LMFDB - Embedded newform 6039.2.a.n.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.77875 q^{2} +5.72143 q^{4} +2.25657 q^{5} +0.879084 q^{7} -10.3409 q^{8} +O(q^{10})$	\(q-2.77875 q^{2} +5.72143 q^{4} +2.25657 q^{5} +0.879084 q^{7} -10.3409 q^{8} -6.27043 q^{10} -1.00000 q^{11} +3.02338 q^{13} -2.44275 q^{14} +17.2919 q^{16} +2.27671 q^{17} +3.93977 q^{19} +12.9108 q^{20} +2.77875 q^{22} -3.56689 q^{23} +0.0920946 q^{25} -8.40121 q^{26} +5.02962 q^{28} -1.57772 q^{29} -9.17553 q^{31} -27.3680 q^{32} -6.32639 q^{34} +1.98371 q^{35} -7.15525 q^{37} -10.9476 q^{38} -23.3350 q^{40} -7.02640 q^{41} +8.40227 q^{43} -5.72143 q^{44} +9.91148 q^{46} -3.46158 q^{47} -6.22721 q^{49} -0.255908 q^{50} +17.2981 q^{52} -1.97988 q^{53} -2.25657 q^{55} -9.09052 q^{56} +4.38408 q^{58} -8.41286 q^{59} -1.00000 q^{61} +25.4965 q^{62} +41.4649 q^{64} +6.82247 q^{65} +0.158699 q^{67} +13.0260 q^{68} -5.51223 q^{70} -11.7036 q^{71} -11.4377 q^{73} +19.8826 q^{74} +22.5411 q^{76} -0.879084 q^{77} +7.62621 q^{79} +39.0203 q^{80} +19.5246 q^{82} -8.04762 q^{83} +5.13754 q^{85} -23.3478 q^{86} +10.3409 q^{88} +6.83581 q^{89} +2.65781 q^{91} -20.4077 q^{92} +9.61884 q^{94} +8.89035 q^{95} -2.44457 q^{97} +17.3038 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.77875	−1.96487	−0.982435	−	0.186605i	$-0.940252\pi$
$2$	−2.77875	−1.96487	−0.982435	+	0.186605i	$0.940252\pi$
$3$	0	0
$3$	0	0
$4$	5.72143	2.86071
$5$	2.25657	1.00917	0.504584	−	0.863363i	$-0.331646\pi$
$5$	2.25657	1.00917	0.504584	+	0.863363i	$0.331646\pi$
$6$	0	0
$7$	0.879084	0.332262	0.166131	−	0.986104i	$-0.446872\pi$
$7$	0.879084	0.332262	0.166131	+	0.986104i	$0.446872\pi$
$8$	−10.3409	−3.65606
$9$	0	0
$10$	−6.27043	−1.98288
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.02338	0.838536	0.419268	−	0.907863i	$-0.362287\pi$
$13$	3.02338	0.838536	0.419268	+	0.907863i	$0.362287\pi$
$14$	−2.44275	−0.652853
$15$	0	0
$16$	17.2919	4.32298
$17$	2.27671	0.552183	0.276091	−	0.961131i	$-0.410961\pi$
$17$	2.27671	0.552183	0.276091	+	0.961131i	$0.410961\pi$
$18$	0	0
$19$	3.93977	0.903845	0.451923	−	0.892057i	$-0.350738\pi$
$19$	3.93977	0.903845	0.451923	+	0.892057i	$0.350738\pi$
$20$	12.9108	2.88694
$21$	0	0
$22$	2.77875	0.592431
$23$	−3.56689	−0.743748	−0.371874	−	0.928283i	$-0.621285\pi$
$23$	−3.56689	−0.743748	−0.371874	+	0.928283i	$0.621285\pi$
$24$	0	0
$25$	0.0920946	0.0184189
$26$	−8.40121	−1.64761
$27$	0	0
$28$	5.02962	0.950508
$29$	−1.57772	−0.292975	−0.146487	−	0.989213i	$-0.546797\pi$
$29$	−1.57772	−0.292975	−0.146487	+	0.989213i	$0.546797\pi$
$30$	0	0
$31$	−9.17553	−1.64797	−0.823987	−	0.566609i	$-0.808255\pi$
$31$	−9.17553	−1.64797	−0.823987	+	0.566609i	$0.808255\pi$
$32$	−27.3680	−4.83802
$33$	0	0
$34$	−6.32639	−1.08497
$35$	1.98371	0.335308
$36$	0	0
$37$	−7.15525	−1.17632	−0.588158	−	0.808746i	$-0.700146\pi$
$37$	−7.15525	−1.17632	−0.588158	+	0.808746i	$0.700146\pi$
$38$	−10.9476	−1.77594
$39$	0	0
$40$	−23.3350	−3.68958
$41$	−7.02640	−1.09734	−0.548669	−	0.836039i	$-0.684865\pi$
$41$	−7.02640	−1.09734	−0.548669	+	0.836039i	$0.684865\pi$
$42$	0	0
$43$	8.40227	1.28133	0.640667	−	0.767819i	$-0.278658\pi$
$43$	8.40227	1.28133	0.640667	+	0.767819i	$0.278658\pi$
$44$	−5.72143	−0.862538
$45$	0	0
$46$	9.91148	1.46137
$47$	−3.46158	−0.504923	−0.252461	−	0.967607i	$-0.581240\pi$
$47$	−3.46158	−0.504923	−0.252461	+	0.967607i	$0.581240\pi$
$48$	0	0
$49$	−6.22721	−0.889602
$50$	−0.255908	−0.0361908
$51$	0	0
$52$	17.2981	2.39881
$53$	−1.97988	−0.271958	−0.135979	−	0.990712i	$-0.543418\pi$
$53$	−1.97988	−0.271958	−0.135979	+	0.990712i	$0.543418\pi$
$54$	0	0
$55$	−2.25657	−0.304275
$56$	−9.09052	−1.21477
$57$	0	0
$58$	4.38408	0.575658
$59$	−8.41286	−1.09526	−0.547631	−	0.836720i	$-0.684470\pi$
$59$	−8.41286	−1.09526	−0.547631	+	0.836720i	$0.684470\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	25.4965	3.23805
$63$	0	0
$64$	41.4649	5.18311
$65$	6.82247	0.846223
$66$	0	0
$67$	0.158699	0.0193881	0.00969407	−	0.999953i	$-0.496914\pi$
$67$	0.158699	0.0193881	0.00969407	+	0.999953i	$0.496914\pi$
$68$	13.0260	1.57964
$69$	0	0
$70$	−5.51223	−0.658838
$71$	−11.7036	−1.38897	−0.694483	−	0.719509i	$-0.744367\pi$
$71$	−11.7036	−1.38897	−0.694483	+	0.719509i	$0.744367\pi$
$72$	0	0
$73$	−11.4377	−1.33869	−0.669343	−	0.742953i	$-0.733424\pi$
$73$	−11.4377	−1.33869	−0.669343	+	0.742953i	$0.733424\pi$
$74$	19.8826	2.31131
$75$	0	0
$76$	22.5411	2.58564
$77$	−0.879084	−0.100181
$78$	0	0
$79$	7.62621	0.858016	0.429008	−	0.903301i	$-0.358863\pi$
$79$	7.62621	0.858016	0.429008	+	0.903301i	$0.358863\pi$
$80$	39.0203	4.36261
$81$	0	0
$82$	19.5246	2.15613
$83$	−8.04762	−0.883341	−0.441670	−	0.897177i	$-0.645614\pi$
$83$	−8.04762	−0.883341	−0.441670	+	0.897177i	$0.645614\pi$
$84$	0	0
$85$	5.13754	0.557245
$86$	−23.3478	−2.51765
$87$	0	0
$88$	10.3409	1.10234
$89$	6.83581	0.724594	0.362297	−	0.932063i	$-0.381993\pi$
$89$	6.83581	0.724594	0.362297	+	0.932063i	$0.381993\pi$
$90$	0	0
$91$	2.65781	0.278614
$92$	−20.4077	−2.12765
$93$	0	0
$94$	9.61884	0.992107
$95$	8.89035	0.912131
$96$	0	0
$97$	−2.44457	−0.248209	−0.124104	−	0.992269i	$-0.539606\pi$
$97$	−2.44457	−0.248209	−0.124104	+	0.992269i	$0.539606\pi$
$98$	17.3038	1.74795
$99$	0	0
$100$	0.526913	0.0526913
$101$	−6.30033	−0.626907	−0.313453	−	0.949604i	$-0.601486\pi$
$101$	−6.30033	−0.626907	−0.313453	+	0.949604i	$0.601486\pi$
$102$	0	0
$103$	−11.2542	−1.10891	−0.554456	−	0.832213i	$-0.687074\pi$
$103$	−11.2542	−1.10891	−0.554456	+	0.832213i	$0.687074\pi$
$104$	−31.2645	−3.06574
$105$	0	0
$106$	5.50159	0.534361
$107$	−4.55757	−0.440597	−0.220299	−	0.975432i	$-0.570703\pi$
$107$	−4.55757	−0.440597	−0.220299	+	0.975432i	$0.570703\pi$
$108$	0	0
$109$	−15.6253	−1.49663	−0.748316	−	0.663343i	$-0.769137\pi$
$109$	−15.6253	−1.49663	−0.748316	+	0.663343i	$0.769137\pi$
$110$	6.27043	0.597862
$111$	0	0
$112$	15.2010	1.43636
$113$	11.4344	1.07566	0.537829	−	0.843054i	$-0.319245\pi$
$113$	11.4344	1.07566	0.537829	+	0.843054i	$0.319245\pi$
$114$	0	0
$115$	−8.04893	−0.750566
$116$	−9.02681	−0.838118
$117$	0	0
$118$	23.3772	2.15205
$119$	2.00142	0.183469
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.77875	0.251576
$123$	0	0
$124$	−52.4971	−4.71438
$125$	−11.0750	−0.990580
$126$	0	0
$127$	9.33193	0.828075	0.414037	−	0.910260i	$-0.364118\pi$
$127$	9.33193	0.828075	0.414037	+	0.910260i	$0.364118\pi$
$128$	−60.4844	−5.34612
$129$	0	0
$130$	−18.9579	−1.66272
$131$	5.74566	0.502000	0.251000	−	0.967987i	$-0.419241\pi$
$131$	5.74566	0.502000	0.251000	+	0.967987i	$0.419241\pi$
$132$	0	0
$133$	3.46339	0.300314
$134$	−0.440983	−0.0380952
$135$	0	0
$136$	−23.5432	−2.01881
$137$	−16.8792	−1.44209	−0.721043	−	0.692890i	$-0.756337\pi$
$137$	−16.8792	−1.44209	−0.721043	+	0.692890i	$0.756337\pi$
$138$	0	0
$139$	−9.84489	−0.835033	−0.417516	−	0.908669i	$-0.637099\pi$
$139$	−9.84489	−0.835033	−0.417516	+	0.908669i	$0.637099\pi$
$140$	11.3497	0.959222
$141$	0	0
$142$	32.5214	2.72914
$143$	−3.02338	−0.252828
$144$	0	0
$145$	−3.56023	−0.295661
$146$	31.7826	2.63035
$147$	0	0
$148$	−40.9382	−3.36510
$149$	−20.1578	−1.65139	−0.825696	−	0.564116i	$-0.809217\pi$
$149$	−20.1578	−1.65139	−0.825696	+	0.564116i	$0.809217\pi$
$150$	0	0
$151$	−13.1601	−1.07095	−0.535475	−	0.844551i	$-0.679867\pi$
$151$	−13.1601	−1.07095	−0.535475	+	0.844551i	$0.679867\pi$
$152$	−40.7408	−3.30452
$153$	0	0
$154$	2.44275	0.196842
$155$	−20.7052	−1.66308
$156$	0	0
$157$	−12.3050	−0.982049	−0.491024	−	0.871146i	$-0.663377\pi$
$157$	−12.3050	−0.982049	−0.491024	+	0.871146i	$0.663377\pi$
$158$	−21.1913	−1.68589
$159$	0	0
$160$	−61.7577	−4.88237
$161$	−3.13559	−0.247119
$162$	0	0
$163$	21.3437	1.67177	0.835883	−	0.548908i	$-0.184956\pi$
$163$	21.3437	1.67177	0.835883	+	0.548908i	$0.184956\pi$
$164$	−40.2010	−3.13917
$165$	0	0
$166$	22.3623	1.73565
$167$	24.8180	1.92048	0.960239	−	0.279181i	$-0.0900629\pi$
$167$	24.8180	1.92048	0.960239	+	0.279181i	$0.0900629\pi$
$168$	0	0
$169$	−3.85916	−0.296858
$170$	−14.2759	−1.09491
$171$	0	0
$172$	48.0730	3.66553
$173$	9.38080	0.713209	0.356604	−	0.934255i	$-0.383934\pi$
$173$	9.38080	0.713209	0.356604	+	0.934255i	$0.383934\pi$
$174$	0	0
$175$	0.0809589	0.00611992
$176$	−17.2919	−1.30343
$177$	0	0
$178$	−18.9950	−1.42373
$179$	2.46165	0.183992	0.0919961	−	0.995759i	$-0.470675\pi$
$179$	2.46165	0.183992	0.0919961	+	0.995759i	$0.470675\pi$
$180$	0	0
$181$	6.43652	0.478423	0.239211	−	0.970968i	$-0.423111\pi$
$181$	6.43652	0.478423	0.239211	+	0.970968i	$0.423111\pi$
$182$	−7.38537	−0.547440
$183$	0	0
$184$	36.8849	2.71919
$185$	−16.1463	−1.18710
$186$	0	0
$187$	−2.27671	−0.166489
$188$	−19.8052	−1.44444
$189$	0	0
$190$	−24.7040	−1.79222
$191$	−19.9200	−1.44136	−0.720679	−	0.693269i	$-0.756170\pi$
$191$	−19.9200	−1.44136	−0.720679	+	0.693269i	$0.756170\pi$
$192$	0	0
$193$	−14.3141	−1.03035	−0.515177	−	0.857084i	$-0.672274\pi$
$193$	−14.3141	−1.03035	−0.515177	+	0.857084i	$0.672274\pi$
$194$	6.79284	0.487698
$195$	0	0
$196$	−35.6286	−2.54490
$197$	13.5977	0.968800	0.484400	−	0.874847i	$-0.339038\pi$
$197$	13.5977	0.968800	0.484400	+	0.874847i	$0.339038\pi$
$198$	0	0
$199$	−2.45588	−0.174093	−0.0870463	−	0.996204i	$-0.527743\pi$
$199$	−2.45588	−0.174093	−0.0870463	+	0.996204i	$0.527743\pi$
$200$	−0.952342	−0.0673408
$201$	0	0
$202$	17.5070	1.23179
$203$	−1.38695	−0.0973446
$204$	0	0
$205$	−15.8555	−1.10740
$206$	31.2727	2.17887
$207$	0	0
$208$	52.2800	3.62497
$209$	−3.93977	−0.272520
$210$	0	0
$211$	9.68992	0.667082	0.333541	−	0.942736i	$-0.391757\pi$
$211$	9.68992	0.667082	0.333541	+	0.942736i	$0.391757\pi$
$212$	−11.3277	−0.777993
$213$	0	0
$214$	12.6643	0.865716
$215$	18.9603	1.29308
$216$	0	0
$217$	−8.06606	−0.547560
$218$	43.4187	2.94069
$219$	0	0
$220$	−12.9108	−0.870445
$221$	6.88336	0.463025
$222$	0	0
$223$	−8.57339	−0.574117	−0.287058	−	0.957913i	$-0.592677\pi$
$223$	−8.57339	−0.574117	−0.287058	+	0.957913i	$0.592677\pi$
$224$	−24.0587	−1.60749
$225$	0	0
$226$	−31.7733	−2.11353
$227$	−3.02895	−0.201039	−0.100519	−	0.994935i	$-0.532050\pi$
$227$	−3.02895	−0.201039	−0.100519	+	0.994935i	$0.532050\pi$
$228$	0	0
$229$	28.1165	1.85799	0.928995	−	0.370091i	$-0.120674\pi$
$229$	28.1165	1.85799	0.928995	+	0.370091i	$0.120674\pi$
$230$	22.3659	1.47477
$231$	0	0
$232$	16.3150	1.07114
$233$	8.79084	0.575907	0.287954	−	0.957644i	$-0.407025\pi$
$233$	8.79084	0.575907	0.287954	+	0.957644i	$0.407025\pi$
$234$	0	0
$235$	−7.81128	−0.509552
$236$	−48.1336	−3.13323
$237$	0	0
$238$	−5.56143	−0.360494
$239$	20.3562	1.31673	0.658366	−	0.752698i	$-0.271248\pi$
$239$	20.3562	1.31673	0.658366	+	0.752698i	$0.271248\pi$
$240$	0	0
$241$	−9.21136	−0.593356	−0.296678	−	0.954978i	$-0.595879\pi$
$241$	−9.21136	−0.593356	−0.296678	+	0.954978i	$0.595879\pi$
$242$	−2.77875	−0.178625
$243$	0	0
$244$	−5.72143	−0.366277
$245$	−14.0521	−0.897757
$246$	0	0
$247$	11.9114	0.757906
$248$	94.8833	6.02509
$249$	0	0
$250$	30.7747	1.94636
$251$	−22.9928	−1.45129	−0.725645	−	0.688069i	$-0.758458\pi$
$251$	−22.9928	−1.45129	−0.725645	+	0.688069i	$0.758458\pi$
$252$	0	0
$253$	3.56689	0.224248
$254$	−25.9311	−1.62706
$255$	0	0
$256$	85.1410	5.32131
$257$	11.9298	0.744162	0.372081	−	0.928200i	$-0.378644\pi$
$257$	11.9298	0.744162	0.372081	+	0.928200i	$0.378644\pi$
$258$	0	0
$259$	−6.29006	−0.390845
$260$	39.0343	2.42080
$261$	0	0
$262$	−15.9657	−0.986366
$263$	4.34851	0.268141	0.134070	−	0.990972i	$-0.457195\pi$
$263$	4.34851	0.268141	0.134070	+	0.990972i	$0.457195\pi$
$264$	0	0
$265$	−4.46773	−0.274451
$266$	−9.62387	−0.590078
$267$	0	0
$268$	0.907984	0.0554639
$269$	13.3407	0.813396	0.406698	−	0.913563i	$-0.366680\pi$
$269$	13.3407	0.813396	0.406698	+	0.913563i	$0.366680\pi$
$270$	0	0
$271$	22.1121	1.34321	0.671606	−	0.740908i	$-0.265605\pi$
$271$	22.1121	1.34321	0.671606	+	0.740908i	$0.265605\pi$
$272$	39.3686	2.38707
$273$	0	0
$274$	46.9030	2.83351
$275$	−0.0920946	−0.00555351
$276$	0	0
$277$	−5.92537	−0.356021	−0.178010	−	0.984029i	$-0.556966\pi$
$277$	−5.92537	−0.356021	−0.178010	+	0.984029i	$0.556966\pi$
$278$	27.3565	1.64073
$279$	0	0
$280$	−20.5134	−1.22591
$281$	2.16254	0.129006	0.0645030	−	0.997918i	$-0.479454\pi$
$281$	2.16254	0.129006	0.0645030	+	0.997918i	$0.479454\pi$
$282$	0	0
$283$	−7.36447	−0.437772	−0.218886	−	0.975750i	$-0.570242\pi$
$283$	−7.36447	−0.437772	−0.218886	+	0.975750i	$0.570242\pi$
$284$	−66.9615	−3.97343
$285$	0	0
$286$	8.40121	0.496774
$287$	−6.17679	−0.364604
$288$	0	0
$289$	−11.8166	−0.695094
$290$	9.89297	0.580935
$291$	0	0
$292$	−65.4403	−3.82960
$293$	7.72221	0.451136	0.225568	−	0.974227i	$-0.427576\pi$
$293$	7.72221	0.451136	0.225568	+	0.974227i	$0.427576\pi$
$294$	0	0
$295$	−18.9842	−1.10530
$296$	73.9917	4.30068
$297$	0	0
$298$	56.0134	3.24477
$299$	−10.7841	−0.623659
$300$	0	0
$301$	7.38630	0.425739
$302$	36.5684	2.10428
$303$	0	0
$304$	68.1261	3.90730
$305$	−2.25657	−0.129211
$306$	0	0
$307$	7.14107	0.407562	0.203781	−	0.979016i	$-0.434677\pi$
$307$	7.14107	0.407562	0.203781	+	0.979016i	$0.434677\pi$
$308$	−5.02962	−0.286589
$309$	0	0
$310$	57.5345	3.26774
$311$	27.0501	1.53387	0.766935	−	0.641725i	$-0.221781\pi$
$311$	27.0501	1.53387	0.766935	+	0.641725i	$0.221781\pi$
$312$	0	0
$313$	13.2759	0.750400	0.375200	−	0.926944i	$-0.377574\pi$
$313$	13.2759	0.750400	0.375200	+	0.926944i	$0.377574\pi$
$314$	34.1926	1.92960
$315$	0	0
$316$	43.6328	2.45454
$317$	−11.4523	−0.643224	−0.321612	−	0.946872i	$-0.604225\pi$
$317$	−11.4523	−0.643224	−0.321612	+	0.946872i	$0.604225\pi$
$318$	0	0
$319$	1.57772	0.0883353
$320$	93.5683	5.23063
$321$	0	0
$322$	8.71302	0.485558
$323$	8.96970	0.499088
$324$	0	0
$325$	0.278437	0.0154449
$326$	−59.3087	−3.28480
$327$	0	0
$328$	72.6593	4.01194
$329$	−3.04301	−0.167767
$330$	0	0
$331$	−31.3131	−1.72113	−0.860563	−	0.509344i	$-0.829888\pi$
$331$	−31.3131	−1.72113	−0.860563	+	0.509344i	$0.829888\pi$
$332$	−46.0439	−2.52699
$333$	0	0
$334$	−68.9630	−3.77349
$335$	0.358114	0.0195659
$336$	0	0
$337$	17.2152	0.937769	0.468885	−	0.883259i	$-0.344656\pi$
$337$	17.2152	0.937769	0.468885	+	0.883259i	$0.344656\pi$
$338$	10.7236	0.583288
$339$	0	0
$340$	29.3941	1.59412
$341$	9.17553	0.496883
$342$	0	0
$343$	−11.6278	−0.627844
$344$	−86.8871	−4.68464
$345$	0	0
$346$	−26.0669	−1.40136
$347$	21.8365	1.17224	0.586122	−	0.810223i	$-0.300654\pi$
$347$	21.8365	1.17224	0.586122	+	0.810223i	$0.300654\pi$
$348$	0	0
$349$	−19.1065	−1.02275	−0.511374	−	0.859358i	$-0.670863\pi$
$349$	−19.1065	−1.02275	−0.511374	+	0.859358i	$0.670863\pi$
$350$	−0.224964	−0.0120248
$351$	0	0
$352$	27.3680	1.45872
$353$	13.5571	0.721570	0.360785	−	0.932649i	$-0.382509\pi$
$353$	13.5571	0.721570	0.360785	+	0.932649i	$0.382509\pi$
$354$	0	0
$355$	−26.4100	−1.40170
$356$	39.1106	2.07286
$357$	0	0
$358$	−6.84029	−0.361521
$359$	21.1484	1.11617	0.558085	−	0.829784i	$-0.311536\pi$
$359$	21.1484	1.11617	0.558085	+	0.829784i	$0.311536\pi$
$360$	0	0
$361$	−3.47821	−0.183064
$362$	−17.8854	−0.940038
$363$	0	0
$364$	15.2065	0.797035
$365$	−25.8100	−1.35096
$366$	0	0
$367$	19.8595	1.03666	0.518328	−	0.855182i	$-0.326555\pi$
$367$	19.8595	1.03666	0.518328	+	0.855182i	$0.326555\pi$
$368$	−61.6783	−3.21520
$369$	0	0
$370$	44.8664	2.33250
$371$	−1.74048	−0.0903613
$372$	0	0
$373$	−18.5109	−0.958458	−0.479229	−	0.877690i	$-0.659084\pi$
$373$	−18.5109	−0.958458	−0.479229	+	0.877690i	$0.659084\pi$
$374$	6.32639	0.327130
$375$	0	0
$376$	35.7958	1.84603
$377$	−4.77005	−0.245670
$378$	0	0
$379$	−23.8501	−1.22510	−0.612549	−	0.790432i	$-0.709856\pi$
$379$	−23.8501	−1.22510	−0.612549	+	0.790432i	$0.709856\pi$
$380$	50.8655	2.60935
$381$	0	0
$382$	55.3525	2.83208
$383$	14.7628	0.754342	0.377171	−	0.926144i	$-0.376897\pi$
$383$	14.7628	0.754342	0.377171	+	0.926144i	$0.376897\pi$
$384$	0	0
$385$	−1.98371	−0.101099
$386$	39.7754	2.02451
$387$	0	0
$388$	−13.9864	−0.710054
$389$	24.8340	1.25913	0.629566	−	0.776947i	$-0.283233\pi$
$389$	24.8340	1.25913	0.629566	+	0.776947i	$0.283233\pi$
$390$	0	0
$391$	−8.12076	−0.410685
$392$	64.3950	3.25244
$393$	0	0
$394$	−37.7847	−1.90357
$395$	17.2091	0.865881
$396$	0	0
$397$	−9.11446	−0.457442	−0.228721	−	0.973492i	$-0.573454\pi$
$397$	−9.11446	−0.457442	−0.228721	+	0.973492i	$0.573454\pi$
$398$	6.82426	0.342069
$399$	0	0
$400$	1.59249	0.0796246
$401$	−20.7748	−1.03744	−0.518722	−	0.854943i	$-0.673592\pi$
$401$	−20.7748	−1.03744	−0.518722	+	0.854943i	$0.673592\pi$
$402$	0	0
$403$	−27.7411	−1.38188
$404$	−36.0469	−1.79340
$405$	0	0
$406$	3.85397	0.191269
$407$	7.15525	0.354672
$408$	0	0
$409$	−14.7084	−0.727283	−0.363642	−	0.931539i	$-0.618467\pi$
$409$	−14.7084	−0.727283	−0.363642	+	0.931539i	$0.618467\pi$
$410$	44.0585	2.17589
$411$	0	0
$412$	−64.3903	−3.17228
$413$	−7.39561	−0.363914
$414$	0	0
$415$	−18.1600	−0.891439
$416$	−82.7439	−4.05685
$417$	0	0
$418$	10.9476	0.535466
$419$	11.7498	0.574014	0.287007	−	0.957928i	$-0.407340\pi$
$419$	11.7498	0.574014	0.287007	+	0.957928i	$0.407340\pi$
$420$	0	0
$421$	−25.8941	−1.26200	−0.631001	−	0.775782i	$-0.717355\pi$
$421$	−25.8941	−1.26200	−0.631001	+	0.775782i	$0.717355\pi$
$422$	−26.9258	−1.31073
$423$	0	0
$424$	20.4738	0.994294
$425$	0.209672	0.0101706
$426$	0	0
$427$	−0.879084	−0.0425418
$428$	−26.0758	−1.26042
$429$	0	0
$430$	−52.6858	−2.54074
$431$	−8.87109	−0.427305	−0.213653	−	0.976910i	$-0.568536\pi$
$431$	−8.87109	−0.427305	−0.213653	+	0.976910i	$0.568536\pi$
$432$	0	0
$433$	5.89007	0.283059	0.141529	−	0.989934i	$-0.454798\pi$
$433$	5.89007	0.283059	0.141529	+	0.989934i	$0.454798\pi$
$434$	22.4135	1.07588
$435$	0	0
$436$	−89.3990	−4.28144
$437$	−14.0527	−0.672233
$438$	0	0
$439$	13.5706	0.647690	0.323845	−	0.946110i	$-0.395024\pi$
$439$	13.5706	0.647690	0.323845	+	0.946110i	$0.395024\pi$
$440$	23.3350	1.11245
$441$	0	0
$442$	−19.1271	−0.909783
$443$	14.0440	0.667250	0.333625	−	0.942706i	$-0.391728\pi$
$443$	14.0440	0.667250	0.333625	+	0.942706i	$0.391728\pi$
$444$	0	0
$445$	15.4255	0.731237
$446$	23.8233	1.12806
$447$	0	0
$448$	36.4511	1.72215
$449$	35.5152	1.67607	0.838033	−	0.545620i	$-0.183706\pi$
$449$	35.5152	1.67607	0.838033	+	0.545620i	$0.183706\pi$
$450$	0	0
$451$	7.02640	0.330860
$452$	65.4211	3.07715
$453$	0	0
$454$	8.41669	0.395015
$455$	5.99752	0.281168
$456$	0	0
$457$	−16.1311	−0.754582	−0.377291	−	0.926095i	$-0.623144\pi$
$457$	−16.1311	−0.754582	−0.377291	+	0.926095i	$0.623144\pi$
$458$	−78.1286	−3.65071
$459$	0	0
$460$	−46.0514	−2.14716
$461$	−3.05165	−0.142129	−0.0710647	−	0.997472i	$-0.522640\pi$
$461$	−3.05165	−0.142129	−0.0710647	+	0.997472i	$0.522640\pi$
$462$	0	0
$463$	31.8522	1.48030	0.740149	−	0.672443i	$-0.234755\pi$
$463$	31.8522	1.48030	0.740149	+	0.672443i	$0.234755\pi$
$464$	−27.2818	−1.26652
$465$	0	0
$466$	−24.4275	−1.13158
$467$	−8.39262	−0.388364	−0.194182	−	0.980966i	$-0.562205\pi$
$467$	−8.39262	−0.388364	−0.194182	+	0.980966i	$0.562205\pi$
$468$	0	0
$469$	0.139509	0.00644195
$470$	21.7056	1.00120
$471$	0	0
$472$	86.9967	4.00435
$473$	−8.40227	−0.386337
$474$	0	0
$475$	0.362832	0.0166479
$476$	11.4510	0.524854
$477$	0	0
$478$	−56.5647	−2.58721
$479$	24.3059	1.11056	0.555282	−	0.831662i	$-0.312610\pi$
$479$	24.3059	1.11056	0.555282	+	0.831662i	$0.312610\pi$
$480$	0	0
$481$	−21.6330	−0.986382
$482$	25.5960	1.16587
$483$	0	0
$484$	5.72143	0.260065
$485$	−5.51634	−0.250484
$486$	0	0
$487$	13.4231	0.608258	0.304129	−	0.952631i	$-0.401635\pi$
$487$	13.4231	0.608258	0.304129	+	0.952631i	$0.401635\pi$
$488$	10.3409	0.468111
$489$	0	0
$490$	39.0473	1.76398
$491$	−5.83539	−0.263347	−0.131674	−	0.991293i	$-0.542035\pi$
$491$	−5.83539	−0.263347	−0.131674	+	0.991293i	$0.542035\pi$
$492$	0	0
$493$	−3.59200	−0.161776
$494$	−33.0989	−1.48919
$495$	0	0
$496$	−158.662	−7.12415
$497$	−10.2885	−0.461501
$498$	0	0
$499$	37.5072	1.67905	0.839527	−	0.543318i	$-0.182832\pi$
$499$	37.5072	1.67905	0.839527	+	0.543318i	$0.182832\pi$
$500$	−63.3649	−2.83377
$501$	0	0
$502$	63.8910	2.85160
$503$	10.5334	0.469659	0.234829	−	0.972037i	$-0.424547\pi$
$503$	10.5334	0.469659	0.234829	+	0.972037i	$0.424547\pi$
$504$	0	0
$505$	−14.2171	−0.632654
$506$	−9.91148	−0.440619
$507$	0	0
$508$	53.3920	2.36889
$509$	8.86365	0.392874	0.196437	−	0.980516i	$-0.437063\pi$
$509$	8.86365	0.392874	0.196437	+	0.980516i	$0.437063\pi$
$510$	0	0
$511$	−10.0547	−0.444795
$512$	−115.617	−5.10958
$513$	0	0
$514$	−33.1499	−1.46218
$515$	−25.3959	−1.11908
$516$	0	0
$517$	3.46158	0.152240
$518$	17.4785	0.767960
$519$	0	0
$520$	−70.5505	−3.09384
$521$	−6.01653	−0.263589	−0.131795	−	0.991277i	$-0.542074\pi$
$521$	−6.01653	−0.263589	−0.131795	+	0.991277i	$0.542074\pi$
$522$	0	0
$523$	5.67338	0.248079	0.124040	−	0.992277i	$-0.460415\pi$
$523$	5.67338	0.248079	0.124040	+	0.992277i	$0.460415\pi$
$524$	32.8734	1.43608
$525$	0	0
$526$	−12.0834	−0.526862
$527$	−20.8900	−0.909982
$528$	0	0
$529$	−10.2773	−0.446839
$530$	12.4147	0.539260
$531$	0	0
$532$	19.8155	0.859112
$533$	−21.2435	−0.920158
$534$	0	0
$535$	−10.2845	−0.444636
$536$	−1.64109	−0.0708842
$537$	0	0
$538$	−37.0704	−1.59822
$539$	6.22721	0.268225
$540$	0	0
$541$	5.96416	0.256419	0.128210	−	0.991747i	$-0.459077\pi$
$541$	5.96416	0.256419	0.128210	+	0.991747i	$0.459077\pi$
$542$	−61.4438	−2.63924
$543$	0	0
$544$	−62.3089	−2.67147
$545$	−35.2595	−1.51035
$546$	0	0
$547$	11.0062	0.470591	0.235295	−	0.971924i	$-0.424394\pi$
$547$	11.0062	0.470591	0.235295	+	0.971924i	$0.424394\pi$
$548$	−96.5731	−4.12540
$549$	0	0
$550$	0.255908	0.0109119
$551$	−6.21585	−0.264804
$552$	0	0
$553$	6.70408	0.285086
$554$	16.4651	0.699535
$555$	0	0
$556$	−56.3269	−2.38879
$557$	32.2939	1.36834	0.684168	−	0.729324i	$-0.260165\pi$
$557$	32.2939	1.36834	0.684168	+	0.729324i	$0.260165\pi$
$558$	0	0
$559$	25.4033	1.07444
$560$	34.3021	1.44953
$561$	0	0
$562$	−6.00914	−0.253480
$563$	−11.4773	−0.483711	−0.241855	−	0.970312i	$-0.577756\pi$
$563$	−11.4773	−0.483711	−0.241855	+	0.970312i	$0.577756\pi$
$564$	0	0
$565$	25.8025	1.08552
$566$	20.4640	0.860165
$567$	0	0
$568$	121.026	5.07815
$569$	−34.3692	−1.44083	−0.720416	−	0.693543i	$-0.756049\pi$
$569$	−34.3692	−1.44083	−0.720416	+	0.693543i	$0.756049\pi$
$570$	0	0
$571$	−7.06686	−0.295739	−0.147869	−	0.989007i	$-0.547242\pi$
$571$	−7.06686	−0.295739	−0.147869	+	0.989007i	$0.547242\pi$
$572$	−17.2981	−0.723269
$573$	0	0
$574$	17.1637	0.716400
$575$	−0.328491	−0.0136990
$576$	0	0
$577$	24.2118	1.00795	0.503974	−	0.863719i	$-0.331871\pi$
$577$	24.2118	1.00795	0.503974	+	0.863719i	$0.331871\pi$
$578$	32.8353	1.36577
$579$	0	0
$580$	−20.3696	−0.845801
$581$	−7.07453	−0.293501
$582$	0	0
$583$	1.97988	0.0819983
$584$	118.277	4.89432
$585$	0	0
$586$	−21.4581	−0.886424
$587$	6.28317	0.259334	0.129667	−	0.991558i	$-0.458609\pi$
$587$	6.28317	0.259334	0.129667	+	0.991558i	$0.458609\pi$
$588$	0	0
$589$	−36.1495	−1.48951
$590$	52.7522	2.17178
$591$	0	0
$592$	−123.728	−5.08518
$593$	46.2325	1.89854	0.949271	−	0.314458i	$-0.101823\pi$
$593$	46.2325	1.89854	0.949271	+	0.314458i	$0.101823\pi$
$594$	0	0
$595$	4.51633	0.185151
$596$	−115.331	−4.72416
$597$	0	0
$598$	29.9662	1.22541
$599$	−29.1635	−1.19159	−0.595793	−	0.803138i	$-0.703162\pi$
$599$	−29.1635	−1.19159	−0.595793	+	0.803138i	$0.703162\pi$
$600$	0	0
$601$	−19.8979	−0.811652	−0.405826	−	0.913950i	$-0.633016\pi$
$601$	−19.8979	−0.811652	−0.405826	+	0.913950i	$0.633016\pi$
$602$	−20.5246	−0.836522
$603$	0	0
$604$	−75.2943	−3.06368
$605$	2.25657	0.0917425
$606$	0	0
$607$	30.2608	1.22825	0.614124	−	0.789210i	$-0.289509\pi$
$607$	30.2608	1.22825	0.614124	+	0.789210i	$0.289509\pi$
$608$	−107.824	−4.37282
$609$	0	0
$610$	6.27043	0.253882
$611$	−10.4657	−0.423396
$612$	0	0
$613$	7.53520	0.304344	0.152172	−	0.988354i	$-0.451373\pi$
$613$	7.53520	0.304344	0.152172	+	0.988354i	$0.451373\pi$
$614$	−19.8432	−0.800807
$615$	0	0
$616$	9.09052	0.366268
$617$	−38.7822	−1.56131	−0.780656	−	0.624961i	$-0.785115\pi$
$617$	−38.7822	−1.56131	−0.780656	+	0.624961i	$0.785115\pi$
$618$	0	0
$619$	36.4645	1.46563	0.732816	−	0.680427i	$-0.238206\pi$
$619$	36.4645	1.46563	0.732816	+	0.680427i	$0.238206\pi$
$620$	−118.463	−4.75760
$621$	0	0
$622$	−75.1653	−3.01386
$623$	6.00925	0.240755
$624$	0	0
$625$	−25.4520	−1.01808
$626$	−36.8904	−1.47444
$627$	0	0
$628$	−70.4024	−2.80936
$629$	−16.2904	−0.649541
$630$	0	0
$631$	28.8562	1.14875	0.574374	−	0.818593i	$-0.305245\pi$
$631$	28.8562	1.14875	0.574374	+	0.818593i	$0.305245\pi$
$632$	−78.8619	−3.13696
$633$	0	0
$634$	31.8230	1.26385
$635$	21.0581	0.835666
$636$	0	0
$637$	−18.8272	−0.745963
$638$	−4.38408	−0.173567
$639$	0	0
$640$	−136.487	−5.39513
$641$	16.0399	0.633539	0.316770	−	0.948503i	$-0.397402\pi$
$641$	16.0399	0.633539	0.316770	+	0.948503i	$0.397402\pi$
$642$	0	0
$643$	−29.1694	−1.15033	−0.575165	−	0.818037i	$-0.695062\pi$
$643$	−29.1694	−1.15033	−0.575165	+	0.818037i	$0.695062\pi$
$644$	−17.9401	−0.706938
$645$	0	0
$646$	−24.9245	−0.980642
$647$	25.1828	0.990038	0.495019	−	0.868882i	$-0.335161\pi$
$647$	25.1828	0.990038	0.495019	+	0.868882i	$0.335161\pi$
$648$	0	0
$649$	8.41286	0.330234
$650$	−0.773707	−0.0303473
$651$	0	0
$652$	122.116	4.78245
$653$	36.6206	1.43307	0.716537	−	0.697549i	$-0.245726\pi$
$653$	36.6206	1.43307	0.716537	+	0.697549i	$0.245726\pi$
$654$	0	0
$655$	12.9655	0.506603
$656$	−121.500	−4.74377
$657$	0	0
$658$	8.45576	0.329640
$659$	9.47607	0.369135	0.184568	−	0.982820i	$-0.440912\pi$
$659$	9.47607	0.369135	0.184568	+	0.982820i	$0.440912\pi$
$660$	0	0
$661$	−43.5020	−1.69203	−0.846017	−	0.533157i	$-0.821006\pi$
$661$	−43.5020	−1.69203	−0.846017	+	0.533157i	$0.821006\pi$
$662$	87.0113	3.38179
$663$	0	0
$664$	83.2197	3.22955
$665$	7.81537	0.303067
$666$	0	0
$667$	5.62755	0.217900
$668$	141.995	5.49394
$669$	0	0
$670$	−0.995109	−0.0384444
$671$	1.00000	0.0386046
$672$	0	0
$673$	3.23998	0.124892	0.0624460	−	0.998048i	$-0.480110\pi$
$673$	3.23998	0.124892	0.0624460	+	0.998048i	$0.480110\pi$
$674$	−47.8365	−1.84259
$675$	0	0
$676$	−22.0799	−0.849226
$677$	−21.3051	−0.818823	−0.409411	−	0.912350i	$-0.634266\pi$
$677$	−21.3051	−0.818823	−0.409411	+	0.912350i	$0.634266\pi$
$678$	0	0
$679$	−2.14898	−0.0824704
$680$	−53.1268	−2.03732
$681$	0	0
$682$	−25.4965	−0.976310
$683$	−27.8345	−1.06506	−0.532528	−	0.846412i	$-0.678758\pi$
$683$	−27.8345	−1.06506	−0.532528	+	0.846412i	$0.678758\pi$
$684$	0	0
$685$	−38.0890	−1.45531
$686$	32.3108	1.23363
$687$	0	0
$688$	145.291	5.53917
$689$	−5.98594	−0.228046
$690$	0	0
$691$	25.7435	0.979328	0.489664	−	0.871911i	$-0.337119\pi$
$691$	25.7435	0.979328	0.489664	+	0.871911i	$0.337119\pi$
$692$	53.6716	2.04029
$693$	0	0
$694$	−60.6781	−2.30331
$695$	−22.2157	−0.842688
$696$	0	0
$697$	−15.9970	−0.605931
$698$	53.0921	2.00957
$699$	0	0
$700$	0.463201	0.0175073
$701$	−23.0700	−0.871341	−0.435670	−	0.900106i	$-0.643489\pi$
$701$	−23.0700	−0.871341	−0.435670	+	0.900106i	$0.643489\pi$
$702$	0	0
$703$	−28.1900	−1.06321
$704$	−41.4649	−1.56277
$705$	0	0
$706$	−37.6717	−1.41779
$707$	−5.53852	−0.208298
$708$	0	0
$709$	−28.8169	−1.08224	−0.541120	−	0.840945i	$-0.682000\pi$
$709$	−28.8169	−1.08224	−0.541120	+	0.840945i	$0.682000\pi$
$710$	73.3868	2.75416
$711$	0	0
$712$	−70.6885	−2.64916
$713$	32.7281	1.22568
$714$	0	0
$715$	−6.82247	−0.255146
$716$	14.0841	0.526349
$717$	0	0
$718$	−58.7660	−2.19313
$719$	−2.93062	−0.109294	−0.0546468	−	0.998506i	$-0.517403\pi$
$719$	−2.93062	−0.109294	−0.0546468	+	0.998506i	$0.517403\pi$
$720$	0	0
$721$	−9.89342	−0.368450
$722$	9.66507	0.359697
$723$	0	0
$724$	36.8261	1.36863
$725$	−0.145299	−0.00539628
$726$	0	0
$727$	−46.7285	−1.73306	−0.866532	−	0.499121i	$-0.833656\pi$
$727$	−46.7285	−1.73306	−0.866532	+	0.499121i	$0.833656\pi$
$728$	−27.4841	−1.01863
$729$	0	0
$730$	71.7195	2.65446
$731$	19.1295	0.707530
$732$	0	0
$733$	−49.8212	−1.84019	−0.920093	−	0.391699i	$-0.871887\pi$
$733$	−49.8212	−1.84019	−0.920093	+	0.391699i	$0.871887\pi$
$734$	−55.1844	−2.03689
$735$	0	0
$736$	97.6186	3.59827
$737$	−0.158699	−0.00584574
$738$	0	0
$739$	−0.538578	−0.0198119	−0.00990596	−	0.999951i	$-0.503153\pi$
$739$	−0.538578	−0.0198119	−0.00990596	+	0.999951i	$0.503153\pi$
$740$	−92.3799	−3.39595
$741$	0	0
$742$	4.83635	0.177548
$743$	−32.7026	−1.19974	−0.599871	−	0.800097i	$-0.704781\pi$
$743$	−32.7026	−1.19974	−0.599871	+	0.800097i	$0.704781\pi$
$744$	0	0
$745$	−45.4874	−1.66653
$746$	51.4371	1.88325
$747$	0	0
$748$	−13.0260	−0.476278
$749$	−4.00649	−0.146394
$750$	0	0
$751$	−27.7851	−1.01389	−0.506946	−	0.861978i	$-0.669226\pi$
$751$	−27.7851	−1.01389	−0.506946	+	0.861978i	$0.669226\pi$
$752$	−59.8572	−2.18277
$753$	0	0
$754$	13.2548	0.482710
$755$	−29.6965	−1.08077
$756$	0	0
$757$	48.0371	1.74594	0.872969	−	0.487777i	$-0.162192\pi$
$757$	48.0371	1.74594	0.872969	+	0.487777i	$0.162192\pi$
$758$	66.2734	2.40716
$759$	0	0
$760$	−91.9344	−3.33481
$761$	0.218663	0.00792654	0.00396327	−	0.999992i	$-0.498738\pi$
$761$	0.218663	0.00792654	0.00396327	+	0.999992i	$0.498738\pi$
$762$	0	0
$763$	−13.7359	−0.497274
$764$	−113.971	−4.12331
$765$	0	0
$766$	−41.0220	−1.48218
$767$	−25.4353	−0.918416
$768$	0	0
$769$	49.0610	1.76919	0.884593	−	0.466364i	$-0.154436\pi$
$769$	49.0610	1.76919	0.884593	+	0.466364i	$0.154436\pi$
$770$	5.51223	0.198647
$771$	0	0
$772$	−81.8974	−2.94755
$773$	33.1821	1.19348	0.596739	−	0.802436i	$-0.296463\pi$
$773$	33.1821	1.19348	0.596739	+	0.802436i	$0.296463\pi$
$774$	0	0
$775$	−0.845017	−0.0303539
$776$	25.2791	0.907467
$777$	0	0
$778$	−69.0073	−2.47403
$779$	−27.6824	−0.991824
$780$	0	0
$781$	11.7036	0.418789
$782$	22.5655	0.806942
$783$	0	0
$784$	−107.680	−3.84573
$785$	−27.7671	−0.991052
$786$	0	0
$787$	−27.0123	−0.962885	−0.481442	−	0.876478i	$-0.659887\pi$
$787$	−27.0123	−0.962885	−0.481442	+	0.876478i	$0.659887\pi$
$788$	77.7986	2.77146
$789$	0	0
$790$	−47.8196	−1.70134
$791$	10.0518	0.357401
$792$	0	0
$793$	−3.02338	−0.107363
$794$	25.3268	0.898814
$795$	0	0
$796$	−14.0511	−0.498029
$797$	−6.35336	−0.225047	−0.112524	−	0.993649i	$-0.535893\pi$
$797$	−6.35336	−0.225047	−0.112524	+	0.993649i	$0.535893\pi$
$798$	0	0
$799$	−7.88099	−0.278809
$800$	−2.52044	−0.0891112
$801$	0	0
$802$	57.7279	2.03844
$803$	11.4377	0.403629
$804$	0	0
$805$	−7.07568	−0.249385
$806$	77.0856	2.71522
$807$	0	0
$808$	65.1512	2.29201
$809$	17.5958	0.618635	0.309318	−	0.950959i	$-0.399899\pi$
$809$	17.5958	0.618635	0.309318	+	0.950959i	$0.399899\pi$
$810$	0	0
$811$	−21.9218	−0.769778	−0.384889	−	0.922963i	$-0.625760\pi$
$811$	−21.9218	−0.769778	−0.384889	+	0.922963i	$0.625760\pi$
$812$	−7.93532	−0.278475
$813$	0	0
$814$	−19.8826	−0.696885
$815$	48.1634	1.68709
$816$	0	0
$817$	33.1030	1.15813
$818$	40.8709	1.42902
$819$	0	0
$820$	−90.7163	−3.16795
$821$	−17.8530	−0.623074	−0.311537	−	0.950234i	$-0.600844\pi$
$821$	−17.8530	−0.623074	−0.311537	+	0.950234i	$0.600844\pi$
$822$	0	0
$823$	34.0433	1.18667	0.593337	−	0.804954i	$-0.297810\pi$
$823$	34.0433	1.18667	0.593337	+	0.804954i	$0.297810\pi$
$824$	116.379	4.05426
$825$	0	0
$826$	20.5505	0.715044
$827$	−42.2928	−1.47067	−0.735333	−	0.677706i	$-0.762974\pi$
$827$	−42.2928	−1.47067	−0.735333	+	0.677706i	$0.762974\pi$
$828$	0	0
$829$	−30.0478	−1.04360	−0.521802	−	0.853067i	$-0.674740\pi$
$829$	−30.0478	−1.04360	−0.521802	+	0.853067i	$0.674740\pi$
$830$	50.4620	1.75156
$831$	0	0
$832$	125.364	4.34622
$833$	−14.1775	−0.491222
$834$	0	0
$835$	56.0036	1.93808
$836$	−22.5411	−0.779601
$837$	0	0
$838$	−32.6496	−1.12786
$839$	35.2533	1.21708	0.608540	−	0.793524i	$-0.291756\pi$
$839$	35.2533	1.21708	0.608540	+	0.793524i	$0.291756\pi$
$840$	0	0
$841$	−26.5108	−0.914166
$842$	71.9531	2.47967
$843$	0	0
$844$	55.4402	1.90833
$845$	−8.70844	−0.299580
$846$	0	0
$847$	0.879084	0.0302057
$848$	−34.2359	−1.17567
$849$	0	0
$850$	−0.582626	−0.0199839
$851$	25.5220	0.874882
$852$	0	0
$853$	20.9798	0.718334	0.359167	−	0.933273i	$-0.383061\pi$
$853$	20.9798	0.718334	0.359167	+	0.933273i	$0.383061\pi$
$854$	2.44275	0.0835892
$855$	0	0
$856$	47.1294	1.61085
$857$	31.3817	1.07198	0.535989	−	0.844225i	$-0.319939\pi$
$857$	31.3817	1.07198	0.535989	+	0.844225i	$0.319939\pi$
$858$	0	0
$859$	−1.85276	−0.0632154	−0.0316077	−	0.999500i	$-0.510063\pi$
$859$	−1.85276	−0.0632154	−0.0316077	+	0.999500i	$0.510063\pi$
$860$	108.480	3.69913
$861$	0	0
$862$	24.6505	0.839599
$863$	31.3460	1.06703	0.533515	−	0.845791i	$-0.320871\pi$
$863$	31.3460	1.06703	0.533515	+	0.845791i	$0.320871\pi$
$864$	0	0
$865$	21.1684	0.719747
$866$	−16.3670	−0.556173
$867$	0	0
$868$	−46.1494	−1.56641
$869$	−7.62621	−0.258701
$870$	0	0
$871$	0.479807	0.0162576
$872$	161.580	5.47178
$873$	0	0
$874$	39.0490	1.32085
$875$	−9.73587	−0.329132
$876$	0	0
$877$	1.86853	0.0630958	0.0315479	−	0.999502i	$-0.489956\pi$
$877$	1.86853	0.0630958	0.0315479	+	0.999502i	$0.489956\pi$
$878$	−37.7093	−1.27263
$879$	0	0
$880$	−39.0203	−1.31538
$881$	−37.2566	−1.25521	−0.627603	−	0.778533i	$-0.715964\pi$
$881$	−37.2566	−1.25521	−0.627603	+	0.778533i	$0.715964\pi$
$882$	0	0
$883$	−28.7788	−0.968483	−0.484241	−	0.874934i	$-0.660904\pi$
$883$	−28.7788	−0.968483	−0.484241	+	0.874934i	$0.660904\pi$
$884$	39.3826	1.32458
$885$	0	0
$886$	−39.0247	−1.31106
$887$	3.79475	0.127415	0.0637076	−	0.997969i	$-0.479707\pi$
$887$	3.79475	0.127415	0.0637076	+	0.997969i	$0.479707\pi$
$888$	0	0
$889$	8.20355	0.275138
$890$	−42.8634	−1.43679
$891$	0	0
$892$	−49.0520	−1.64238
$893$	−13.6378	−0.456372
$894$	0	0
$895$	5.55487	0.185679
$896$	−53.1709	−1.77631
$897$	0	0
$898$	−98.6877	−3.29325
$899$	14.4764	0.482815
$900$	0	0
$901$	−4.50761	−0.150170
$902$	−19.5246	−0.650097
$903$	0	0
$904$	−118.242	−3.93267
$905$	14.5244	0.482809
$906$	0	0
$907$	40.8561	1.35660	0.678302	−	0.734783i	$-0.262716\pi$
$907$	40.8561	1.35660	0.678302	+	0.734783i	$0.262716\pi$
$908$	−17.3299	−0.575114
$909$	0	0
$910$	−16.6656	−0.552459
$911$	−23.8729	−0.790944	−0.395472	−	0.918478i	$-0.629419\pi$
$911$	−23.8729	−0.790944	−0.395472	+	0.918478i	$0.629419\pi$
$912$	0	0
$913$	8.04762	0.266337
$914$	44.8243	1.48266
$915$	0	0
$916$	160.867	5.31518
$917$	5.05091	0.166796
$918$	0	0
$919$	−13.9390	−0.459807	−0.229903	−	0.973213i	$-0.573841\pi$
$919$	−13.9390	−0.459807	−0.229903	+	0.973213i	$0.573841\pi$
$920$	83.2332	2.74412
$921$	0	0
$922$	8.47975	0.279266
$923$	−35.3846	−1.16470
$924$	0	0
$925$	−0.658960	−0.0216665
$926$	−88.5092	−2.90859
$927$	0	0
$928$	43.1790	1.41742
$929$	−40.8111	−1.33897	−0.669484	−	0.742826i	$-0.733485\pi$
$929$	−40.8111	−1.33897	−0.669484	+	0.742826i	$0.733485\pi$
$930$	0	0
$931$	−24.5338	−0.804062
$932$	50.2962	1.64751
$933$	0	0
$934$	23.3209	0.763085
$935$	−5.13754	−0.168016
$936$	0	0
$937$	−44.3010	−1.44725	−0.723626	−	0.690193i	$-0.757526\pi$
$937$	−44.3010	−1.44725	−0.723626	+	0.690193i	$0.757526\pi$
$938$	−0.387661	−0.0126576
$939$	0	0
$940$	−44.6917	−1.45768
$941$	−31.7141	−1.03385	−0.516925	−	0.856031i	$-0.672923\pi$
$941$	−31.7141	−1.03385	−0.516925	+	0.856031i	$0.672923\pi$
$942$	0	0
$943$	25.0624	0.816143
$944$	−145.474	−4.73479
$945$	0	0
$946$	23.3478	0.759101
$947$	−48.3193	−1.57017	−0.785083	−	0.619391i	$-0.787380\pi$
$947$	−48.3193	−1.57017	−0.785083	+	0.619391i	$0.787380\pi$
$948$	0	0
$949$	−34.5807	−1.12254
$950$	−1.00822	−0.0327109
$951$	0	0
$952$	−20.6965	−0.670776
$953$	−18.7099	−0.606073	−0.303037	−	0.952979i	$-0.598000\pi$
$953$	−18.7099	−0.606073	−0.303037	+	0.952979i	$0.598000\pi$
$954$	0	0
$955$	−44.9507	−1.45457
$956$	116.467	3.76680
$957$	0	0
$958$	−67.5399	−2.18211
$959$	−14.8382	−0.479151
$960$	0	0
$961$	53.1903	1.71582
$962$	60.1127	1.93811
$963$	0	0
$964$	−52.7021	−1.69742
$965$	−32.3008	−1.03980
$966$	0	0
$967$	34.4139	1.10668	0.553339	−	0.832956i	$-0.313353\pi$
$967$	34.4139	1.10668	0.553339	+	0.832956i	$0.313353\pi$
$968$	−10.3409	−0.332369
$969$	0	0
$970$	15.3285	0.492169
$971$	−47.9372	−1.53838	−0.769189	−	0.639022i	$-0.779339\pi$
$971$	−47.9372	−1.53838	−0.769189	+	0.639022i	$0.779339\pi$
$972$	0	0
$973$	−8.65448	−0.277450
$974$	−37.2993	−1.19515
$975$	0	0
$976$	−17.2919	−0.553500
$977$	−8.54037	−0.273231	−0.136615	−	0.990624i	$-0.543622\pi$
$977$	−8.54037	−0.273231	−0.136615	+	0.990624i	$0.543622\pi$
$978$	0	0
$979$	−6.83581	−0.218473
$980$	−80.3982	−2.56823
$981$	0	0
$982$	16.2151	0.517443
$983$	−24.0428	−0.766847	−0.383423	−	0.923573i	$-0.625255\pi$
$983$	−24.0428	−0.766847	−0.383423	+	0.923573i	$0.625255\pi$
$984$	0	0
$985$	30.6842	0.977681
$986$	9.98126	0.317868
$987$	0	0
$988$	68.1504	2.16815
$989$	−29.9700	−0.952989
$990$	0	0
$991$	11.2069	0.356000	0.178000	−	0.984030i	$-0.443037\pi$
$991$	11.2069	0.356000	0.178000	+	0.984030i	$0.443037\pi$
$992$	251.116	7.97293
$993$	0	0
$994$	28.5891	0.906790
$995$	−5.54185	−0.175688
$996$	0	0
$997$	−58.1589	−1.84191	−0.920955	−	0.389668i	$-0.872590\pi$
$997$	−58.1589	−1.84191	−0.920955	+	0.389668i	$0.872590\pi$
$998$	−104.223	−3.29912
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.n.1.1	✓	25
3.2	odd	2		6039.2.a.o.1.25	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.1	✓	25	1.1	even	1	trivial
6039.2.a.o.1.25	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)

\(f(q)\)	\(=\)	\(q-2.77875 q^{2} +5.72143 q^{4} +2.25657 q^{5} +0.879084 q^{7} -10.3409 q^{8} +O(q^{10})\)	\(q-2.77875 q^{2} +5.72143 q^{4} +2.25657 q^{5} +0.879084 q^{7} -10.3409 q^{8} -6.27043 q^{10} -1.00000 q^{11} +3.02338 q^{13} -2.44275 q^{14} +17.2919 q^{16} +2.27671 q^{17} +3.93977 q^{19} +12.9108 q^{20} +2.77875 q^{22} -3.56689 q^{23} +0.0920946 q^{25} -8.40121 q^{26} +5.02962 q^{28} -1.57772 q^{29} -9.17553 q^{31} -27.3680 q^{32} -6.32639 q^{34} +1.98371 q^{35} -7.15525 q^{37} -10.9476 q^{38} -23.3350 q^{40} -7.02640 q^{41} +8.40227 q^{43} -5.72143 q^{44} +9.91148 q^{46} -3.46158 q^{47} -6.22721 q^{49} -0.255908 q^{50} +17.2981 q^{52} -1.97988 q^{53} -2.25657 q^{55} -9.09052 q^{56} +4.38408 q^{58} -8.41286 q^{59} -1.00000 q^{61} +25.4965 q^{62} +41.4649 q^{64} +6.82247 q^{65} +0.158699 q^{67} +13.0260 q^{68} -5.51223 q^{70} -11.7036 q^{71} -11.4377 q^{73} +19.8826 q^{74} +22.5411 q^{76} -0.879084 q^{77} +7.62621 q^{79} +39.0203 q^{80} +19.5246 q^{82} -8.04762 q^{83} +5.13754 q^{85} -23.3478 q^{86} +10.3409 q^{88} +6.83581 q^{89} +2.65781 q^{91} -20.4077 q^{92} +9.61884 q^{94} +8.89035 q^{95} -2.44457 q^{97} +17.3038 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) 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

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Database

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