LMFDB - Embedded newform 6039.2.a.i.1.12

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 $ x^13 - 2x^12 - 19x^11 + 35x^10 + 136x^9 - 220x^8 - 469x^7 + 610x^6 + 841x^5 - 760x^4 - 742x^3 + 366x^2 + 236x - 47
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.37960$ of defining polynomial
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)

$f(q)$	$=$	$q+2.37960 q^{2} +3.66252 q^{4} -2.55185 q^{5} +1.67712 q^{7} +3.95613 q^{8} +O(q^{10})$	\(q+2.37960 q^{2} +3.66252 q^{4} -2.55185 q^{5} +1.67712 q^{7} +3.95613 q^{8} -6.07239 q^{10} -1.00000 q^{11} +5.65147 q^{13} +3.99087 q^{14} +2.08899 q^{16} +4.01907 q^{17} +2.77230 q^{19} -9.34619 q^{20} -2.37960 q^{22} -6.13877 q^{23} +1.51194 q^{25} +13.4483 q^{26} +6.14247 q^{28} -1.25605 q^{29} +5.37742 q^{31} -2.94129 q^{32} +9.56379 q^{34} -4.27975 q^{35} +2.89012 q^{37} +6.59699 q^{38} -10.0954 q^{40} -6.31673 q^{41} +9.33950 q^{43} -3.66252 q^{44} -14.6078 q^{46} +8.72405 q^{47} -4.18728 q^{49} +3.59781 q^{50} +20.6986 q^{52} +4.90573 q^{53} +2.55185 q^{55} +6.63489 q^{56} -2.98889 q^{58} +4.68512 q^{59} -1.00000 q^{61} +12.7961 q^{62} -11.1771 q^{64} -14.4217 q^{65} +5.67098 q^{67} +14.7199 q^{68} -10.1841 q^{70} +2.71844 q^{71} +15.1095 q^{73} +6.87735 q^{74} +10.1536 q^{76} -1.67712 q^{77} -1.47612 q^{79} -5.33079 q^{80} -15.0313 q^{82} +2.19058 q^{83} -10.2561 q^{85} +22.2243 q^{86} -3.95613 q^{88} +13.1550 q^{89} +9.47818 q^{91} -22.4833 q^{92} +20.7598 q^{94} -7.07450 q^{95} -7.84526 q^{97} -9.96407 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * 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 $

$2$	2.37960	1.68263	0.841317	−	0.540542i	$-0.181781\pi$
$2$	2.37960	1.68263	0.841317	+	0.540542i	$0.181781\pi$
$3$	0	0
$3$	0	0
$4$	3.66252	1.83126
$5$	−2.55185	−1.14122	−0.570611	−	0.821221i	$-0.693294\pi$
$5$	−2.55185	−1.14122	−0.570611	+	0.821221i	$0.693294\pi$
$6$	0	0
$7$	1.67712	0.633891	0.316945	−	0.948444i	$-0.397343\pi$
$7$	1.67712	0.633891	0.316945	+	0.948444i	$0.397343\pi$
$8$	3.95613	1.39870
$9$	0	0
$10$	−6.07239	−1.92026
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.65147	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$13$	5.65147	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$14$	3.99087	1.06661
$15$	0	0
$16$	2.08899	0.522248
$17$	4.01907	0.974767	0.487383	−	0.873188i	$-0.337951\pi$
$17$	4.01907	0.974767	0.487383	+	0.873188i	$0.337951\pi$
$18$	0	0
$19$	2.77230	0.636010	0.318005	−	0.948089i	$-0.396987\pi$
$19$	2.77230	0.636010	0.318005	+	0.948089i	$0.396987\pi$
$20$	−9.34619	−2.08987
$21$	0	0
$22$	−2.37960	−0.507333
$23$	−6.13877	−1.28002	−0.640011	−	0.768366i	$-0.721070\pi$
$23$	−6.13877	−1.28002	−0.640011	+	0.768366i	$0.721070\pi$
$24$	0	0
$25$	1.51194	0.302387
$26$	13.4483	2.63742
$27$	0	0
$28$	6.14247	1.16082
$29$	−1.25605	−0.233242	−0.116621	−	0.993177i	$-0.537206\pi$
$29$	−1.25605	−0.233242	−0.116621	+	0.993177i	$0.537206\pi$
$30$	0	0
$31$	5.37742	0.965814	0.482907	−	0.875672i	$-0.339581\pi$
$31$	5.37742	0.965814	0.482907	+	0.875672i	$0.339581\pi$
$32$	−2.94129	−0.519951
$33$	0	0
$34$	9.56379	1.64018
$35$	−4.27975	−0.723410
$36$	0	0
$37$	2.89012	0.475134	0.237567	−	0.971371i	$-0.423650\pi$
$37$	2.89012	0.475134	0.237567	+	0.971371i	$0.423650\pi$
$38$	6.59699	1.07017
$39$	0	0
$40$	−10.0954	−1.59623
$41$	−6.31673	−0.986508	−0.493254	−	0.869885i	$-0.664193\pi$
$41$	−6.31673	−0.986508	−0.493254	+	0.869885i	$0.664193\pi$
$42$	0	0
$43$	9.33950	1.42426	0.712130	−	0.702047i	$-0.247731\pi$
$43$	9.33950	1.42426	0.712130	+	0.702047i	$0.247731\pi$
$44$	−3.66252	−0.552145
$45$	0	0
$46$	−14.6078	−2.15381
$47$	8.72405	1.27253	0.636267	−	0.771469i	$-0.280478\pi$
$47$	8.72405	1.27253	0.636267	+	0.771469i	$0.280478\pi$
$48$	0	0
$49$	−4.18728	−0.598183
$50$	3.59781	0.508807
$51$	0	0
$52$	20.6986	2.87038
$53$	4.90573	0.673854	0.336927	−	0.941531i	$-0.390612\pi$
$53$	4.90573	0.673854	0.336927	+	0.941531i	$0.390612\pi$
$54$	0	0
$55$	2.55185	0.344091
$56$	6.63489	0.886625
$57$	0	0
$58$	−2.98889	−0.392461
$59$	4.68512	0.609951	0.304976	−	0.952360i	$-0.401352\pi$
$59$	4.68512	0.609951	0.304976	+	0.952360i	$0.401352\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	12.7961	1.62511
$63$	0	0
$64$	−11.1771	−1.39714
$65$	−14.4217	−1.78879
$66$	0	0
$67$	5.67098	0.692820	0.346410	−	0.938083i	$-0.387401\pi$
$67$	5.67098	0.692820	0.346410	+	0.938083i	$0.387401\pi$
$68$	14.7199	1.78505
$69$	0	0
$70$	−10.1841	−1.21723
$71$	2.71844	0.322620	0.161310	−	0.986904i	$-0.448428\pi$
$71$	2.71844	0.322620	0.161310	+	0.986904i	$0.448428\pi$
$72$	0	0
$73$	15.1095	1.76843	0.884215	−	0.467080i	$-0.154694\pi$
$73$	15.1095	1.76843	0.884215	+	0.467080i	$0.154694\pi$
$74$	6.87735	0.799476
$75$	0	0
$76$	10.1536	1.16470
$77$	−1.67712	−0.191125
$78$	0	0
$79$	−1.47612	−0.166076	−0.0830381	−	0.996546i	$-0.526462\pi$
$79$	−1.47612	−0.166076	−0.0830381	+	0.996546i	$0.526462\pi$
$80$	−5.33079	−0.596001
$81$	0	0
$82$	−15.0313	−1.65993
$83$	2.19058	0.240448	0.120224	−	0.992747i	$-0.461639\pi$
$83$	2.19058	0.240448	0.120224	+	0.992747i	$0.461639\pi$
$84$	0	0
$85$	−10.2561	−1.11243
$86$	22.2243	2.39651
$87$	0	0
$88$	−3.95613	−0.421725
$89$	13.1550	1.39442	0.697212	−	0.716865i	$-0.254424\pi$
$89$	13.1550	1.39442	0.697212	+	0.716865i	$0.254424\pi$
$90$	0	0
$91$	9.47818	0.993583
$92$	−22.4833	−2.34405
$93$	0	0
$94$	20.7598	2.14121
$95$	−7.07450	−0.725829
$96$	0	0
$97$	−7.84526	−0.796565	−0.398283	−	0.917263i	$-0.630394\pi$
$97$	−7.84526	−0.796565	−0.398283	+	0.917263i	$0.630394\pi$
$98$	−9.96407	−1.00652
$99$	0	0
$100$	5.53749	0.553749
$101$	3.23484	0.321879	0.160939	−	0.986964i	$-0.448548\pi$
$101$	3.23484	0.321879	0.160939	+	0.986964i	$0.448548\pi$
$102$	0	0
$103$	−7.86347	−0.774811	−0.387405	−	0.921909i	$-0.626629\pi$
$103$	−7.86347	−0.774811	−0.387405	+	0.921909i	$0.626629\pi$
$104$	22.3580	2.19238
$105$	0	0
$106$	11.6737	1.13385
$107$	18.7522	1.81284	0.906420	−	0.422378i	$-0.138805\pi$
$107$	18.7522	1.81284	0.906420	+	0.422378i	$0.138805\pi$
$108$	0	0
$109$	−19.5666	−1.87414	−0.937070	−	0.349141i	$-0.886473\pi$
$109$	−19.5666	−1.87414	−0.937070	+	0.349141i	$0.886473\pi$
$110$	6.07239	0.578980
$111$	0	0
$112$	3.50348	0.331048
$113$	−11.4456	−1.07671	−0.538356	−	0.842718i	$-0.680954\pi$
$113$	−11.4456	−1.07671	−0.538356	+	0.842718i	$0.680954\pi$
$114$	0	0
$115$	15.6652	1.46079
$116$	−4.60029	−0.427126
$117$	0	0
$118$	11.1487	1.02632
$119$	6.74045	0.617896
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.37960	−0.215439
$123$	0	0
$124$	19.6949	1.76865
$125$	8.90101	0.796131
$126$	0	0
$127$	1.63206	0.144822	0.0724109	−	0.997375i	$-0.476931\pi$
$127$	1.63206	0.144822	0.0724109	+	0.997375i	$0.476931\pi$
$128$	−20.7145	−1.83092
$129$	0	0
$130$	−34.3179	−3.00988
$131$	−5.81437	−0.508004	−0.254002	−	0.967204i	$-0.581747\pi$
$131$	−5.81437	−0.508004	−0.254002	+	0.967204i	$0.581747\pi$
$132$	0	0
$133$	4.64948	0.403161
$134$	13.4947	1.16576
$135$	0	0
$136$	15.9000	1.36341
$137$	0.124547	0.0106407	0.00532037	−	0.999986i	$-0.498306\pi$
$137$	0.124547	0.0106407	0.00532037	+	0.999986i	$0.498306\pi$
$138$	0	0
$139$	5.70439	0.483840	0.241920	−	0.970296i	$-0.422223\pi$
$139$	5.70439	0.483840	0.241920	+	0.970296i	$0.422223\pi$
$140$	−15.6747	−1.32475
$141$	0	0
$142$	6.46881	0.542851
$143$	−5.65147	−0.472600
$144$	0	0
$145$	3.20524	0.266181
$146$	35.9546	2.97562
$147$	0	0
$148$	10.5851	0.870092
$149$	20.0285	1.64080	0.820399	−	0.571792i	$-0.193751\pi$
$149$	20.0285	1.64080	0.820399	+	0.571792i	$0.193751\pi$
$150$	0	0
$151$	8.05186	0.655251	0.327626	−	0.944808i	$-0.393752\pi$
$151$	8.05186	0.655251	0.327626	+	0.944808i	$0.393752\pi$
$152$	10.9676	0.889589
$153$	0	0
$154$	−3.99087	−0.321594
$155$	−13.7224	−1.10221
$156$	0	0
$157$	10.2781	0.820281	0.410141	−	0.912022i	$-0.365480\pi$
$157$	10.2781	0.820281	0.410141	+	0.912022i	$0.365480\pi$
$158$	−3.51258	−0.279446
$159$	0	0
$160$	7.50572	0.593379
$161$	−10.2954	−0.811394
$162$	0	0
$163$	−6.68649	−0.523727	−0.261863	−	0.965105i	$-0.584337\pi$
$163$	−6.68649	−0.523727	−0.261863	+	0.965105i	$0.584337\pi$
$164$	−23.1351	−1.80655
$165$	0	0
$166$	5.21272	0.404585
$167$	−14.0284	−1.08555	−0.542773	−	0.839879i	$-0.682626\pi$
$167$	−14.0284	−1.08555	−0.542773	+	0.839879i	$0.682626\pi$
$168$	0	0
$169$	18.9391	1.45686
$170$	−24.4054	−1.87180
$171$	0	0
$172$	34.2061	2.60819
$173$	−10.3570	−0.787430	−0.393715	−	0.919233i	$-0.628810\pi$
$173$	−10.3570	−0.787430	−0.393715	+	0.919233i	$0.628810\pi$
$174$	0	0
$175$	2.53569	0.191680
$176$	−2.08899	−0.157464
$177$	0	0
$178$	31.3036	2.34630
$179$	7.34446	0.548951	0.274476	−	0.961594i	$-0.411496\pi$
$179$	7.34446	0.548951	0.274476	+	0.961594i	$0.411496\pi$
$180$	0	0
$181$	−0.531235	−0.0394864	−0.0197432	−	0.999805i	$-0.506285\pi$
$181$	−0.531235	−0.0394864	−0.0197432	+	0.999805i	$0.506285\pi$
$182$	22.5543	1.67184
$183$	0	0
$184$	−24.2858	−1.79037
$185$	−7.37516	−0.542233
$186$	0	0
$187$	−4.01907	−0.293903
$188$	31.9520	2.33034
$189$	0	0
$190$	−16.8345	−1.22130
$191$	−16.5231	−1.19557	−0.597785	−	0.801656i	$-0.703952\pi$
$191$	−16.5231	−1.19557	−0.597785	+	0.801656i	$0.703952\pi$
$192$	0	0
$193$	17.9043	1.28878	0.644391	−	0.764697i	$-0.277111\pi$
$193$	17.9043	1.28878	0.644391	+	0.764697i	$0.277111\pi$
$194$	−18.6686	−1.34033
$195$	0	0
$196$	−15.3360	−1.09543
$197$	−9.54976	−0.680392	−0.340196	−	0.940355i	$-0.610493\pi$
$197$	−9.54976	−0.680392	−0.340196	+	0.940355i	$0.610493\pi$
$198$	0	0
$199$	7.91086	0.560786	0.280393	−	0.959885i	$-0.409535\pi$
$199$	7.91086	0.560786	0.280393	+	0.959885i	$0.409535\pi$
$200$	5.98142	0.422950
$201$	0	0
$202$	7.69764	0.541604
$203$	−2.10654	−0.147850
$204$	0	0
$205$	16.1194	1.12582
$206$	−18.7120	−1.30372
$207$	0	0
$208$	11.8059	0.818590
$209$	−2.77230	−0.191764
$210$	0	0
$211$	−8.64980	−0.595477	−0.297738	−	0.954648i	$-0.596232\pi$
$211$	−8.64980	−0.595477	−0.297738	+	0.954648i	$0.596232\pi$
$212$	17.9673	1.23400
$213$	0	0
$214$	44.6227	3.05035
$215$	−23.8330	−1.62540
$216$	0	0
$217$	9.01857	0.612220
$218$	−46.5608	−3.15349
$219$	0	0
$220$	9.34619	0.630120
$221$	22.7136	1.52788
$222$	0	0
$223$	8.83430	0.591588	0.295794	−	0.955252i	$-0.404416\pi$
$223$	8.83430	0.591588	0.295794	+	0.955252i	$0.404416\pi$
$224$	−4.93288	−0.329592
$225$	0	0
$226$	−27.2360	−1.81171
$227$	−15.6307	−1.03744	−0.518722	−	0.854943i	$-0.673592\pi$
$227$	−15.6307	−1.03744	−0.518722	+	0.854943i	$0.673592\pi$
$228$	0	0
$229$	−13.1402	−0.868329	−0.434164	−	0.900834i	$-0.642956\pi$
$229$	−13.1402	−0.868329	−0.434164	+	0.900834i	$0.642956\pi$
$230$	37.2770	2.45797
$231$	0	0
$232$	−4.96908	−0.326236
$233$	19.2138	1.25874	0.629368	−	0.777108i	$-0.283314\pi$
$233$	19.2138	1.25874	0.629368	+	0.777108i	$0.283314\pi$
$234$	0	0
$235$	−22.2625	−1.45224
$236$	17.1593	1.11698
$237$	0	0
$238$	16.0396	1.03969
$239$	−13.3971	−0.866585	−0.433293	−	0.901253i	$-0.642648\pi$
$239$	−13.3971	−0.866585	−0.433293	+	0.901253i	$0.642648\pi$
$240$	0	0
$241$	−25.2756	−1.62814	−0.814072	−	0.580765i	$-0.802754\pi$
$241$	−25.2756	−1.62814	−0.814072	+	0.580765i	$0.802754\pi$
$242$	2.37960	0.152967
$243$	0	0
$244$	−3.66252	−0.234469
$245$	10.6853	0.682659
$246$	0	0
$247$	15.6676	0.996905
$248$	21.2738	1.35089
$249$	0	0
$250$	21.1809	1.33960
$251$	−16.2067	−1.02296	−0.511480	−	0.859295i	$-0.670903\pi$
$251$	−16.2067	−1.02296	−0.511480	+	0.859295i	$0.670903\pi$
$252$	0	0
$253$	6.13877	0.385941
$254$	3.88366	0.243682
$255$	0	0
$256$	−26.9380	−1.68363
$257$	16.7068	1.04214	0.521072	−	0.853513i	$-0.325532\pi$
$257$	16.7068	1.04214	0.521072	+	0.853513i	$0.325532\pi$
$258$	0	0
$259$	4.84708	0.301183
$260$	−52.8197	−3.27574
$261$	0	0
$262$	−13.8359	−0.854785
$263$	−10.1887	−0.628263	−0.314132	−	0.949379i	$-0.601713\pi$
$263$	−10.1887	−0.628263	−0.314132	+	0.949379i	$0.601713\pi$
$264$	0	0
$265$	−12.5187	−0.769017
$266$	11.0639	0.678372
$267$	0	0
$268$	20.7700	1.26873
$269$	−1.61445	−0.0984350	−0.0492175	−	0.998788i	$-0.515673\pi$
$269$	−1.61445	−0.0984350	−0.0492175	+	0.998788i	$0.515673\pi$
$270$	0	0
$271$	25.4804	1.54782	0.773912	−	0.633294i	$-0.218297\pi$
$271$	25.4804	1.54782	0.773912	+	0.633294i	$0.218297\pi$
$272$	8.39580	0.509070
$273$	0	0
$274$	0.296372	0.0179045
$275$	−1.51194	−0.0911732
$276$	0	0
$277$	17.4548	1.04876	0.524378	−	0.851486i	$-0.324298\pi$
$277$	17.4548	1.04876	0.524378	+	0.851486i	$0.324298\pi$
$278$	13.5742	0.814125
$279$	0	0
$280$	−16.9312	−1.01184
$281$	16.5498	0.987275	0.493638	−	0.869668i	$-0.335667\pi$
$281$	16.5498	0.987275	0.493638	+	0.869668i	$0.335667\pi$
$282$	0	0
$283$	21.5787	1.28272	0.641359	−	0.767241i	$-0.278371\pi$
$283$	21.5787	1.28272	0.641359	+	0.767241i	$0.278371\pi$
$284$	9.95633	0.590800
$285$	0	0
$286$	−13.4483	−0.795213
$287$	−10.5939	−0.625338
$288$	0	0
$289$	−0.847100	−0.0498294
$290$	7.62720	0.447885
$291$	0	0
$292$	55.3387	3.23845
$293$	3.17063	0.185230	0.0926150	−	0.995702i	$-0.470477\pi$
$293$	3.17063	0.185230	0.0926150	+	0.995702i	$0.470477\pi$
$294$	0	0
$295$	−11.9557	−0.696090
$296$	11.4337	0.664571
$297$	0	0
$298$	47.6599	2.76086
$299$	−34.6931	−2.00635
$300$	0	0
$301$	15.6634	0.902825
$302$	19.1602	1.10255
$303$	0	0
$304$	5.79132	0.332155
$305$	2.55185	0.146118
$306$	0	0
$307$	11.3935	0.650259	0.325130	−	0.945669i	$-0.394592\pi$
$307$	11.3935	0.650259	0.325130	+	0.945669i	$0.394592\pi$
$308$	−6.14247	−0.350000
$309$	0	0
$310$	−32.6538	−1.85461
$311$	−13.0228	−0.738456	−0.369228	−	0.929339i	$-0.620378\pi$
$311$	−13.0228	−0.738456	−0.369228	+	0.929339i	$0.620378\pi$
$312$	0	0
$313$	−31.9422	−1.80548	−0.902740	−	0.430186i	$-0.858448\pi$
$313$	−31.9422	−1.80548	−0.902740	+	0.430186i	$0.858448\pi$
$314$	24.4578	1.38023
$315$	0	0
$316$	−5.40631	−0.304129
$317$	−4.27975	−0.240375	−0.120187	−	0.992751i	$-0.538350\pi$
$317$	−4.27975	−0.240375	−0.120187	+	0.992751i	$0.538350\pi$
$318$	0	0
$319$	1.25605	0.0703250
$320$	28.5222	1.59444
$321$	0	0
$322$	−24.4991	−1.36528
$323$	11.1421	0.619962
$324$	0	0
$325$	8.54466	0.473973
$326$	−15.9112	−0.881240
$327$	0	0
$328$	−24.9898	−1.37983
$329$	14.6313	0.806647
$330$	0	0
$331$	−30.6153	−1.68277	−0.841385	−	0.540436i	$-0.818259\pi$
$331$	−30.6153	−1.68277	−0.841385	+	0.540436i	$0.818259\pi$
$332$	8.02304	0.440321
$333$	0	0
$334$	−33.3819	−1.82658
$335$	−14.4715	−0.790662
$336$	0	0
$337$	20.1363	1.09689	0.548447	−	0.836185i	$-0.315219\pi$
$337$	20.1363	1.09689	0.548447	+	0.836185i	$0.315219\pi$
$338$	45.0676	2.45136
$339$	0	0
$340$	−37.5630	−2.03714
$341$	−5.37742	−0.291204
$342$	0	0
$343$	−18.7624	−1.01307
$344$	36.9483	1.99212
$345$	0	0
$346$	−24.6456	−1.32496
$347$	31.7798	1.70603	0.853015	−	0.521887i	$-0.174772\pi$
$347$	31.7798	1.70603	0.853015	+	0.521887i	$0.174772\pi$
$348$	0	0
$349$	−28.4164	−1.52110	−0.760548	−	0.649282i	$-0.775069\pi$
$349$	−28.4164	−1.52110	−0.760548	+	0.649282i	$0.775069\pi$
$350$	6.03395	0.322528
$351$	0	0
$352$	2.94129	0.156771
$353$	−13.9859	−0.744392	−0.372196	−	0.928154i	$-0.621395\pi$
$353$	−13.9859	−0.744392	−0.372196	+	0.928154i	$0.621395\pi$
$354$	0	0
$355$	−6.93705	−0.368180
$356$	48.1803	2.55355
$357$	0	0
$358$	17.4769	0.923684
$359$	−9.57884	−0.505552	−0.252776	−	0.967525i	$-0.581344\pi$
$359$	−9.57884	−0.505552	−0.252776	+	0.967525i	$0.581344\pi$
$360$	0	0
$361$	−11.3143	−0.595491
$362$	−1.26413	−0.0664412
$363$	0	0
$364$	34.7140	1.81951
$365$	−38.5571	−2.01817
$366$	0	0
$367$	−20.5751	−1.07401	−0.537006	−	0.843578i	$-0.680445\pi$
$367$	−20.5751	−1.07401	−0.537006	+	0.843578i	$0.680445\pi$
$368$	−12.8238	−0.668489
$369$	0	0
$370$	−17.5500	−0.912379
$371$	8.22749	0.427150
$372$	0	0
$373$	27.0493	1.40056	0.700280	−	0.713868i	$-0.253058\pi$
$373$	27.0493	1.40056	0.700280	+	0.713868i	$0.253058\pi$
$374$	−9.56379	−0.494532
$375$	0	0
$376$	34.5135	1.77990
$377$	−7.09850	−0.365592
$378$	0	0
$379$	−18.5659	−0.953665	−0.476832	−	0.878994i	$-0.658215\pi$
$379$	−18.5659	−0.953665	−0.476832	+	0.878994i	$0.658215\pi$
$380$	−25.9105	−1.32918
$381$	0	0
$382$	−39.3185	−2.01171
$383$	−3.36015	−0.171696	−0.0858478	−	0.996308i	$-0.527360\pi$
$383$	−3.36015	−0.171696	−0.0858478	+	0.996308i	$0.527360\pi$
$384$	0	0
$385$	4.27975	0.218116
$386$	42.6052	2.16855
$387$	0	0
$388$	−28.7334	−1.45872
$389$	−10.0729	−0.510716	−0.255358	−	0.966847i	$-0.582193\pi$
$389$	−10.0729	−0.510716	−0.255358	+	0.966847i	$0.582193\pi$
$390$	0	0
$391$	−24.6721	−1.24772
$392$	−16.5654	−0.836680
$393$	0	0
$394$	−22.7246	−1.14485
$395$	3.76683	0.189530
$396$	0	0
$397$	−11.9242	−0.598461	−0.299230	−	0.954181i	$-0.596730\pi$
$397$	−11.9242	−0.598461	−0.299230	+	0.954181i	$0.596730\pi$
$398$	18.8247	0.943598
$399$	0	0
$400$	3.15842	0.157921
$401$	−11.2101	−0.559804	−0.279902	−	0.960029i	$-0.590302\pi$
$401$	−11.2101	−0.559804	−0.279902	+	0.960029i	$0.590302\pi$
$402$	0	0
$403$	30.3904	1.51385
$404$	11.8477	0.589443
$405$	0	0
$406$	−5.01272	−0.248777
$407$	−2.89012	−0.143258
$408$	0	0
$409$	−29.5649	−1.46189	−0.730944	−	0.682437i	$-0.760920\pi$
$409$	−29.5649	−1.46189	−0.730944	+	0.682437i	$0.760920\pi$
$410$	38.3577	1.89435
$411$	0	0
$412$	−28.8001	−1.41888
$413$	7.85750	0.386642
$414$	0	0
$415$	−5.59003	−0.274404
$416$	−16.6226	−0.814990
$417$	0	0
$418$	−6.59699	−0.322669
$419$	−13.1249	−0.641194	−0.320597	−	0.947216i	$-0.603884\pi$
$419$	−13.1249	−0.641194	−0.320597	+	0.947216i	$0.603884\pi$
$420$	0	0
$421$	−39.4833	−1.92430	−0.962149	−	0.272524i	$-0.912142\pi$
$421$	−39.4833	−1.92430	−0.962149	+	0.272524i	$0.912142\pi$
$422$	−20.5831	−1.00197
$423$	0	0
$424$	19.4077	0.942522
$425$	6.07657	0.294757
$426$	0	0
$427$	−1.67712	−0.0811614
$428$	68.6801	3.31978
$429$	0	0
$430$	−56.7131	−2.73495
$431$	−8.99935	−0.433483	−0.216742	−	0.976229i	$-0.569543\pi$
$431$	−8.99935	−0.433483	−0.216742	+	0.976229i	$0.569543\pi$
$432$	0	0
$433$	−19.8497	−0.953916	−0.476958	−	0.878926i	$-0.658261\pi$
$433$	−19.8497	−0.953916	−0.476958	+	0.878926i	$0.658261\pi$
$434$	21.4606	1.03014
$435$	0	0
$436$	−71.6630	−3.43203
$437$	−17.0185	−0.814107
$438$	0	0
$439$	24.7912	1.18322	0.591609	−	0.806225i	$-0.298493\pi$
$439$	24.7912	1.18322	0.591609	+	0.806225i	$0.298493\pi$
$440$	10.0954	0.481282
$441$	0	0
$442$	54.0495	2.57087
$443$	−11.8657	−0.563758	−0.281879	−	0.959450i	$-0.590958\pi$
$443$	−11.8657	−0.563758	−0.281879	+	0.959450i	$0.590958\pi$
$444$	0	0
$445$	−33.5695	−1.59135
$446$	21.0221	0.995427
$447$	0	0
$448$	−18.7453	−0.885631
$449$	−5.23662	−0.247132	−0.123566	−	0.992336i	$-0.539433\pi$
$449$	−5.23662	−0.247132	−0.123566	+	0.992336i	$0.539433\pi$
$450$	0	0
$451$	6.31673	0.297443
$452$	−41.9197	−1.97174
$453$	0	0
$454$	−37.1948	−1.74564
$455$	−24.1869	−1.13390
$456$	0	0
$457$	9.12526	0.426862	0.213431	−	0.976958i	$-0.431536\pi$
$457$	9.12526	0.426862	0.213431	+	0.976958i	$0.431536\pi$
$458$	−31.2685	−1.46108
$459$	0	0
$460$	57.3741	2.67508
$461$	−2.08183	−0.0969604	−0.0484802	−	0.998824i	$-0.515438\pi$
$461$	−2.08183	−0.0969604	−0.0484802	+	0.998824i	$0.515438\pi$
$462$	0	0
$463$	27.5553	1.28060	0.640302	−	0.768124i	$-0.278809\pi$
$463$	27.5553	1.28060	0.640302	+	0.768124i	$0.278809\pi$
$464$	−2.62387	−0.121810
$465$	0	0
$466$	45.7211	2.11799
$467$	−27.9889	−1.29517	−0.647587	−	0.761992i	$-0.724222\pi$
$467$	−27.9889	−1.29517	−0.647587	+	0.761992i	$0.724222\pi$
$468$	0	0
$469$	9.51089	0.439172
$470$	−52.9759	−2.44359
$471$	0	0
$472$	18.5350	0.853141
$473$	−9.33950	−0.429431
$474$	0	0
$475$	4.19155	0.192321
$476$	24.6870	1.13153
$477$	0	0
$478$	−31.8798	−1.45815
$479$	−0.953001	−0.0435437	−0.0217719	−	0.999763i	$-0.506931\pi$
$479$	−0.953001	−0.0435437	−0.0217719	+	0.999763i	$0.506931\pi$
$480$	0	0
$481$	16.3335	0.744741
$482$	−60.1459	−2.73957
$483$	0	0
$484$	3.66252	0.166478
$485$	20.0199	0.909058
$486$	0	0
$487$	−12.8346	−0.581591	−0.290796	−	0.956785i	$-0.593920\pi$
$487$	−12.8346	−0.581591	−0.290796	+	0.956785i	$0.593920\pi$
$488$	−3.95613	−0.179086
$489$	0	0
$490$	25.4268	1.14867
$491$	−39.6400	−1.78893	−0.894465	−	0.447138i	$-0.852443\pi$
$491$	−39.6400	−1.78893	−0.894465	+	0.447138i	$0.852443\pi$
$492$	0	0
$493$	−5.04813	−0.227356
$494$	37.2827	1.67743
$495$	0	0
$496$	11.2334	0.504394
$497$	4.55914	0.204505
$498$	0	0
$499$	−35.1235	−1.57234	−0.786172	−	0.618007i	$-0.787940\pi$
$499$	−35.1235	−1.57234	−0.786172	+	0.618007i	$0.787940\pi$
$500$	32.6001	1.45792
$501$	0	0
$502$	−38.5656	−1.72127
$503$	−2.69005	−0.119943	−0.0599717	−	0.998200i	$-0.519101\pi$
$503$	−2.69005	−0.119943	−0.0599717	+	0.998200i	$0.519101\pi$
$504$	0	0
$505$	−8.25482	−0.367335
$506$	14.6078	0.649398
$507$	0	0
$508$	5.97745	0.265206
$509$	−22.4595	−0.995500	−0.497750	−	0.867321i	$-0.665840\pi$
$509$	−22.4595	−0.995500	−0.497750	+	0.867321i	$0.665840\pi$
$510$	0	0
$511$	25.3403	1.12099
$512$	−22.6730	−1.00201
$513$	0	0
$514$	39.7557	1.75355
$515$	20.0664	0.884231
$516$	0	0
$517$	−8.72405	−0.383683
$518$	11.5341	0.506780
$519$	0	0
$520$	−57.0541	−2.50199
$521$	−24.9298	−1.09219	−0.546096	−	0.837722i	$-0.683887\pi$
$521$	−24.9298	−1.09219	−0.546096	+	0.837722i	$0.683887\pi$
$522$	0	0
$523$	−22.1551	−0.968775	−0.484387	−	0.874854i	$-0.660957\pi$
$523$	−22.1551	−0.968775	−0.484387	+	0.874854i	$0.660957\pi$
$524$	−21.2952	−0.930287
$525$	0	0
$526$	−24.2451	−1.05714
$527$	21.6122	0.941444
$528$	0	0
$529$	14.6845	0.638456
$530$	−29.7895	−1.29397
$531$	0	0
$532$	17.0288	0.738292
$533$	−35.6988	−1.54629
$534$	0	0
$535$	−47.8527	−2.06885
$536$	22.4351	0.969050
$537$	0	0
$538$	−3.84176	−0.165630
$539$	4.18728	0.180359
$540$	0	0
$541$	34.9763	1.50375	0.751874	−	0.659307i	$-0.229150\pi$
$541$	34.9763	1.50375	0.751874	+	0.659307i	$0.229150\pi$
$542$	60.6332	2.60442
$543$	0	0
$544$	−11.8212	−0.506831
$545$	49.9310	2.13881
$546$	0	0
$547$	41.5082	1.77476	0.887382	−	0.461035i	$-0.152522\pi$
$547$	41.5082	1.77476	0.887382	+	0.461035i	$0.152522\pi$
$548$	0.456154	0.0194859
$549$	0	0
$550$	−3.59781	−0.153411
$551$	−3.48214	−0.148344
$552$	0	0
$553$	−2.47562	−0.105274
$554$	41.5354	1.76467
$555$	0	0
$556$	20.8924	0.886036
$557$	13.4014	0.567836	0.283918	−	0.958849i	$-0.408366\pi$
$557$	13.4014	0.567836	0.283918	+	0.958849i	$0.408366\pi$
$558$	0	0
$559$	52.7819	2.23244
$560$	−8.94036	−0.377799
$561$	0	0
$562$	39.3819	1.66122
$563$	10.6144	0.447343	0.223671	−	0.974665i	$-0.428196\pi$
$563$	10.6144	0.447343	0.223671	+	0.974665i	$0.428196\pi$
$564$	0	0
$565$	29.2074	1.22877
$566$	51.3486	2.15834
$567$	0	0
$568$	10.7545	0.451249
$569$	25.0719	1.05107	0.525535	−	0.850772i	$-0.323865\pi$
$569$	25.0719	1.05107	0.525535	+	0.850772i	$0.323865\pi$
$570$	0	0
$571$	−1.81795	−0.0760790	−0.0380395	−	0.999276i	$-0.512111\pi$
$571$	−1.81795	−0.0760790	−0.0380395	+	0.999276i	$0.512111\pi$
$572$	−20.6986	−0.865452
$573$	0	0
$574$	−25.2093	−1.05222
$575$	−9.28143	−0.387062
$576$	0	0
$577$	32.7010	1.36136	0.680681	−	0.732580i	$-0.261684\pi$
$577$	32.7010	1.36136	0.680681	+	0.732580i	$0.261684\pi$
$578$	−2.01576	−0.0838447
$579$	0	0
$580$	11.7392	0.487445
$581$	3.67386	0.152417
$582$	0	0
$583$	−4.90573	−0.203175
$584$	59.7750	2.47351
$585$	0	0
$586$	7.54484	0.311674
$587$	−24.0218	−0.991488	−0.495744	−	0.868469i	$-0.665104\pi$
$587$	−24.0218	−0.991488	−0.495744	+	0.868469i	$0.665104\pi$
$588$	0	0
$589$	14.9079	0.614268
$590$	−28.4499	−1.17126
$591$	0	0
$592$	6.03745	0.248138
$593$	3.62291	0.148775	0.0743875	−	0.997229i	$-0.476300\pi$
$593$	3.62291	0.148775	0.0743875	+	0.997229i	$0.476300\pi$
$594$	0	0
$595$	−17.2006	−0.705156
$596$	73.3547	3.00472
$597$	0	0
$598$	−82.5558	−3.37596
$599$	−19.0751	−0.779389	−0.389695	−	0.920944i	$-0.627419\pi$
$599$	−19.0751	−0.779389	−0.389695	+	0.920944i	$0.627419\pi$
$600$	0	0
$601$	−9.56199	−0.390042	−0.195021	−	0.980799i	$-0.562477\pi$
$601$	−9.56199	−0.390042	−0.195021	+	0.980799i	$0.562477\pi$
$602$	37.2728	1.51912
$603$	0	0
$604$	29.4901	1.19993
$605$	−2.55185	−0.103747
$606$	0	0
$607$	−33.3670	−1.35432	−0.677162	−	0.735834i	$-0.736790\pi$
$607$	−33.3670	−1.35432	−0.677162	+	0.735834i	$0.736790\pi$
$608$	−8.15414	−0.330694
$609$	0	0
$610$	6.07239	0.245864
$611$	49.3037	1.99462
$612$	0	0
$613$	21.0516	0.850268	0.425134	−	0.905131i	$-0.360227\pi$
$613$	21.0516	0.850268	0.425134	+	0.905131i	$0.360227\pi$
$614$	27.1119	1.09415
$615$	0	0
$616$	−6.63489	−0.267327
$617$	−3.39567	−0.136705	−0.0683523	−	0.997661i	$-0.521774\pi$
$617$	−3.39567	−0.136705	−0.0683523	+	0.997661i	$0.521774\pi$
$618$	0	0
$619$	−11.3446	−0.455980	−0.227990	−	0.973664i	$-0.573215\pi$
$619$	−11.3446	−0.455980	−0.227990	+	0.973664i	$0.573215\pi$
$620$	−50.2584	−2.01843
$621$	0	0
$622$	−30.9892	−1.24255
$623$	22.0624	0.883912
$624$	0	0
$625$	−30.2737	−1.21095
$626$	−76.0098	−3.03796
$627$	0	0
$628$	37.6437	1.50215
$629$	11.6156	0.463144
$630$	0	0
$631$	44.0848	1.75499	0.877494	−	0.479587i	$-0.159214\pi$
$631$	44.0848	1.75499	0.877494	+	0.479587i	$0.159214\pi$
$632$	−5.83972	−0.232291
$633$	0	0
$634$	−10.1841	−0.404463
$635$	−4.16477	−0.165274
$636$	0	0
$637$	−23.6643	−0.937613
$638$	2.98889	0.118331
$639$	0	0
$640$	52.8602	2.08948
$641$	5.41854	0.214019	0.107010	−	0.994258i	$-0.465872\pi$
$641$	5.41854	0.214019	0.107010	+	0.994258i	$0.465872\pi$
$642$	0	0
$643$	19.3418	0.762768	0.381384	−	0.924417i	$-0.375448\pi$
$643$	19.3418	0.762768	0.381384	+	0.924417i	$0.375448\pi$
$644$	−37.7072	−1.48587
$645$	0	0
$646$	26.5137	1.04317
$647$	−21.8968	−0.860852	−0.430426	−	0.902626i	$-0.641637\pi$
$647$	−21.8968	−0.860852	−0.430426	+	0.902626i	$0.641637\pi$
$648$	0	0
$649$	−4.68512	−0.183907
$650$	20.3329	0.797523
$651$	0	0
$652$	−24.4894	−0.959078
$653$	4.80308	0.187959	0.0939795	−	0.995574i	$-0.470041\pi$
$653$	4.80308	0.187959	0.0939795	+	0.995574i	$0.470041\pi$
$654$	0	0
$655$	14.8374	0.579746
$656$	−13.1956	−0.515202
$657$	0	0
$658$	34.8166	1.35729
$659$	29.9093	1.16510	0.582550	−	0.812795i	$-0.302055\pi$
$659$	29.9093	1.16510	0.582550	+	0.812795i	$0.302055\pi$
$660$	0	0
$661$	−33.4154	−1.29971	−0.649854	−	0.760059i	$-0.725170\pi$
$661$	−33.4154	−1.29971	−0.649854	+	0.760059i	$0.725170\pi$
$662$	−72.8523	−2.83149
$663$	0	0
$664$	8.66623	0.336315
$665$	−11.8648	−0.460096
$666$	0	0
$667$	7.71057	0.298555
$668$	−51.3791	−1.98792
$669$	0	0
$670$	−34.4364	−1.33039
$671$	1.00000	0.0386046
$672$	0	0
$673$	−7.76575	−0.299348	−0.149674	−	0.988735i	$-0.547822\pi$
$673$	−7.76575	−0.299348	−0.149674	+	0.988735i	$0.547822\pi$
$674$	47.9165	1.84567
$675$	0	0
$676$	69.3649	2.66788
$677$	−32.9643	−1.26692	−0.633460	−	0.773775i	$-0.718366\pi$
$677$	−32.9643	−1.26692	−0.633460	+	0.773775i	$0.718366\pi$
$678$	0	0
$679$	−13.1574	−0.504935
$680$	−40.5743	−1.55595
$681$	0	0
$682$	−12.7961	−0.489990
$683$	16.7372	0.640429	0.320215	−	0.947345i	$-0.396245\pi$
$683$	16.7372	0.640429	0.320215	+	0.947345i	$0.396245\pi$
$684$	0	0
$685$	−0.317824	−0.0121434
$686$	−44.6470	−1.70463
$687$	0	0
$688$	19.5101	0.743817
$689$	27.7246	1.05622
$690$	0	0
$691$	−17.4790	−0.664931	−0.332465	−	0.943115i	$-0.607880\pi$
$691$	−17.4790	−0.664931	−0.332465	+	0.943115i	$0.607880\pi$
$692$	−37.9328	−1.44199
$693$	0	0
$694$	75.6233	2.87062
$695$	−14.5567	−0.552169
$696$	0	0
$697$	−25.3874	−0.961616
$698$	−67.6198	−2.55945
$699$	0	0
$700$	9.28702	0.351016
$701$	−20.9278	−0.790434	−0.395217	−	0.918588i	$-0.629331\pi$
$701$	−20.9278	−0.790434	−0.395217	+	0.918588i	$0.629331\pi$
$702$	0	0
$703$	8.01230	0.302190
$704$	11.1771	0.421252
$705$	0	0
$706$	−33.2808	−1.25254
$707$	5.42520	0.204036
$708$	0	0
$709$	−24.1466	−0.906845	−0.453423	−	0.891296i	$-0.649797\pi$
$709$	−24.1466	−0.906845	−0.453423	+	0.891296i	$0.649797\pi$
$710$	−16.5074	−0.619513
$711$	0	0
$712$	52.0427	1.95038
$713$	−33.0108	−1.23626
$714$	0	0
$715$	14.4217	0.539341
$716$	26.8992	1.00527
$717$	0	0
$718$	−22.7938	−0.850659
$719$	32.0699	1.19601	0.598003	−	0.801494i	$-0.295961\pi$
$719$	32.0699	1.19601	0.598003	+	0.801494i	$0.295961\pi$
$720$	0	0
$721$	−13.1880	−0.491145
$722$	−26.9236	−1.00199
$723$	0	0
$724$	−1.94566	−0.0723098
$725$	−1.89906	−0.0705293
$726$	0	0
$727$	−22.5250	−0.835406	−0.417703	−	0.908584i	$-0.637165\pi$
$727$	−22.5250	−0.835406	−0.417703	+	0.908584i	$0.637165\pi$
$728$	37.4969	1.38973
$729$	0	0
$730$	−91.7506	−3.39584
$731$	37.5361	1.38832
$732$	0	0
$733$	45.0196	1.66284	0.831418	−	0.555648i	$-0.187530\pi$
$733$	45.0196	1.66284	0.831418	+	0.555648i	$0.187530\pi$
$734$	−48.9606	−1.80717
$735$	0	0
$736$	18.0559	0.665549
$737$	−5.67098	−0.208893
$738$	0	0
$739$	−38.6313	−1.42108	−0.710539	−	0.703658i	$-0.751549\pi$
$739$	−38.6313	−1.42108	−0.710539	+	0.703658i	$0.751549\pi$
$740$	−27.0117	−0.992968
$741$	0	0
$742$	19.5782	0.718737
$743$	40.6081	1.48977	0.744884	−	0.667194i	$-0.232505\pi$
$743$	40.6081	1.48977	0.744884	+	0.667194i	$0.232505\pi$
$744$	0	0
$745$	−51.1097	−1.87251
$746$	64.3666	2.35663
$747$	0	0
$748$	−14.7199	−0.538213
$749$	31.4496	1.14914
$750$	0	0
$751$	24.7212	0.902088	0.451044	−	0.892502i	$-0.351052\pi$
$751$	24.7212	0.902088	0.451044	+	0.892502i	$0.351052\pi$
$752$	18.2245	0.664578
$753$	0	0
$754$	−16.8916	−0.615157
$755$	−20.5471	−0.747787
$756$	0	0
$757$	22.0864	0.802743	0.401371	−	0.915915i	$-0.368534\pi$
$757$	22.0864	0.802743	0.401371	+	0.915915i	$0.368534\pi$
$758$	−44.1794	−1.60467
$759$	0	0
$760$	−27.9877	−1.01522
$761$	4.10116	0.148667	0.0743334	−	0.997233i	$-0.476317\pi$
$761$	4.10116	0.148667	0.0743334	+	0.997233i	$0.476317\pi$
$762$	0	0
$763$	−32.8155	−1.18800
$764$	−60.5161	−2.18940
$765$	0	0
$766$	−7.99583	−0.288901
$767$	26.4778	0.956060
$768$	0	0
$769$	−1.84340	−0.0664747	−0.0332374	−	0.999447i	$-0.510582\pi$
$769$	−1.84340	−0.0664747	−0.0332374	+	0.999447i	$0.510582\pi$
$770$	10.1841	0.367010
$771$	0	0
$772$	65.5749	2.36009
$773$	46.4688	1.67137	0.835684	−	0.549211i	$-0.185072\pi$
$773$	46.4688	1.67137	0.835684	+	0.549211i	$0.185072\pi$
$774$	0	0
$775$	8.13032	0.292050
$776$	−31.0369	−1.11416
$777$	0	0
$778$	−23.9695	−0.859348
$779$	−17.5119	−0.627429
$780$	0	0
$781$	−2.71844	−0.0972734
$782$	−58.7099	−2.09946
$783$	0	0
$784$	−8.74719	−0.312400
$785$	−26.2281	−0.936123
$786$	0	0
$787$	−26.6731	−0.950795	−0.475397	−	0.879771i	$-0.657696\pi$
$787$	−26.6731	−0.950795	−0.475397	+	0.879771i	$0.657696\pi$
$788$	−34.9761	−1.24597
$789$	0	0
$790$	8.96357	0.318909
$791$	−19.1956	−0.682517
$792$	0	0
$793$	−5.65147	−0.200690
$794$	−28.3750	−1.00699
$795$	0	0
$796$	28.9737	1.02694
$797$	22.8323	0.808763	0.404381	−	0.914590i	$-0.367487\pi$
$797$	22.8323	0.808763	0.404381	+	0.914590i	$0.367487\pi$
$798$	0	0
$799$	35.0625	1.24042
$800$	−4.44704	−0.157227
$801$	0	0
$802$	−26.6755	−0.941945
$803$	−15.1095	−0.533202
$804$	0	0
$805$	26.2724	0.925980
$806$	72.3170	2.54726
$807$	0	0
$808$	12.7974	0.450213
$809$	−5.39701	−0.189749	−0.0948744	−	0.995489i	$-0.530245\pi$
$809$	−5.39701	−0.189749	−0.0948744	+	0.995489i	$0.530245\pi$
$810$	0	0
$811$	−49.8393	−1.75009	−0.875047	−	0.484037i	$-0.839170\pi$
$811$	−49.8393	−1.75009	−0.875047	+	0.484037i	$0.839170\pi$
$812$	−7.71522	−0.270751
$813$	0	0
$814$	−6.87735	−0.241051
$815$	17.0629	0.597688
$816$	0	0
$817$	25.8919	0.905844
$818$	−70.3527	−2.45982
$819$	0	0
$820$	59.0374	2.06168
$821$	−17.3451	−0.605347	−0.302674	−	0.953094i	$-0.597879\pi$
$821$	−17.3451	−0.605347	−0.302674	+	0.953094i	$0.597879\pi$
$822$	0	0
$823$	−30.8810	−1.07644	−0.538221	−	0.842804i	$-0.680904\pi$
$823$	−30.8810	−1.07644	−0.538221	+	0.842804i	$0.680904\pi$
$824$	−31.1089	−1.08373
$825$	0	0
$826$	18.6977	0.650578
$827$	−52.0029	−1.80832	−0.904160	−	0.427195i	$-0.859502\pi$
$827$	−52.0029	−1.80832	−0.904160	+	0.427195i	$0.859502\pi$
$828$	0	0
$829$	48.0581	1.66913	0.834563	−	0.550912i	$-0.185720\pi$
$829$	48.0581	1.66913	0.834563	+	0.550912i	$0.185720\pi$
$830$	−13.3021	−0.461722
$831$	0	0
$832$	−63.1670	−2.18992
$833$	−16.8290	−0.583089
$834$	0	0
$835$	35.7983	1.23885
$836$	−10.1536	−0.351170
$837$	0	0
$838$	−31.2321	−1.07890
$839$	34.1026	1.17735	0.588676	−	0.808369i	$-0.299649\pi$
$839$	34.1026	1.17735	0.588676	+	0.808369i	$0.299649\pi$
$840$	0	0
$841$	−27.4223	−0.945598
$842$	−93.9546	−3.23789
$843$	0	0
$844$	−31.6800	−1.09047
$845$	−48.3298	−1.66260
$846$	0	0
$847$	1.67712	0.0576264
$848$	10.2480	0.351919
$849$	0	0
$850$	14.4598	0.495968
$851$	−17.7418	−0.608181
$852$	0	0
$853$	−1.66881	−0.0571391	−0.0285695	−	0.999592i	$-0.509095\pi$
$853$	−1.66881	−0.0571391	−0.0285695	+	0.999592i	$0.509095\pi$
$854$	−3.99087	−0.136565
$855$	0	0
$856$	74.1860	2.53562
$857$	52.5109	1.79374	0.896869	−	0.442297i	$-0.145836\pi$
$857$	52.5109	1.79374	0.896869	+	0.442297i	$0.145836\pi$
$858$	0	0
$859$	7.22377	0.246472	0.123236	−	0.992377i	$-0.460673\pi$
$859$	7.22377	0.246472	0.123236	+	0.992377i	$0.460673\pi$
$860$	−87.2887	−2.97652
$861$	0	0
$862$	−21.4149	−0.729394
$863$	−47.7485	−1.62538	−0.812688	−	0.582698i	$-0.801997\pi$
$863$	−47.7485	−1.62538	−0.812688	+	0.582698i	$0.801997\pi$
$864$	0	0
$865$	26.4296	0.898632
$866$	−47.2344	−1.60509
$867$	0	0
$868$	33.0307	1.12113
$869$	1.47612	0.0500739
$870$	0	0
$871$	32.0494	1.08595
$872$	−77.4080	−2.62137
$873$	0	0
$874$	−40.4974	−1.36984
$875$	14.9280	0.504660
$876$	0	0
$877$	−17.6923	−0.597427	−0.298714	−	0.954343i	$-0.596558\pi$
$877$	−17.6923	−0.597427	−0.298714	+	0.954343i	$0.596558\pi$
$878$	58.9932	1.99092
$879$	0	0
$880$	5.33079	0.179701
$881$	−18.9137	−0.637219	−0.318610	−	0.947886i	$-0.603216\pi$
$881$	−18.9137	−0.637219	−0.318610	+	0.947886i	$0.603216\pi$
$882$	0	0
$883$	59.1263	1.98976	0.994878	−	0.101079i	$-0.0322294\pi$
$883$	59.1263	1.98976	0.994878	+	0.101079i	$0.0322294\pi$
$884$	83.1891	2.79795
$885$	0	0
$886$	−28.2357	−0.948599
$887$	34.1377	1.14623	0.573117	−	0.819474i	$-0.305734\pi$
$887$	34.1377	1.14623	0.573117	+	0.819474i	$0.305734\pi$
$888$	0	0
$889$	2.73716	0.0918012
$890$	−79.8821	−2.67765
$891$	0	0
$892$	32.3558	1.08335
$893$	24.1857	0.809344
$894$	0	0
$895$	−18.7420	−0.626475
$896$	−34.7406	−1.16060
$897$	0	0
$898$	−12.4611	−0.415832
$899$	−6.75429	−0.225268
$900$	0	0
$901$	19.7165	0.656851
$902$	15.0313	0.500488
$903$	0	0
$904$	−45.2803	−1.50600
$905$	1.35563	0.0450628
$906$	0	0
$907$	8.40299	0.279017	0.139508	−	0.990221i	$-0.455448\pi$
$907$	8.40299	0.279017	0.139508	+	0.990221i	$0.455448\pi$
$908$	−57.2476	−1.89983
$909$	0	0
$910$	−57.5552	−1.90794
$911$	12.5927	0.417214	0.208607	−	0.978000i	$-0.433107\pi$
$911$	12.5927	0.417214	0.208607	+	0.978000i	$0.433107\pi$
$912$	0	0
$913$	−2.19058	−0.0724977
$914$	21.7145	0.718252
$915$	0	0
$916$	−48.1262	−1.59013
$917$	−9.75138	−0.322019
$918$	0	0
$919$	51.7102	1.70576	0.852881	−	0.522105i	$-0.174853\pi$
$919$	51.7102	1.70576	0.852881	+	0.522105i	$0.174853\pi$
$920$	61.9736	2.04321
$921$	0	0
$922$	−4.95393	−0.163149
$923$	15.3632	0.505685
$924$	0	0
$925$	4.36968	0.143674
$926$	65.5707	2.15479
$927$	0	0
$928$	3.69439	0.121274
$929$	43.9138	1.44076	0.720382	−	0.693577i	$-0.243966\pi$
$929$	43.9138	1.44076	0.720382	+	0.693577i	$0.243966\pi$
$930$	0	0
$931$	−11.6084	−0.380450
$932$	70.3707	2.30507
$933$	0	0
$934$	−66.6026	−2.17930
$935$	10.2561	0.335409
$936$	0	0
$937$	14.6324	0.478021	0.239010	−	0.971017i	$-0.423177\pi$
$937$	14.6324	0.478021	0.239010	+	0.971017i	$0.423177\pi$
$938$	22.6322	0.738966
$939$	0	0
$940$	−81.5366	−2.65943
$941$	−46.6148	−1.51960	−0.759799	−	0.650158i	$-0.774703\pi$
$941$	−46.6148	−1.51960	−0.759799	+	0.650158i	$0.774703\pi$
$942$	0	0
$943$	38.7770	1.26275
$944$	9.78719	0.318546
$945$	0	0
$946$	−22.2243	−0.722575
$947$	−6.99474	−0.227299	−0.113649	−	0.993521i	$-0.536254\pi$
$947$	−6.99474	−0.227299	−0.113649	+	0.993521i	$0.536254\pi$
$948$	0	0
$949$	85.3907	2.77190
$950$	9.97422	0.323607
$951$	0	0
$952$	26.6661	0.864253
$953$	−39.6795	−1.28534	−0.642672	−	0.766142i	$-0.722174\pi$
$953$	−39.6795	−1.28534	−0.642672	+	0.766142i	$0.722174\pi$
$954$	0	0
$955$	42.1645	1.36441
$956$	−49.0670	−1.58694
$957$	0	0
$958$	−2.26777	−0.0732682
$959$	0.208879	0.00674506
$960$	0	0
$961$	−2.08330	−0.0672033
$962$	38.8672	1.25313
$963$	0	0
$964$	−92.5722	−2.98155
$965$	−45.6891	−1.47079
$966$	0	0
$967$	−56.5820	−1.81956	−0.909778	−	0.415096i	$-0.863748\pi$
$967$	−56.5820	−1.81956	−0.909778	+	0.415096i	$0.863748\pi$
$968$	3.95613	0.127155
$969$	0	0
$970$	47.6395	1.52961
$971$	−23.7722	−0.762885	−0.381443	−	0.924392i	$-0.624573\pi$
$971$	−23.7722	−0.762885	−0.381443	+	0.924392i	$0.624573\pi$
$972$	0	0
$973$	9.56692	0.306702
$974$	−30.5413	−0.978605
$975$	0	0
$976$	−2.08899	−0.0668670
$977$	43.0284	1.37660	0.688300	−	0.725426i	$-0.258357\pi$
$977$	43.0284	1.37660	0.688300	+	0.725426i	$0.258357\pi$
$978$	0	0
$979$	−13.1550	−0.420434
$980$	39.1351	1.25013
$981$	0	0
$982$	−94.3276	−3.01011
$983$	−28.8215	−0.919263	−0.459631	−	0.888110i	$-0.652018\pi$
$983$	−28.8215	−0.919263	−0.459631	+	0.888110i	$0.652018\pi$
$984$	0	0
$985$	24.3695	0.776478
$986$	−12.0126	−0.382558
$987$	0	0
$988$	57.3828	1.82559
$989$	−57.3330	−1.82308
$990$	0	0
$991$	5.66098	0.179827	0.0899135	−	0.995950i	$-0.471341\pi$
$991$	5.66098	0.179827	0.0899135	+	0.995950i	$0.471341\pi$
$992$	−15.8165	−0.502176
$993$	0	0
$994$	10.8490	0.344108
$995$	−20.1873	−0.639981
$996$	0	0
$997$	−50.8029	−1.60894	−0.804472	−	0.593991i	$-0.797552\pi$
$997$	−50.8029	−1.60894	−0.804472	+	0.593991i	$0.797552\pi$
$998$	−83.5801	−2.64568
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.12		13
3.2	odd	2		2013.2.a.e.1.2	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.2	✓	13	3.2	odd	2
6039.2.a.i.1.12		13	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)

\(f(q)\)	\(=\)	\(q+2.37960 q^{2} +3.66252 q^{4} -2.55185 q^{5} +1.67712 q^{7} +3.95613 q^{8} +O(q^{10})\)	\(q+2.37960 q^{2} +3.66252 q^{4} -2.55185 q^{5} +1.67712 q^{7} +3.95613 q^{8} -6.07239 q^{10} -1.00000 q^{11} +5.65147 q^{13} +3.99087 q^{14} +2.08899 q^{16} +4.01907 q^{17} +2.77230 q^{19} -9.34619 q^{20} -2.37960 q^{22} -6.13877 q^{23} +1.51194 q^{25} +13.4483 q^{26} +6.14247 q^{28} -1.25605 q^{29} +5.37742 q^{31} -2.94129 q^{32} +9.56379 q^{34} -4.27975 q^{35} +2.89012 q^{37} +6.59699 q^{38} -10.0954 q^{40} -6.31673 q^{41} +9.33950 q^{43} -3.66252 q^{44} -14.6078 q^{46} +8.72405 q^{47} -4.18728 q^{49} +3.59781 q^{50} +20.6986 q^{52} +4.90573 q^{53} +2.55185 q^{55} +6.63489 q^{56} -2.98889 q^{58} +4.68512 q^{59} -1.00000 q^{61} +12.7961 q^{62} -11.1771 q^{64} -14.4217 q^{65} +5.67098 q^{67} +14.7199 q^{68} -10.1841 q^{70} +2.71844 q^{71} +15.1095 q^{73} +6.87735 q^{74} +10.1536 q^{76} -1.67712 q^{77} -1.47612 q^{79} -5.33079 q^{80} -15.0313 q^{82} +2.19058 q^{83} -10.2561 q^{85} +22.2243 q^{86} -3.95613 q^{88} +13.1550 q^{89} +9.47818 q^{91} -22.4833 q^{92} +20.7598 q^{94} -7.07450 q^{95} -7.84526 q^{97} -9.96407 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more