$931$	−76.7330	−2.51482
$932$	84.8778	2.78026
$933$	0	0
$934$	8.47567	0.277332
$935$	−22.0257	−0.720318
$936$	0	0
$937$	−12.8282	−0.419079	−0.209540	−	0.977800i	$-0.567197\pi$
$937$	−12.8282	−0.419079	−0.209540	+	0.977800i	$0.567197\pi$
$938$	41.5373	1.35624
$939$	0	0
$940$	34.6874	1.13138
$941$	−11.7037	−0.381530	−0.190765	−	0.981636i	$-0.561097\pi$
$941$	−11.7037	−0.381530	−0.190765	+	0.981636i	$0.561097\pi$
$942$	0	0
$943$	77.9497	2.53839
$944$	−7.99791	−0.260310
$945$	0	0
$946$	−9.80367	−0.318745
$947$	52.5003	1.70603	0.853015	−	0.521887i	$-0.174772\pi$
$947$	52.5003	1.70603	0.853015	+	0.521887i	$0.174772\pi$
$948$	0	0
$949$	−2.51578	−0.0816656
$950$	56.0633	1.81893
$951$	0	0
$952$	−158.419	−5.13440
$953$	10.6622	0.345382	0.172691	−	0.984976i	$-0.444754\pi$
$953$	10.6622	0.345382	0.172691	+	0.984976i	$0.444754\pi$
$954$	0	0
$955$	−25.1016	−0.812269
$956$	−18.6171	−0.602119
$957$	0	0
$958$	20.7649	0.670884
$959$	−75.5604	−2.43997
$960$	0	0
$961$	−5.18132	−0.167139
$962$	6.76626	0.218153
$963$	0	0
$964$	−42.1155	−1.35645
$965$	4.55630	0.146673
$966$	0	0
$967$	−37.5446	−1.20735	−0.603677	−	0.797229i	$-0.706298\pi$
$967$	−37.5446	−1.20735	−0.603677	+	0.797229i	$0.706298\pi$
$968$	−4.72373	−0.151827
$969$	0	0
$970$	−118.832	−3.81548
$971$	−22.2280	−0.713332	−0.356666	−	0.934232i	$-0.616087\pi$
$971$	−22.2280	−0.713332	−0.356666	+	0.934232i	$0.616087\pi$
$972$	0	0
$973$	49.2936	1.58028
$974$	−14.5966	−0.467704
$975$	0	0
$976$	3.63436	0.116333
$977$	−17.4662	−0.558792	−0.279396	−	0.960176i	$-0.590134\pi$
$977$	−17.4662	−0.558792	−0.279396	+	0.960176i	$0.590134\pi$
$978$	0	0
$979$	−17.2281	−0.550613
$980$	−182.966	−5.84462
$981$	0	0
$982$	17.3647	0.554130
$983$	−4.15210	−0.132431	−0.0662157	−	0.997805i	$-0.521093\pi$
$983$	−4.15210	−0.132431	−0.0662157	+	0.997805i	$0.521093\pi$
$984$	0	0
$985$	4.62166	0.147258
$986$	−128.105	−4.07970
$987$	0	0
$988$	−10.3725	−0.329992
$989$	−33.5945	−1.06824
$990$	0	0
$991$	38.7236	1.23010	0.615048	−	0.788490i	$-0.289137\pi$
$991$	38.7236	1.23010	0.615048	+	0.788490i	$0.289137\pi$
$992$	3.00267	0.0953348
$993$	0	0
$994$	190.552	6.04395
$995$	−36.7833	−1.16611
$996$	0	0
$997$	44.8317	1.41983	0.709917	−	0.704285i	$-0.248732\pi$
$997$	44.8317	1.41983	0.709917	+	0.704285i	$0.248732\pi$
$998$	89.3610	2.82867
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.4	✓	25
3.2	odd	2		6039.2.a.p.1.22	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.4	✓	25	1.1	even	1	trivial
6039.2.a.p.1.22	yes	25	3.2	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43689 q^{2} +3.93843 q^{4} -3.08470 q^{5} +4.69684 q^{7} -4.72373 q^{8} +O(q^{10})\)	\(q-2.43689 q^{2} +3.93843 q^{4} -3.08470 q^{5} +4.69684 q^{7} -4.72373 q^{8} +7.51707 q^{10} +1.00000 q^{11} +0.516905 q^{13} -11.4457 q^{14} +3.63436 q^{16} +7.14031 q^{17} -5.09505 q^{19} -12.1489 q^{20} -2.43689 q^{22} -8.35055 q^{23} +4.51538 q^{25} -1.25964 q^{26} +18.4982 q^{28} +7.36230 q^{29} +5.08121 q^{31} +0.590936 q^{32} -17.4001 q^{34} -14.4883 q^{35} -5.37159 q^{37} +12.4161 q^{38} +14.5713 q^{40} -9.33468 q^{41} +4.02303 q^{43} +3.93843 q^{44} +20.3494 q^{46} -2.85519 q^{47} +15.0603 q^{49} -11.0035 q^{50} +2.03579 q^{52} -5.63336 q^{53} -3.08470 q^{55} -22.1866 q^{56} -17.9411 q^{58} -2.20064 q^{59} +1.00000 q^{61} -12.3823 q^{62} -8.70877 q^{64} -1.59450 q^{65} -3.62908 q^{67} +28.1216 q^{68} +35.3065 q^{70} -16.6484 q^{71} -4.86700 q^{73} +13.0900 q^{74} -20.0665 q^{76} +4.69684 q^{77} -1.20008 q^{79} -11.2109 q^{80} +22.7476 q^{82} +0.616496 q^{83} -22.0257 q^{85} -9.80367 q^{86} -4.72373 q^{88} -17.2281 q^{89} +2.42782 q^{91} -32.8880 q^{92} +6.95779 q^{94} +15.7167 q^{95} -15.8083 q^{97} -36.7003 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Introduction

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