LMFDB - Embedded newform 6039.2.a.o.1.11

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:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q+0.0238689 q^{2} -1.99943 q^{4} -2.56388 q^{5} +0.894211 q^{7} -0.0954618 q^{8} +O(q^{10})$	\(q+0.0238689 q^{2} -1.99943 q^{4} -2.56388 q^{5} +0.894211 q^{7} -0.0954618 q^{8} -0.0611970 q^{10} +1.00000 q^{11} -2.49879 q^{13} +0.0213438 q^{14} +3.99658 q^{16} +2.00462 q^{17} -3.11648 q^{19} +5.12631 q^{20} +0.0238689 q^{22} +0.753762 q^{23} +1.57350 q^{25} -0.0596433 q^{26} -1.78791 q^{28} -8.63363 q^{29} +5.83697 q^{31} +0.286318 q^{32} +0.0478481 q^{34} -2.29265 q^{35} -3.01770 q^{37} -0.0743869 q^{38} +0.244753 q^{40} +4.20757 q^{41} -8.27693 q^{43} -1.99943 q^{44} +0.0179914 q^{46} +3.17581 q^{47} -6.20039 q^{49} +0.0375575 q^{50} +4.99616 q^{52} -8.04781 q^{53} -2.56388 q^{55} -0.0853630 q^{56} -0.206075 q^{58} +4.47017 q^{59} -1.00000 q^{61} +0.139322 q^{62} -7.98633 q^{64} +6.40661 q^{65} +0.916574 q^{67} -4.00810 q^{68} -0.0547230 q^{70} +8.16659 q^{71} -3.98703 q^{73} -0.0720291 q^{74} +6.23119 q^{76} +0.894211 q^{77} +5.34694 q^{79} -10.2468 q^{80} +0.100430 q^{82} -0.882213 q^{83} -5.13962 q^{85} -0.197561 q^{86} -0.0954618 q^{88} -14.4054 q^{89} -2.23445 q^{91} -1.50709 q^{92} +0.0758029 q^{94} +7.99029 q^{95} -17.2558 q^{97} -0.147996 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 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 $

$2$	0.0238689	0.0168778	0.00843892	−	0.999964i	$-0.497314\pi$
$2$	0.0238689	0.0168778	0.00843892	+	0.999964i	$0.497314\pi$
$3$	0	0
$3$	0	0
$4$	−1.99943	−0.999715
$5$	−2.56388	−1.14660	−0.573302	−	0.819344i	$-0.694338\pi$
$5$	−2.56388	−1.14660	−0.573302	+	0.819344i	$0.694338\pi$
$6$	0	0
$7$	0.894211	0.337980	0.168990	−	0.985618i	$-0.445949\pi$
$7$	0.894211	0.337980	0.168990	+	0.985618i	$0.445949\pi$
$8$	−0.0954618	−0.0337509
$9$	0	0
$10$	−0.0611970	−0.0193522
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.49879	−0.693041	−0.346520	−	0.938042i	$-0.612637\pi$
$13$	−2.49879	−0.693041	−0.346520	+	0.938042i	$0.612637\pi$
$14$	0.0213438	0.00570437
$15$	0	0
$16$	3.99658	0.999145
$17$	2.00462	0.486192	0.243096	−	0.970002i	$-0.421837\pi$
$17$	2.00462	0.486192	0.243096	+	0.970002i	$0.421837\pi$
$18$	0	0
$19$	−3.11648	−0.714970	−0.357485	−	0.933919i	$-0.616366\pi$
$19$	−3.11648	−0.714970	−0.357485	+	0.933919i	$0.616366\pi$
$20$	5.12631	1.14628
$21$	0	0
$22$	0.0238689	0.00508886
$23$	0.753762	0.157170	0.0785851	−	0.996907i	$-0.474960\pi$
$23$	0.753762	0.157170	0.0785851	+	0.996907i	$0.474960\pi$
$24$	0	0
$25$	1.57350	0.314699
$26$	−0.0596433	−0.0116970
$27$	0	0
$28$	−1.78791	−0.337884
$29$	−8.63363	−1.60322	−0.801612	−	0.597845i	$-0.796024\pi$
$29$	−8.63363	−1.60322	−0.801612	+	0.597845i	$0.796024\pi$
$30$	0	0
$31$	5.83697	1.04835	0.524176	−	0.851610i	$-0.324374\pi$
$31$	5.83697	1.04835	0.524176	+	0.851610i	$0.324374\pi$
$32$	0.286318	0.0506143
$33$	0	0
$34$	0.0478481	0.00820587
$35$	−2.29265	−0.387529
$36$	0	0
$37$	−3.01770	−0.496107	−0.248053	−	0.968746i	$-0.579791\pi$
$37$	−3.01770	−0.496107	−0.248053	+	0.968746i	$0.579791\pi$
$38$	−0.0743869	−0.0120671
$39$	0	0
$40$	0.244753	0.0386988
$41$	4.20757	0.657113	0.328556	−	0.944484i	$-0.393438\pi$
$41$	4.20757	0.657113	0.328556	+	0.944484i	$0.393438\pi$
$42$	0	0
$43$	−8.27693	−1.26222	−0.631110	−	0.775693i	$-0.717400\pi$
$43$	−8.27693	−1.26222	−0.631110	+	0.775693i	$0.717400\pi$
$44$	−1.99943	−0.301425
$45$	0	0
$46$	0.0179914	0.00265269
$47$	3.17581	0.463239	0.231620	−	0.972806i	$-0.425598\pi$
$47$	3.17581	0.463239	0.231620	+	0.972806i	$0.425598\pi$
$48$	0	0
$49$	−6.20039	−0.885770
$50$	0.0375575	0.00531144
$51$	0	0
$52$	4.99616	0.692843
$53$	−8.04781	−1.10545	−0.552726	−	0.833363i	$-0.686412\pi$
$53$	−8.04781	−1.10545	−0.552726	+	0.833363i	$0.686412\pi$
$54$	0	0
$55$	−2.56388	−0.345714
$56$	−0.0853630	−0.0114071
$57$	0	0
$58$	−0.206075	−0.0270589
$59$	4.47017	0.581967	0.290983	−	0.956728i	$-0.406018\pi$
$59$	4.47017	0.581967	0.290983	+	0.956728i	$0.406018\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0.139322	0.0176939
$63$	0	0
$64$	−7.98633	−0.998291
$65$	6.40661	0.794643
$66$	0	0
$67$	0.916574	0.111977	0.0559887	−	0.998431i	$-0.482169\pi$
$67$	0.916574	0.111977	0.0559887	+	0.998431i	$0.482169\pi$
$68$	−4.00810	−0.486054
$69$	0	0
$70$	−0.0547230	−0.00654065
$71$	8.16659	0.969196	0.484598	−	0.874737i	$-0.338966\pi$
$71$	8.16659	0.969196	0.484598	+	0.874737i	$0.338966\pi$
$72$	0	0
$73$	−3.98703	−0.466647	−0.233323	−	0.972399i	$-0.574960\pi$
$73$	−3.98703	−0.466647	−0.233323	+	0.972399i	$0.574960\pi$
$74$	−0.0720291	−0.00837321
$75$	0	0
$76$	6.23119	0.714766
$77$	0.894211	0.101905
$78$	0	0
$79$	5.34694	0.601577	0.300789	−	0.953691i	$-0.402750\pi$
$79$	5.34694	0.601577	0.300789	+	0.953691i	$0.402750\pi$
$80$	−10.2468	−1.14562
$81$	0	0
$82$	0.100430	0.0110906
$83$	−0.882213	−0.0968355	−0.0484177	−	0.998827i	$-0.515418\pi$
$83$	−0.882213	−0.0968355	−0.0484177	+	0.998827i	$0.515418\pi$
$84$	0	0
$85$	−5.13962	−0.557470
$86$	−0.197561	−0.0213035
$87$	0	0
$88$	−0.0954618	−0.0101763
$89$	−14.4054	−1.52697	−0.763486	−	0.645825i	$-0.776514\pi$
$89$	−14.4054	−1.52697	−0.763486	+	0.645825i	$0.776514\pi$
$90$	0	0
$91$	−2.23445	−0.234234
$92$	−1.50709	−0.157126
$93$	0	0
$94$	0.0758029	0.00781847
$95$	7.99029	0.819787
$96$	0	0
$97$	−17.2558	−1.75206	−0.876031	−	0.482255i	$-0.839818\pi$
$97$	−17.2558	−1.75206	−0.876031	+	0.482255i	$0.839818\pi$
$98$	−0.147996	−0.0149499
$99$	0	0
$100$	−3.14610	−0.314610
$101$	17.3480	1.72619	0.863097	−	0.505038i	$-0.168522\pi$
$101$	17.3480	1.72619	0.863097	+	0.505038i	$0.168522\pi$
$102$	0	0
$103$	−14.1160	−1.39089	−0.695447	−	0.718577i	$-0.744794\pi$
$103$	−14.1160	−1.39089	−0.695447	+	0.718577i	$0.744794\pi$
$104$	0.238539	0.0233907
$105$	0	0
$106$	−0.192092	−0.0186576
$107$	−14.9273	−1.44308	−0.721539	−	0.692373i	$-0.756565\pi$
$107$	−14.9273	−1.44308	−0.721539	+	0.692373i	$0.756565\pi$
$108$	0	0
$109$	−2.33450	−0.223604	−0.111802	−	0.993730i	$-0.535662\pi$
$109$	−2.33450	−0.223604	−0.111802	+	0.993730i	$0.535662\pi$
$110$	−0.0611970	−0.00583490
$111$	0	0
$112$	3.57379	0.337691
$113$	11.0296	1.03758	0.518790	−	0.854902i	$-0.326383\pi$
$113$	11.0296	1.03758	0.518790	+	0.854902i	$0.326383\pi$
$114$	0	0
$115$	−1.93256	−0.180212
$116$	17.2623	1.60277
$117$	0	0
$118$	0.106698	0.00982234
$119$	1.79255	0.164323
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.0238689	−0.00216099
$123$	0	0
$124$	−11.6706	−1.04805
$125$	8.78516	0.785768
$126$	0	0
$127$	20.1027	1.78382	0.891912	−	0.452210i	$-0.149364\pi$
$127$	20.1027	1.78382	0.891912	+	0.452210i	$0.149364\pi$
$128$	−0.763260	−0.0674633
$129$	0	0
$130$	0.152919	0.0134118
$131$	8.50404	0.743002	0.371501	−	0.928433i	$-0.378843\pi$
$131$	8.50404	0.743002	0.371501	+	0.928433i	$0.378843\pi$
$132$	0	0
$133$	−2.78679	−0.241645
$134$	0.0218776	0.00188993
$135$	0	0
$136$	−0.191365	−0.0164094
$137$	12.3469	1.05487	0.527435	−	0.849595i	$-0.323154\pi$
$137$	12.3469	1.05487	0.527435	+	0.849595i	$0.323154\pi$
$138$	0	0
$139$	15.3465	1.30168	0.650838	−	0.759217i	$-0.274418\pi$
$139$	15.3465	1.30168	0.650838	+	0.759217i	$0.274418\pi$
$140$	4.58400	0.387418
$141$	0	0
$142$	0.194927	0.0163579
$143$	−2.49879	−0.208960
$144$	0	0
$145$	22.1356	1.83826
$146$	−0.0951659	−0.00787599
$147$	0	0
$148$	6.03368	0.495966
$149$	4.03344	0.330432	0.165216	−	0.986257i	$-0.447168\pi$
$149$	4.03344	0.330432	0.165216	+	0.986257i	$0.447168\pi$
$150$	0	0
$151$	−0.166611	−0.0135586	−0.00677929	−	0.999977i	$-0.502158\pi$
$151$	−0.166611	−0.0135586	−0.00677929	+	0.999977i	$0.502158\pi$
$152$	0.297505	0.0241308
$153$	0	0
$154$	0.0213438	0.00171993
$155$	−14.9653	−1.20204
$156$	0	0
$157$	12.1602	0.970488	0.485244	−	0.874379i	$-0.338731\pi$
$157$	12.1602	0.970488	0.485244	+	0.874379i	$0.338731\pi$
$158$	0.127625	0.0101533
$159$	0	0
$160$	−0.734085	−0.0580345
$161$	0.674022	0.0531204
$162$	0	0
$163$	−6.42816	−0.503492	−0.251746	−	0.967793i	$-0.581005\pi$
$163$	−6.42816	−0.503492	−0.251746	+	0.967793i	$0.581005\pi$
$164$	−8.41275	−0.656925
$165$	0	0
$166$	−0.0210574	−0.00163437
$167$	5.47366	0.423565	0.211782	−	0.977317i	$-0.432073\pi$
$167$	5.47366	0.423565	0.211782	+	0.977317i	$0.432073\pi$
$168$	0	0
$169$	−6.75603	−0.519695
$170$	−0.122677	−0.00940888
$171$	0	0
$172$	16.5491	1.26186
$173$	−22.3897	−1.70226	−0.851129	−	0.524957i	$-0.824082\pi$
$173$	−22.3897	−1.70226	−0.851129	+	0.524957i	$0.824082\pi$
$174$	0	0
$175$	1.40704	0.106362
$176$	3.99658	0.301254
$177$	0	0
$178$	−0.343841	−0.0257720
$179$	10.3046	0.770201	0.385101	−	0.922875i	$-0.374167\pi$
$179$	10.3046	0.770201	0.385101	+	0.922875i	$0.374167\pi$
$180$	0	0
$181$	−14.8183	−1.10144	−0.550718	−	0.834692i	$-0.685646\pi$
$181$	−14.8183	−1.10144	−0.550718	+	0.834692i	$0.685646\pi$
$182$	−0.0533337	−0.00395336
$183$	0	0
$184$	−0.0719555	−0.00530463
$185$	7.73703	0.568838
$186$	0	0
$187$	2.00462	0.146593
$188$	−6.34980	−0.463107
$189$	0	0
$190$	0.190719	0.0138362
$191$	0.355642	0.0257333	0.0128667	−	0.999917i	$-0.495904\pi$
$191$	0.355642	0.0257333	0.0128667	+	0.999917i	$0.495904\pi$
$192$	0	0
$193$	3.29854	0.237434	0.118717	−	0.992928i	$-0.462122\pi$
$193$	3.29854	0.237434	0.118717	+	0.992928i	$0.462122\pi$
$194$	−0.411876	−0.0295710
$195$	0	0
$196$	12.3972	0.885517
$197$	14.3781	1.02440	0.512198	−	0.858867i	$-0.328831\pi$
$197$	14.3781	1.02440	0.512198	+	0.858867i	$0.328831\pi$
$198$	0	0
$199$	23.1821	1.64333	0.821666	−	0.569969i	$-0.193045\pi$
$199$	23.1821	1.64333	0.821666	+	0.569969i	$0.193045\pi$
$200$	−0.150209	−0.0106214
$201$	0	0
$202$	0.414078	0.0291344
$203$	−7.72028	−0.541857
$204$	0	0
$205$	−10.7877	−0.753447
$206$	−0.336934	−0.0234753
$207$	0	0
$208$	−9.98663	−0.692448
$209$	−3.11648	−0.215572
$210$	0	0
$211$	0.493640	0.0339836	0.0169918	−	0.999856i	$-0.494591\pi$
$211$	0.493640	0.0339836	0.0169918	+	0.999856i	$0.494591\pi$
$212$	16.0910	1.10514
$213$	0	0
$214$	−0.356298	−0.0243560
$215$	21.2211	1.44727
$216$	0	0
$217$	5.21948	0.354322
$218$	−0.0557218	−0.00377395
$219$	0	0
$220$	5.12631	0.345615
$221$	−5.00914	−0.336951
$222$	0	0
$223$	22.6728	1.51828	0.759141	−	0.650926i	$-0.225619\pi$
$223$	22.6728	1.51828	0.759141	+	0.650926i	$0.225619\pi$
$224$	0.256028	0.0171066
$225$	0	0
$226$	0.263264	0.0175121
$227$	14.4682	0.960289	0.480145	−	0.877189i	$-0.340584\pi$
$227$	14.4682	0.960289	0.480145	+	0.877189i	$0.340584\pi$
$228$	0	0
$229$	−18.5393	−1.22511	−0.612556	−	0.790427i	$-0.709859\pi$
$229$	−18.5393	−1.22511	−0.612556	+	0.790427i	$0.709859\pi$
$230$	−0.0461280	−0.00304159
$231$	0	0
$232$	0.824182	0.0541102
$233$	−3.61475	−0.236810	−0.118405	−	0.992965i	$-0.537778\pi$
$233$	−3.61475	−0.236810	−0.118405	+	0.992965i	$0.537778\pi$
$234$	0	0
$235$	−8.14240	−0.531151
$236$	−8.93779	−0.581801
$237$	0	0
$238$	0.0427862	0.00277342
$239$	26.3686	1.70564	0.852821	−	0.522203i	$-0.174890\pi$
$239$	26.3686	1.70564	0.852821	+	0.522203i	$0.174890\pi$
$240$	0	0
$241$	5.94792	0.383139	0.191570	−	0.981479i	$-0.438642\pi$
$241$	5.94792	0.383139	0.191570	+	0.981479i	$0.438642\pi$
$242$	0.0238689	0.00153435
$243$	0	0
$244$	1.99943	0.128000
$245$	15.8971	1.01563
$246$	0	0
$247$	7.78744	0.495503
$248$	−0.557208	−0.0353828
$249$	0	0
$250$	0.209692	0.0132621
$251$	23.1327	1.46012	0.730062	−	0.683381i	$-0.239491\pi$
$251$	23.1327	1.46012	0.730062	+	0.683381i	$0.239491\pi$
$252$	0	0
$253$	0.753762	0.0473886
$254$	0.479828	0.0301071
$255$	0	0
$256$	15.9544	0.997153
$257$	8.16619	0.509393	0.254696	−	0.967021i	$-0.418024\pi$
$257$	8.16619	0.509393	0.254696	+	0.967021i	$0.418024\pi$
$258$	0	0
$259$	−2.69846	−0.167674
$260$	−12.8096	−0.794416
$261$	0	0
$262$	0.202982	0.0125403
$263$	−15.7192	−0.969285	−0.484642	−	0.874712i	$-0.661050\pi$
$263$	−15.7192	−0.969285	−0.484642	+	0.874712i	$0.661050\pi$
$264$	0	0
$265$	20.6336	1.26751
$266$	−0.0665175	−0.00407845
$267$	0	0
$268$	−1.83263	−0.111945
$269$	−0.233662	−0.0142466	−0.00712331	−	0.999975i	$-0.502267\pi$
$269$	−0.233662	−0.0142466	−0.00712331	+	0.999975i	$0.502267\pi$
$270$	0	0
$271$	−7.22743	−0.439035	−0.219517	−	0.975609i	$-0.570448\pi$
$271$	−7.22743	−0.439035	−0.219517	+	0.975609i	$0.570448\pi$
$272$	8.01164	0.485777
$273$	0	0
$274$	0.294707	0.0178039
$275$	1.57350	0.0948854
$276$	0	0
$277$	−25.9978	−1.56206	−0.781029	−	0.624495i	$-0.785305\pi$
$277$	−25.9978	−1.56206	−0.781029	+	0.624495i	$0.785305\pi$
$278$	0.366304	0.0219695
$279$	0	0
$280$	0.218861	0.0130794
$281$	−0.889956	−0.0530903	−0.0265452	−	0.999648i	$-0.508451\pi$
$281$	−0.889956	−0.0530903	−0.0265452	+	0.999648i	$0.508451\pi$
$282$	0	0
$283$	28.7335	1.70803	0.854016	−	0.520247i	$-0.174160\pi$
$283$	28.7335	1.70803	0.854016	+	0.520247i	$0.174160\pi$
$284$	−16.3285	−0.968920
$285$	0	0
$286$	−0.0596433	−0.00352679
$287$	3.76246	0.222091
$288$	0	0
$289$	−12.9815	−0.763617
$290$	0.528352	0.0310259
$291$	0	0
$292$	7.97179	0.466514
$293$	26.9276	1.57313	0.786564	−	0.617509i	$-0.211858\pi$
$293$	26.9276	1.57313	0.786564	+	0.617509i	$0.211858\pi$
$294$	0	0
$295$	−11.4610	−0.667285
$296$	0.288075	0.0167440
$297$	0	0
$298$	0.0962736	0.00557698
$299$	−1.88350	−0.108925
$300$	0	0
$301$	−7.40132	−0.426605
$302$	−0.00397681	−0.000228839 0
$303$	0	0
$304$	−12.4553	−0.714359
$305$	2.56388	0.146808
$306$	0	0
$307$	−31.1211	−1.77618	−0.888088	−	0.459674i	$-0.847966\pi$
$307$	−31.1211	−1.77618	−0.888088	+	0.459674i	$0.847966\pi$
$308$	−1.78791	−0.101876
$309$	0	0
$310$	−0.357205	−0.0202879
$311$	26.2864	1.49056	0.745281	−	0.666750i	$-0.232315\pi$
$311$	26.2864	1.49056	0.745281	+	0.666750i	$0.232315\pi$
$312$	0	0
$313$	17.8297	1.00779	0.503896	−	0.863764i	$-0.331900\pi$
$313$	17.8297	1.00779	0.503896	+	0.863764i	$0.331900\pi$
$314$	0.290250	0.0163797
$315$	0	0
$316$	−10.6908	−0.601406
$317$	30.6975	1.72414	0.862071	−	0.506787i	$-0.169167\pi$
$317$	30.6975	1.72414	0.862071	+	0.506787i	$0.169167\pi$
$318$	0	0
$319$	−8.63363	−0.483390
$320$	20.4760	1.14464
$321$	0	0
$322$	0.0160881	0.000896557 0
$323$	−6.24737	−0.347613
$324$	0	0
$325$	−3.93184	−0.218099
$326$	−0.153433	−0.00849786
$327$	0	0
$328$	−0.401663	−0.0221781
$329$	2.83984	0.156565
$330$	0	0
$331$	29.7089	1.63295	0.816476	−	0.577380i	$-0.195925\pi$
$331$	29.7089	1.63295	0.816476	+	0.577380i	$0.195925\pi$
$332$	1.76392	0.0968079
$333$	0	0
$334$	0.130650	0.00714885
$335$	−2.34999	−0.128394
$336$	0	0
$337$	20.5090	1.11720	0.558599	−	0.829438i	$-0.311339\pi$
$337$	20.5090	1.11720	0.558599	+	0.829438i	$0.311339\pi$
$338$	−0.161259	−0.00877132
$339$	0	0
$340$	10.2763	0.557311
$341$	5.83697	0.316090
$342$	0	0
$343$	−11.8039	−0.637352
$344$	0.790131	0.0426010
$345$	0	0
$346$	−0.534417	−0.0287304
$347$	−9.64023	−0.517514	−0.258757	−	0.965942i	$-0.583313\pi$
$347$	−9.64023	−0.517514	−0.258757	+	0.965942i	$0.583313\pi$
$348$	0	0
$349$	−19.2255	−1.02912	−0.514560	−	0.857454i	$-0.672045\pi$
$349$	−19.2255	−1.02912	−0.514560	+	0.857454i	$0.672045\pi$
$350$	0.0335844	0.00179516
$351$	0	0
$352$	0.286318	0.0152608
$353$	−19.3901	−1.03203	−0.516016	−	0.856579i	$-0.672585\pi$
$353$	−19.3901	−1.03203	−0.516016	+	0.856579i	$0.672585\pi$
$354$	0	0
$355$	−20.9382	−1.11128
$356$	28.8026	1.52654
$357$	0	0
$358$	0.245959	0.0129993
$359$	14.7547	0.778725	0.389362	−	0.921085i	$-0.372695\pi$
$359$	14.7547	0.778725	0.389362	+	0.921085i	$0.372695\pi$
$360$	0	0
$361$	−9.28754	−0.488818
$362$	−0.353696	−0.0185898
$363$	0	0
$364$	4.46762	0.234167
$365$	10.2223	0.535059
$366$	0	0
$367$	−32.2955	−1.68581	−0.842907	−	0.538060i	$-0.819158\pi$
$367$	−32.2955	−1.68581	−0.842907	+	0.538060i	$0.819158\pi$
$368$	3.01247	0.157036
$369$	0	0
$370$	0.184674	0.00960075
$371$	−7.19643	−0.373620
$372$	0	0
$373$	25.0651	1.29782	0.648911	−	0.760864i	$-0.275225\pi$
$373$	25.0651	1.29782	0.648911	+	0.760864i	$0.275225\pi$
$374$	0.0478481	0.00247416
$375$	0	0
$376$	−0.303168	−0.0156347
$377$	21.5736	1.11110
$378$	0	0
$379$	−0.998334	−0.0512810	−0.0256405	−	0.999671i	$-0.508163\pi$
$379$	−0.998334	−0.0512810	−0.0256405	+	0.999671i	$0.508163\pi$
$380$	−15.9760	−0.819553
$381$	0	0
$382$	0.00848876	0.000434323 0
$383$	3.19124	0.163065	0.0815325	−	0.996671i	$-0.474019\pi$
$383$	3.19124	0.163065	0.0815325	+	0.996671i	$0.474019\pi$
$384$	0	0
$385$	−2.29265	−0.116844
$386$	0.0787325	0.00400738
$387$	0	0
$388$	34.5018	1.75156
$389$	16.3465	0.828801	0.414401	−	0.910095i	$-0.363991\pi$
$389$	16.3465	0.828801	0.414401	+	0.910095i	$0.363991\pi$
$390$	0	0
$391$	1.51101	0.0764150
$392$	0.591900	0.0298955
$393$	0	0
$394$	0.343188	0.0172896
$395$	−13.7089	−0.689771
$396$	0	0
$397$	−0.685361	−0.0343973	−0.0171987	−	0.999852i	$-0.505475\pi$
$397$	−0.685361	−0.0343973	−0.0171987	+	0.999852i	$0.505475\pi$
$398$	0.553329	0.0277359
$399$	0	0
$400$	6.28860	0.314430
$401$	25.0432	1.25060	0.625299	−	0.780385i	$-0.284977\pi$
$401$	25.0432	1.25060	0.625299	+	0.780385i	$0.284977\pi$
$402$	0	0
$403$	−14.5854	−0.726550
$404$	−34.6862	−1.72570
$405$	0	0
$406$	−0.184274	−0.00914538
$407$	−3.01770	−0.149582
$408$	0	0
$409$	−3.35859	−0.166072	−0.0830358	−	0.996547i	$-0.526462\pi$
$409$	−3.35859	−0.166072	−0.0830358	+	0.996547i	$0.526462\pi$
$410$	−0.257491	−0.0127166
$411$	0	0
$412$	28.2240	1.39050
$413$	3.99727	0.196693
$414$	0	0
$415$	2.26189	0.111032
$416$	−0.715448	−0.0350777
$417$	0	0
$418$	−0.0743869	−0.00363838
$419$	21.6381	1.05709	0.528546	−	0.848905i	$-0.322738\pi$
$419$	21.6381	1.05709	0.528546	+	0.848905i	$0.322738\pi$
$420$	0	0
$421$	−28.8028	−1.40376	−0.701882	−	0.712294i	$-0.747656\pi$
$421$	−28.8028	−1.40376	−0.701882	+	0.712294i	$0.747656\pi$
$422$	0.0117826	0.000573569 0
$423$	0	0
$424$	0.768258	0.0373099
$425$	3.15426	0.153004
$426$	0	0
$427$	−0.894211	−0.0432739
$428$	29.8461	1.44267
$429$	0	0
$430$	0.506523	0.0244267
$431$	−0.0769325	−0.00370571	−0.00185285	−	0.999998i	$-0.500590\pi$
$431$	−0.0769325	−0.00370571	−0.00185285	+	0.999998i	$0.500590\pi$
$432$	0	0
$433$	25.9476	1.24696	0.623481	−	0.781838i	$-0.285718\pi$
$433$	25.9476	1.24696	0.623481	+	0.781838i	$0.285718\pi$
$434$	0.124583	0.00598018
$435$	0	0
$436$	4.66766	0.223541
$437$	−2.34909	−0.112372
$438$	0	0
$439$	24.3938	1.16425	0.582126	−	0.813098i	$-0.302221\pi$
$439$	24.3938	1.16425	0.582126	+	0.813098i	$0.302221\pi$
$440$	0.244753	0.0116681
$441$	0	0
$442$	−0.119562	−0.00568700
$443$	13.8551	0.658275	0.329138	−	0.944282i	$-0.393242\pi$
$443$	13.8551	0.658275	0.329138	+	0.944282i	$0.393242\pi$
$444$	0	0
$445$	36.9338	1.75083
$446$	0.541174	0.0256253
$447$	0	0
$448$	−7.14146	−0.337402
$449$	−0.382410	−0.0180470	−0.00902352	−	0.999959i	$-0.502872\pi$
$449$	−0.382410	−0.0180470	−0.00902352	+	0.999959i	$0.502872\pi$
$450$	0	0
$451$	4.20757	0.198127
$452$	−22.0530	−1.03728
$453$	0	0
$454$	0.345340	0.0162076
$455$	5.72886	0.268573
$456$	0	0
$457$	41.2232	1.92834	0.964170	−	0.265285i	$-0.0854659\pi$
$457$	41.2232	1.92834	0.964170	+	0.265285i	$0.0854659\pi$
$458$	−0.442512	−0.0206772
$459$	0	0
$460$	3.86402	0.180161
$461$	−37.1823	−1.73175	−0.865876	−	0.500259i	$-0.833238\pi$
$461$	−37.1823	−1.73175	−0.865876	+	0.500259i	$0.833238\pi$
$462$	0	0
$463$	−11.7439	−0.545788	−0.272894	−	0.962044i	$-0.587981\pi$
$463$	−11.7439	−0.545788	−0.272894	+	0.962044i	$0.587981\pi$
$464$	−34.5050	−1.60185
$465$	0	0
$466$	−0.0862800	−0.00399684
$467$	−16.3009	−0.754317	−0.377158	−	0.926149i	$-0.623099\pi$
$467$	−16.3009	−0.754317	−0.377158	+	0.926149i	$0.623099\pi$
$468$	0	0
$469$	0.819610	0.0378461
$470$	−0.194350	−0.00896469
$471$	0	0
$472$	−0.426731	−0.0196419
$473$	−8.27693	−0.380574
$474$	0	0
$475$	−4.90377	−0.225000
$476$	−3.58409	−0.164276
$477$	0	0
$478$	0.629388	0.0287875
$479$	−8.40030	−0.383820	−0.191910	−	0.981413i	$-0.561468\pi$
$479$	−8.40030	−0.383820	−0.191910	+	0.981413i	$0.561468\pi$
$480$	0	0
$481$	7.54061	0.343822
$482$	0.141970	0.00646656
$483$	0	0
$484$	−1.99943	−0.0908832
$485$	44.2419	2.00892
$486$	0	0
$487$	−26.1794	−1.18630	−0.593152	−	0.805091i	$-0.702117\pi$
$487$	−26.1794	−1.18630	−0.593152	+	0.805091i	$0.702117\pi$
$488$	0.0954618	0.00432135
$489$	0	0
$490$	0.379445	0.0171416
$491$	32.7819	1.47942	0.739712	−	0.672923i	$-0.234962\pi$
$491$	32.7819	1.47942	0.739712	+	0.672923i	$0.234962\pi$
$492$	0	0
$493$	−17.3072	−0.779475
$494$	0.185877	0.00836302
$495$	0	0
$496$	23.3279	1.04746
$497$	7.30265	0.327569
$498$	0	0
$499$	−4.60411	−0.206108	−0.103054	−	0.994676i	$-0.532861\pi$
$499$	−4.60411	−0.206108	−0.103054	+	0.994676i	$0.532861\pi$
$500$	−17.5653	−0.785544
$501$	0	0
$502$	0.552151	0.0246437
$503$	22.0543	0.983353	0.491676	−	0.870778i	$-0.336384\pi$
$503$	22.0543	0.983353	0.491676	+	0.870778i	$0.336384\pi$
$504$	0	0
$505$	−44.4783	−1.97926
$506$	0.0179914	0.000799817 0
$507$	0	0
$508$	−40.1939	−1.78331
$509$	−17.0215	−0.754463	−0.377232	−	0.926119i	$-0.623124\pi$
$509$	−17.0215	−0.754463	−0.377232	+	0.926119i	$0.623124\pi$
$510$	0	0
$511$	−3.56525	−0.157717
$512$	1.90733	0.0842930
$513$	0	0
$514$	0.194918	0.00859744
$515$	36.1919	1.59480
$516$	0	0
$517$	3.17581	0.139672
$518$	−0.0644092	−0.00282998
$519$	0	0
$520$	−0.611587	−0.0268199
$521$	36.5540	1.60146	0.800729	−	0.599026i	$-0.204445\pi$
$521$	36.5540	1.60146	0.800729	+	0.599026i	$0.204445\pi$
$522$	0	0
$523$	9.02969	0.394841	0.197420	−	0.980319i	$-0.436744\pi$
$523$	9.02969	0.394841	0.197420	+	0.980319i	$0.436744\pi$
$524$	−17.0032	−0.742790
$525$	0	0
$526$	−0.375198	−0.0163594
$527$	11.7009	0.509700
$528$	0	0
$529$	−22.4318	−0.975298
$530$	0.492501	0.0213929
$531$	0	0
$532$	5.57199	0.241577
$533$	−10.5139	−0.455406
$534$	0	0
$535$	38.2719	1.65464
$536$	−0.0874978	−0.00377933
$537$	0	0
$538$	−0.00557724	−0.000240452 0
$539$	−6.20039	−0.267070
$540$	0	0
$541$	−8.72807	−0.375249	−0.187624	−	0.982241i	$-0.560079\pi$
$541$	−8.72807	−0.375249	−0.187624	+	0.982241i	$0.560079\pi$
$542$	−0.172510	−0.00740996
$543$	0	0
$544$	0.573959	0.0246083
$545$	5.98538	0.256385
$546$	0	0
$547$	−13.1225	−0.561078	−0.280539	−	0.959843i	$-0.590513\pi$
$547$	−13.1225	−0.561078	−0.280539	+	0.959843i	$0.590513\pi$
$548$	−24.6868	−1.05457
$549$	0	0
$550$	0.0375575	0.00160146
$551$	26.9065	1.14626
$552$	0	0
$553$	4.78129	0.203321
$554$	−0.620538	−0.0263641
$555$	0	0
$556$	−30.6843	−1.30130
$557$	27.7973	1.17781	0.588905	−	0.808202i	$-0.299559\pi$
$557$	27.7973	1.17781	0.588905	+	0.808202i	$0.299559\pi$
$558$	0	0
$559$	20.6823	0.874770
$560$	−9.16277	−0.387198
$561$	0	0
$562$	−0.0212422	−0.000896050 0
$563$	−4.59168	−0.193516	−0.0967581	−	0.995308i	$-0.530847\pi$
$563$	−4.59168	−0.193516	−0.0967581	+	0.995308i	$0.530847\pi$
$564$	0	0
$565$	−28.2786	−1.18969
$566$	0.685837	0.0288279
$567$	0	0
$568$	−0.779598	−0.0327112
$569$	6.44209	0.270067	0.135033	−	0.990841i	$-0.456886\pi$
$569$	6.44209	0.270067	0.135033	+	0.990841i	$0.456886\pi$
$570$	0	0
$571$	−34.3627	−1.43803	−0.719017	−	0.694992i	$-0.755408\pi$
$571$	−34.3627	−1.43803	−0.719017	+	0.694992i	$0.755408\pi$
$572$	4.99616	0.208900
$573$	0	0
$574$	0.0898055	0.00374841
$575$	1.18604	0.0494614
$576$	0	0
$577$	−13.0220	−0.542113	−0.271056	−	0.962563i	$-0.587373\pi$
$577$	−13.0220	−0.542113	−0.271056	+	0.962563i	$0.587373\pi$
$578$	−0.309853	−0.0128882
$579$	0	0
$580$	−44.2586	−1.83774
$581$	−0.788884	−0.0327284
$582$	0	0
$583$	−8.04781	−0.333306
$584$	0.380609	0.0157497
$585$	0	0
$586$	0.642731	0.0265510
$587$	6.51311	0.268825	0.134412	−	0.990925i	$-0.457085\pi$
$587$	6.51311	0.268825	0.134412	+	0.990925i	$0.457085\pi$
$588$	0	0
$589$	−18.1908	−0.749540
$590$	−0.273561	−0.0112623
$591$	0	0
$592$	−12.0605	−0.495683
$593$	27.3932	1.12491	0.562453	−	0.826829i	$-0.309858\pi$
$593$	27.3932	1.12491	0.562453	+	0.826829i	$0.309858\pi$
$594$	0	0
$595$	−4.59590	−0.188414
$596$	−8.06458	−0.330338
$597$	0	0
$598$	−0.0449569	−0.00183842
$599$	37.0023	1.51187	0.755936	−	0.654646i	$-0.227182\pi$
$599$	37.0023	1.51187	0.755936	+	0.654646i	$0.227182\pi$
$600$	0	0
$601$	31.9443	1.30304	0.651518	−	0.758633i	$-0.274133\pi$
$601$	31.9443	1.30304	0.651518	+	0.758633i	$0.274133\pi$
$602$	−0.176661	−0.00720016
$603$	0	0
$604$	0.333126	0.0135547
$605$	−2.56388	−0.104237
$606$	0	0
$607$	19.0116	0.771659	0.385829	−	0.922570i	$-0.373915\pi$
$607$	19.0116	0.771659	0.385829	+	0.922570i	$0.373915\pi$
$608$	−0.892303	−0.0361877
$609$	0	0
$610$	0.0611970	0.00247779
$611$	−7.93569	−0.321043
$612$	0	0
$613$	−32.3764	−1.30767	−0.653836	−	0.756636i	$-0.726841\pi$
$613$	−32.3764	−1.30767	−0.653836	+	0.756636i	$0.726841\pi$
$614$	−0.742825	−0.0299780
$615$	0	0
$616$	−0.0853630	−0.00343937
$617$	−29.8715	−1.20258	−0.601291	−	0.799030i	$-0.705347\pi$
$617$	−29.8715	−1.20258	−0.601291	+	0.799030i	$0.705347\pi$
$618$	0	0
$619$	46.7226	1.87794	0.938970	−	0.344000i	$-0.111782\pi$
$619$	46.7226	1.87794	0.938970	+	0.344000i	$0.111782\pi$
$620$	29.9221	1.20170
$621$	0	0
$622$	0.627425	0.0251575
$623$	−12.8815	−0.516086
$624$	0	0
$625$	−30.3916	−1.21566
$626$	0.425574	0.0170093
$627$	0	0
$628$	−24.3134	−0.970211
$629$	−6.04935	−0.241203
$630$	0	0
$631$	−19.8543	−0.790386	−0.395193	−	0.918598i	$-0.629322\pi$
$631$	−19.8543	−0.790386	−0.395193	+	0.918598i	$0.629322\pi$
$632$	−0.510428	−0.0203037
$633$	0	0
$634$	0.732714	0.0290998
$635$	−51.5409	−2.04534
$636$	0	0
$637$	15.4935	0.613874
$638$	−0.206075	−0.00815858
$639$	0	0
$640$	1.95691	0.0773536
$641$	−49.0939	−1.93909	−0.969545	−	0.244912i	$-0.921241\pi$
$641$	−49.0939	−1.93909	−0.969545	+	0.244912i	$0.921241\pi$
$642$	0	0
$643$	47.5337	1.87455	0.937274	−	0.348593i	$-0.113341\pi$
$643$	47.5337	1.87455	0.937274	+	0.348593i	$0.113341\pi$
$644$	−1.34766	−0.0531053
$645$	0	0
$646$	−0.149118	−0.00586695
$647$	−6.81219	−0.267815	−0.133907	−	0.990994i	$-0.542753\pi$
$647$	−6.81219	−0.267815	−0.133907	+	0.990994i	$0.542753\pi$
$648$	0	0
$649$	4.47017	0.175470
$650$	−0.0938486	−0.00368104
$651$	0	0
$652$	12.8527	0.503349
$653$	−29.2524	−1.14473	−0.572367	−	0.819998i	$-0.693975\pi$
$653$	−29.2524	−1.14473	−0.572367	+	0.819998i	$0.693975\pi$
$654$	0	0
$655$	−21.8034	−0.851928
$656$	16.8159	0.656551
$657$	0	0
$658$	0.0677837	0.00264249
$659$	−37.7971	−1.47237	−0.736183	−	0.676782i	$-0.763374\pi$
$659$	−37.7971	−1.47237	−0.736183	+	0.676782i	$0.763374\pi$
$660$	0	0
$661$	−10.4933	−0.408143	−0.204071	−	0.978956i	$-0.565417\pi$
$661$	−10.4933	−0.408143	−0.204071	+	0.978956i	$0.565417\pi$
$662$	0.709119	0.0275607
$663$	0	0
$664$	0.0842177	0.00326828
$665$	7.14500	0.277071
$666$	0	0
$667$	−6.50770	−0.251979
$668$	−10.9442	−0.423444
$669$	0	0
$670$	−0.0560915	−0.00216701
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	0.714580	0.0275450	0.0137725	−	0.999905i	$-0.495616\pi$
$673$	0.714580	0.0275450	0.0137725	+	0.999905i	$0.495616\pi$
$674$	0.489527	0.0188559
$675$	0	0
$676$	13.5082	0.519547
$677$	−27.6757	−1.06367	−0.531833	−	0.846849i	$-0.678496\pi$
$677$	−27.6757	−1.06367	−0.531833	+	0.846849i	$0.678496\pi$
$678$	0	0
$679$	−15.4303	−0.592162
$680$	0.490637	0.0188151
$681$	0	0
$682$	0.139322	0.00533491
$683$	12.4574	0.476669	0.238335	−	0.971183i	$-0.423398\pi$
$683$	12.4574	0.476669	0.238335	+	0.971183i	$0.423398\pi$
$684$	0	0
$685$	−31.6561	−1.20952
$686$	−0.281746	−0.0107571
$687$	0	0
$688$	−33.0794	−1.26114
$689$	20.1098	0.766123
$690$	0	0
$691$	−2.54147	−0.0966822	−0.0483411	−	0.998831i	$-0.515393\pi$
$691$	−2.54147	−0.0966822	−0.0483411	+	0.998831i	$0.515393\pi$
$692$	44.7667	1.70177
$693$	0	0
$694$	−0.230101	−0.00873452
$695$	−39.3467	−1.49251
$696$	0	0
$697$	8.43460	0.319483
$698$	−0.458892	−0.0173693
$699$	0	0
$700$	−2.81327	−0.106332
$701$	−37.3676	−1.41136	−0.705678	−	0.708533i	$-0.749357\pi$
$701$	−37.3676	−1.41136	−0.705678	+	0.708533i	$0.749357\pi$
$702$	0	0
$703$	9.40461	0.354701
$704$	−7.98633	−0.300996
$705$	0	0
$706$	−0.462820	−0.0174185
$707$	15.5128	0.583419
$708$	0	0
$709$	10.4559	0.392679	0.196340	−	0.980536i	$-0.437095\pi$
$709$	10.4559	0.392679	0.196340	+	0.980536i	$0.437095\pi$
$710$	−0.499771	−0.0187561
$711$	0	0
$712$	1.37517	0.0515366
$713$	4.39969	0.164770
$714$	0	0
$715$	6.40661	0.239594
$716$	−20.6033	−0.769982
$717$	0	0
$718$	0.352179	0.0131432
$719$	−8.11085	−0.302484	−0.151242	−	0.988497i	$-0.548327\pi$
$719$	−8.11085	−0.302484	−0.151242	+	0.988497i	$0.548327\pi$
$720$	0	0
$721$	−12.6227	−0.470094
$722$	−0.221683	−0.00825019
$723$	0	0
$724$	29.6282	1.10112
$725$	−13.5850	−0.504533
$726$	0	0
$727$	−4.51534	−0.167465	−0.0837323	−	0.996488i	$-0.526684\pi$
$727$	−4.51534	−0.167465	−0.0837323	+	0.996488i	$0.526684\pi$
$728$	0.213304	0.00790559
$729$	0	0
$730$	0.243994	0.00903063
$731$	−16.5921	−0.613682
$732$	0	0
$733$	−36.2178	−1.33774	−0.668868	−	0.743381i	$-0.733221\pi$
$733$	−36.2178	−1.33774	−0.668868	+	0.743381i	$0.733221\pi$
$734$	−0.770858	−0.0284529
$735$	0	0
$736$	0.215815	0.00795506
$737$	0.916574	0.0337624
$738$	0	0
$739$	24.5047	0.901421	0.450711	−	0.892670i	$-0.351171\pi$
$739$	24.5047	0.901421	0.450711	+	0.892670i	$0.351171\pi$
$740$	−15.4697	−0.568676
$741$	0	0
$742$	−0.171771	−0.00630590
$743$	−17.5438	−0.643620	−0.321810	−	0.946804i	$-0.604291\pi$
$743$	−17.5438	−0.643620	−0.321810	+	0.946804i	$0.604291\pi$
$744$	0	0
$745$	−10.3413	−0.378875
$746$	0.598275	0.0219044
$747$	0	0
$748$	−4.00810	−0.146551
$749$	−13.3482	−0.487731
$750$	0	0
$751$	37.9383	1.38439	0.692193	−	0.721712i	$-0.256644\pi$
$751$	37.9383	1.38439	0.692193	+	0.721712i	$0.256644\pi$
$752$	12.6924	0.462843
$753$	0	0
$754$	0.514938	0.0187529
$755$	0.427170	0.0155463
$756$	0	0
$757$	11.0478	0.401540	0.200770	−	0.979638i	$-0.435656\pi$
$757$	11.0478	0.401540	0.200770	+	0.979638i	$0.435656\pi$
$758$	−0.0238291	−0.000865512 0
$759$	0	0
$760$	−0.762768	−0.0276685
$761$	26.2994	0.953351	0.476675	−	0.879079i	$-0.341842\pi$
$761$	26.2994	0.953351	0.476675	+	0.879079i	$0.341842\pi$
$762$	0	0
$763$	−2.08753	−0.0755737
$764$	−0.711081	−0.0257260
$765$	0	0
$766$	0.0761713	0.00275218
$767$	−11.1700	−0.403327
$768$	0	0
$769$	−26.2011	−0.944835	−0.472418	−	0.881375i	$-0.656619\pi$
$769$	−26.2011	−0.944835	−0.472418	+	0.881375i	$0.656619\pi$
$770$	−0.0547230	−0.00197208
$771$	0	0
$772$	−6.59521	−0.237367
$773$	−52.2360	−1.87880	−0.939399	−	0.342826i	$-0.888616\pi$
$773$	−52.2360	−1.87880	−0.939399	+	0.342826i	$0.888616\pi$
$774$	0	0
$775$	9.18445	0.329915
$776$	1.64727	0.0591336
$777$	0	0
$778$	0.390173	0.0139884
$779$	−13.1128	−0.469816
$780$	0	0
$781$	8.16659	0.292224
$782$	0.0360661	0.00128972
$783$	0	0
$784$	−24.7804	−0.885013
$785$	−31.1773	−1.11276
$786$	0	0
$787$	25.3780	0.904627	0.452314	−	0.891859i	$-0.350599\pi$
$787$	25.3780	0.904627	0.452314	+	0.891859i	$0.350599\pi$
$788$	−28.7480	−1.02410
$789$	0	0
$790$	−0.327216	−0.0116418
$791$	9.86280	0.350681
$792$	0	0
$793$	2.49879	0.0887348
$794$	−0.0163588	−0.000580552 0
$795$	0	0
$796$	−46.3509	−1.64286
$797$	2.43952	0.0864121	0.0432060	−	0.999066i	$-0.486243\pi$
$797$	2.43952	0.0864121	0.0432060	+	0.999066i	$0.486243\pi$
$798$	0	0
$799$	6.36629	0.225223
$800$	0.450519	0.0159283
$801$	0	0
$802$	0.597753	0.0211074
$803$	−3.98703	−0.140699
$804$	0	0
$805$	−1.72811	−0.0609080
$806$	−0.348137	−0.0122626
$807$	0	0
$808$	−1.65608	−0.0582605
$809$	32.4569	1.14112	0.570561	−	0.821255i	$-0.306726\pi$
$809$	32.4569	1.14112	0.570561	+	0.821255i	$0.306726\pi$
$810$	0	0
$811$	−10.8153	−0.379775	−0.189887	−	0.981806i	$-0.560812\pi$
$811$	−10.8153	−0.379775	−0.189887	+	0.981806i	$0.560812\pi$
$812$	15.4362	0.541703
$813$	0	0
$814$	−0.0720291	−0.00252462
$815$	16.4810	0.577306
$816$	0	0
$817$	25.7949	0.902449
$818$	−0.0801657	−0.00280293
$819$	0	0
$820$	21.5693	0.753233
$821$	−3.79621	−0.132489	−0.0662444	−	0.997803i	$-0.521102\pi$
$821$	−3.79621	−0.132489	−0.0662444	+	0.997803i	$0.521102\pi$
$822$	0	0
$823$	−7.92959	−0.276408	−0.138204	−	0.990404i	$-0.544133\pi$
$823$	−7.92959	−0.276408	−0.138204	+	0.990404i	$0.544133\pi$
$824$	1.34754	0.0469439
$825$	0	0
$826$	0.0954104	0.00331975
$827$	−36.9782	−1.28586	−0.642929	−	0.765926i	$-0.722281\pi$
$827$	−36.9782	−1.28586	−0.642929	+	0.765926i	$0.722281\pi$
$828$	0	0
$829$	10.8781	0.377812	0.188906	−	0.981995i	$-0.439506\pi$
$829$	10.8781	0.377812	0.188906	+	0.981995i	$0.439506\pi$
$830$	0.0539888	0.00187398
$831$	0	0
$832$	19.9562	0.691856
$833$	−12.4294	−0.430654
$834$	0	0
$835$	−14.0338	−0.485661
$836$	6.23119	0.215510
$837$	0	0
$838$	0.516477	0.0178414
$839$	−29.7679	−1.02770	−0.513851	−	0.857880i	$-0.671781\pi$
$839$	−29.7679	−1.02770	−0.513851	+	0.857880i	$0.671781\pi$
$840$	0	0
$841$	45.5395	1.57033
$842$	−0.687490	−0.0236925
$843$	0	0
$844$	−0.986998	−0.0339739
$845$	17.3217	0.595884
$846$	0	0
$847$	0.894211	0.0307254
$848$	−32.1637	−1.10451
$849$	0	0
$850$	0.0752887	0.00258238
$851$	−2.27463	−0.0779733
$852$	0	0
$853$	−6.84520	−0.234375	−0.117188	−	0.993110i	$-0.537388\pi$
$853$	−6.84520	−0.234375	−0.117188	+	0.993110i	$0.537388\pi$
$854$	−0.0213438	−0.000730369 0
$855$	0	0
$856$	1.42499	0.0487051
$857$	11.3794	0.388712	0.194356	−	0.980931i	$-0.437738\pi$
$857$	11.3794	0.388712	0.194356	+	0.980931i	$0.437738\pi$
$858$	0	0
$859$	12.9232	0.440933	0.220467	−	0.975395i	$-0.429242\pi$
$859$	12.9232	0.440933	0.220467	+	0.975395i	$0.429242\pi$
$860$	−42.4301	−1.44685
$861$	0	0
$862$	−0.00183629	−6.25443e−5 0
$863$	−6.72786	−0.229019	−0.114510	−	0.993422i	$-0.536530\pi$
$863$	−6.72786	−0.229019	−0.114510	+	0.993422i	$0.536530\pi$
$864$	0	0
$865$	57.4046	1.95181
$866$	0.619340	0.0210460
$867$	0	0
$868$	−10.4360	−0.354221
$869$	5.34694	0.181382
$870$	0	0
$871$	−2.29033	−0.0776048
$872$	0.222855	0.00754683
$873$	0	0
$874$	−0.0560700	−0.00189660
$875$	7.85578	0.265574
$876$	0	0
$877$	−40.1457	−1.35562	−0.677812	−	0.735235i	$-0.737072\pi$
$877$	−40.1457	−1.35562	−0.677812	+	0.735235i	$0.737072\pi$
$878$	0.582252	0.0196501
$879$	0	0
$880$	−10.2468	−0.345418
$881$	−38.5882	−1.30007	−0.650034	−	0.759905i	$-0.725245\pi$
$881$	−38.5882	−1.30007	−0.650034	+	0.759905i	$0.725245\pi$
$882$	0	0
$883$	24.0618	0.809745	0.404873	−	0.914373i	$-0.367316\pi$
$883$	24.0618	0.809745	0.404873	+	0.914373i	$0.367316\pi$
$884$	10.0154	0.336855
$885$	0	0
$886$	0.330705	0.0111103
$887$	−30.5817	−1.02683	−0.513416	−	0.858140i	$-0.671620\pi$
$887$	−30.5817	−1.02683	−0.513416	+	0.858140i	$0.671620\pi$
$888$	0	0
$889$	17.9760	0.602896
$890$	0.881568	0.0295502
$891$	0	0
$892$	−45.3327	−1.51785
$893$	−9.89734	−0.331202
$894$	0	0
$895$	−26.4198	−0.883115
$896$	−0.682515	−0.0228012
$897$	0	0
$898$	−0.00912769	−0.000304595 0
$899$	−50.3942	−1.68074
$900$	0	0
$901$	−16.1328	−0.537462
$902$	0.100430	0.00334395
$903$	0	0
$904$	−1.05291	−0.0350192
$905$	37.9924	1.26291
$906$	0	0
$907$	22.6261	0.751288	0.375644	−	0.926764i	$-0.377422\pi$
$907$	22.6261	0.751288	0.375644	+	0.926764i	$0.377422\pi$
$908$	−28.9282	−0.960015
$909$	0	0
$910$	0.136741	0.00453293
$911$	35.3043	1.16968	0.584842	−	0.811147i	$-0.301156\pi$
$911$	35.3043	1.16968	0.584842	+	0.811147i	$0.301156\pi$
$912$	0	0
$913$	−0.882213	−0.0291970
$914$	0.983951	0.0325462
$915$	0	0
$916$	37.0680	1.22476
$917$	7.60440	0.251120
$918$	0	0
$919$	48.5637	1.60197	0.800984	−	0.598686i	$-0.204310\pi$
$919$	48.5637	1.60197	0.800984	+	0.598686i	$0.204310\pi$
$920$	0.184486	0.00608231
$921$	0	0
$922$	−0.887499	−0.0292282
$923$	−20.4066	−0.671692
$924$	0	0
$925$	−4.74834	−0.156124
$926$	−0.280315	−0.00921171
$927$	0	0
$928$	−2.47196	−0.0811460
$929$	18.2131	0.597552	0.298776	−	0.954323i	$-0.403422\pi$
$929$	18.2131	0.597552	0.298776	+	0.954323i	$0.403422\pi$
$930$	0	0
$931$	19.3234	0.633299
$932$	7.22744	0.236743
$933$	0	0
$934$	−0.389084	−0.0127312
$935$	−5.13962	−0.168083
$936$	0	0
$937$	13.0072	0.424927	0.212464	−	0.977169i	$-0.431851\pi$
$937$	13.0072	0.424927	0.212464	+	0.977169i	$0.431851\pi$
$938$	0.0195632	0.000638760 0
$939$	0	0
$940$	16.2802	0.531000
$941$	−14.3701	−0.468450	−0.234225	−	0.972182i	$-0.575255\pi$
$941$	−14.3701	−0.468450	−0.234225	+	0.972182i	$0.575255\pi$
$942$	0	0
$943$	3.17151	0.103279
$944$	17.8654	0.581469
$945$	0	0
$946$	−0.197561	−0.00642326
$947$	5.37409	0.174634	0.0873172	−	0.996181i	$-0.472171\pi$
$947$	5.37409	0.174634	0.0873172	+	0.996181i	$0.472171\pi$
$948$	0	0
$949$	9.96277	0.323405
$950$	−0.117047	−0.00379752
$951$	0	0
$952$	−0.171121	−0.00554605
$953$	52.4771	1.69990	0.849950	−	0.526864i	$-0.176632\pi$
$953$	52.4771	1.69990	0.849950	+	0.526864i	$0.176632\pi$
$954$	0	0
$955$	−0.911824	−0.0295059
$956$	−52.7222	−1.70516
$957$	0	0
$958$	−0.200506	−0.00647804
$959$	11.0408	0.356525
$960$	0	0
$961$	3.07026	0.0990407
$962$	0.179986	0.00580297
$963$	0	0
$964$	−11.8925	−0.383030
$965$	−8.45708	−0.272243
$966$	0	0
$967$	−7.54692	−0.242693	−0.121346	−	0.992610i	$-0.538721\pi$
$967$	−7.54692	−0.242693	−0.121346	+	0.992610i	$0.538721\pi$
$968$	−0.0954618	−0.00306826
$969$	0	0
$970$	1.05600	0.0339062
$971$	39.8176	1.27781	0.638904	−	0.769287i	$-0.279388\pi$
$971$	39.8176	1.27781	0.638904	+	0.769287i	$0.279388\pi$
$972$	0	0
$973$	13.7230	0.439940
$974$	−0.624873	−0.0200222
$975$	0	0
$976$	−3.99658	−0.127927
$977$	−18.6439	−0.596471	−0.298235	−	0.954492i	$-0.596398\pi$
$977$	−18.6439	−0.596471	−0.298235	+	0.954492i	$0.596398\pi$
$978$	0	0
$979$	−14.4054	−0.460399
$980$	−31.7851	−1.01534
$981$	0	0
$982$	0.782465	0.0249695
$983$	14.0626	0.448527	0.224264	−	0.974529i	$-0.428002\pi$
$983$	14.0626	0.448527	0.224264	+	0.974529i	$0.428002\pi$
$984$	0	0
$985$	−36.8637	−1.17458
$986$	−0.413102	−0.0131559
$987$	0	0
$988$	−15.5704	−0.495362
$989$	−6.23884	−0.198383
$990$	0	0
$991$	−15.3577	−0.487855	−0.243927	−	0.969794i	$-0.578436\pi$
$991$	−15.3577	−0.487855	−0.243927	+	0.969794i	$0.578436\pi$
$992$	1.67123	0.0530615
$993$	0	0
$994$	0.174306	0.00552865
$995$	−59.4361	−1.88425
$996$	0	0
$997$	37.3199	1.18193	0.590966	−	0.806696i	$-0.298747\pi$
$997$	37.3199	1.18193	0.590966	+	0.806696i	$0.298747\pi$
$998$	−0.109895	−0.00347866
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.o.1.11	yes	25
3.2	odd	2		6039.2.a.n.1.15	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.15	✓	25	3.2	odd	2
6039.2.a.o.1.11	yes	25	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.0238689 q^{2} -1.99943 q^{4} -2.56388 q^{5} +0.894211 q^{7} -0.0954618 q^{8} +O(q^{10})\)	\(q+0.0238689 q^{2} -1.99943 q^{4} -2.56388 q^{5} +0.894211 q^{7} -0.0954618 q^{8} -0.0611970 q^{10} +1.00000 q^{11} -2.49879 q^{13} +0.0213438 q^{14} +3.99658 q^{16} +2.00462 q^{17} -3.11648 q^{19} +5.12631 q^{20} +0.0238689 q^{22} +0.753762 q^{23} +1.57350 q^{25} -0.0596433 q^{26} -1.78791 q^{28} -8.63363 q^{29} +5.83697 q^{31} +0.286318 q^{32} +0.0478481 q^{34} -2.29265 q^{35} -3.01770 q^{37} -0.0743869 q^{38} +0.244753 q^{40} +4.20757 q^{41} -8.27693 q^{43} -1.99943 q^{44} +0.0179914 q^{46} +3.17581 q^{47} -6.20039 q^{49} +0.0375575 q^{50} +4.99616 q^{52} -8.04781 q^{53} -2.56388 q^{55} -0.0853630 q^{56} -0.206075 q^{58} +4.47017 q^{59} -1.00000 q^{61} +0.139322 q^{62} -7.98633 q^{64} +6.40661 q^{65} +0.916574 q^{67} -4.00810 q^{68} -0.0547230 q^{70} +8.16659 q^{71} -3.98703 q^{73} -0.0720291 q^{74} +6.23119 q^{76} +0.894211 q^{77} +5.34694 q^{79} -10.2468 q^{80} +0.100430 q^{82} -0.882213 q^{83} -5.13962 q^{85} -0.197561 q^{86} -0.0954618 q^{88} -14.4054 q^{89} -2.23445 q^{91} -1.50709 q^{92} +0.0758029 q^{94} +7.99029 q^{95} -17.2558 q^{97} -0.147996 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more