LMFDB - Embedded newform 6039.2.a.n.1.13

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			1.13
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)

$f(q)$	$=$	$q-0.110609 q^{2} -1.98777 q^{4} +0.0761036 q^{5} +3.79648 q^{7} +0.441081 q^{8} +O(q^{10})$	\(q-0.110609 q^{2} -1.98777 q^{4} +0.0761036 q^{5} +3.79648 q^{7} +0.441081 q^{8} -0.00841771 q^{10} -1.00000 q^{11} +2.86557 q^{13} -0.419923 q^{14} +3.92674 q^{16} -4.34025 q^{17} +7.98182 q^{19} -0.151276 q^{20} +0.110609 q^{22} -8.82636 q^{23} -4.99421 q^{25} -0.316956 q^{26} -7.54651 q^{28} -8.77010 q^{29} +2.15132 q^{31} -1.31649 q^{32} +0.480069 q^{34} +0.288926 q^{35} -9.28709 q^{37} -0.882858 q^{38} +0.0335679 q^{40} +3.66441 q^{41} -3.58502 q^{43} +1.98777 q^{44} +0.976271 q^{46} -6.97477 q^{47} +7.41325 q^{49} +0.552402 q^{50} -5.69607 q^{52} +9.56561 q^{53} -0.0761036 q^{55} +1.67456 q^{56} +0.970048 q^{58} +4.57264 q^{59} -1.00000 q^{61} -0.237955 q^{62} -7.70787 q^{64} +0.218080 q^{65} -6.56001 q^{67} +8.62740 q^{68} -0.0319577 q^{70} -7.11249 q^{71} -5.09557 q^{73} +1.02723 q^{74} -15.8660 q^{76} -3.79648 q^{77} +5.57044 q^{79} +0.298839 q^{80} -0.405315 q^{82} -4.90970 q^{83} -0.330309 q^{85} +0.396534 q^{86} -0.441081 q^{88} -6.47445 q^{89} +10.8791 q^{91} +17.5447 q^{92} +0.771469 q^{94} +0.607445 q^{95} +7.14406 q^{97} -0.819969 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.110609	−0.0782121	−0.0391060	−	0.999235i	$-0.512451\pi$
$2$	−0.110609	−0.0782121	−0.0391060	+	0.999235i	$0.512451\pi$
$3$	0	0
$3$	0	0
$4$	−1.98777	−0.993883
$5$	0.0761036	0.0340345	0.0170173	−	0.999855i	$-0.494583\pi$
$5$	0.0761036	0.0340345	0.0170173	+	0.999855i	$0.494583\pi$
$6$	0	0
$7$	3.79648	1.43493	0.717467	−	0.696592i	$-0.245301\pi$
$7$	3.79648	1.43493	0.717467	+	0.696592i	$0.245301\pi$
$8$	0.441081	0.155946
$9$	0	0
$10$	−0.00841771	−0.00266191
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.86557	0.794765	0.397382	−	0.917653i	$-0.369919\pi$
$13$	2.86557	0.794765	0.397382	+	0.917653i	$0.369919\pi$
$14$	−0.419923	−0.112229
$15$	0	0
$16$	3.92674	0.981686
$17$	−4.34025	−1.05267	−0.526333	−	0.850279i	$-0.676433\pi$
$17$	−4.34025	−1.05267	−0.526333	+	0.850279i	$0.676433\pi$
$18$	0	0
$19$	7.98182	1.83116	0.915578	−	0.402141i	$-0.131734\pi$
$19$	7.98182	1.83116	0.915578	+	0.402141i	$0.131734\pi$
$20$	−0.151276	−0.0338264
$21$	0	0
$22$	0.110609	0.0235818
$23$	−8.82636	−1.84042	−0.920212	−	0.391421i	$-0.871984\pi$
$23$	−8.82636	−1.84042	−0.920212	+	0.391421i	$0.871984\pi$
$24$	0	0
$25$	−4.99421	−0.998842
$26$	−0.316956	−0.0621602
$27$	0	0
$28$	−7.54651	−1.42616
$29$	−8.77010	−1.62857	−0.814283	−	0.580468i	$-0.802870\pi$
$29$	−8.77010	−1.62857	−0.814283	+	0.580468i	$0.802870\pi$
$30$	0	0
$31$	2.15132	0.386389	0.193194	−	0.981160i	$-0.438115\pi$
$31$	2.15132	0.386389	0.193194	+	0.981160i	$0.438115\pi$
$32$	−1.31649	−0.232725
$33$	0	0
$34$	0.480069	0.0823312
$35$	0.288926	0.0488373
$36$	0	0
$37$	−9.28709	−1.52679	−0.763394	−	0.645933i	$-0.776468\pi$
$37$	−9.28709	−1.52679	−0.763394	+	0.645933i	$0.776468\pi$
$38$	−0.882858	−0.143219
$39$	0	0
$40$	0.0335679	0.00530754
$41$	3.66441	0.572284	0.286142	−	0.958187i	$-0.407627\pi$
$41$	3.66441	0.572284	0.286142	+	0.958187i	$0.407627\pi$
$42$	0	0
$43$	−3.58502	−0.546710	−0.273355	−	0.961913i	$-0.588133\pi$
$43$	−3.58502	−0.546710	−0.273355	+	0.961913i	$0.588133\pi$
$44$	1.98777	0.299667
$45$	0	0
$46$	0.976271	0.143943
$47$	−6.97477	−1.01737	−0.508687	−	0.860951i	$-0.669869\pi$
$47$	−6.97477	−1.01737	−0.508687	+	0.860951i	$0.669869\pi$
$48$	0	0
$49$	7.41325	1.05904
$50$	0.552402	0.0781215
$51$	0	0
$52$	−5.69607	−0.789903
$53$	9.56561	1.31394	0.656969	−	0.753918i	$-0.271838\pi$
$53$	9.56561	1.31394	0.656969	+	0.753918i	$0.271838\pi$
$54$	0	0
$55$	−0.0761036	−0.0102618
$56$	1.67456	0.223772
$57$	0	0
$58$	0.970048	0.127374
$59$	4.57264	0.595307	0.297653	−	0.954674i	$-0.403796\pi$
$59$	4.57264	0.595307	0.297653	+	0.954674i	$0.403796\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−0.237955	−0.0302203
$63$	0	0
$64$	−7.70787	−0.963484
$65$	0.218080	0.0270495
$66$	0	0
$67$	−6.56001	−0.801433	−0.400716	−	0.916202i	$-0.631239\pi$
$67$	−6.56001	−0.801433	−0.400716	+	0.916202i	$0.631239\pi$
$68$	8.62740	1.04623
$69$	0	0
$70$	−0.0319577	−0.00381967
$71$	−7.11249	−0.844097	−0.422049	−	0.906573i	$-0.638689\pi$
$71$	−7.11249	−0.844097	−0.422049	+	0.906573i	$0.638689\pi$
$72$	0	0
$73$	−5.09557	−0.596392	−0.298196	−	0.954505i	$-0.596385\pi$
$73$	−5.09557	−0.596392	−0.298196	+	0.954505i	$0.596385\pi$
$74$	1.02723	0.119413
$75$	0	0
$76$	−15.8660	−1.81995
$77$	−3.79648	−0.432649
$78$	0	0
$79$	5.57044	0.626724	0.313362	−	0.949634i	$-0.398545\pi$
$79$	5.57044	0.626724	0.313362	+	0.949634i	$0.398545\pi$
$80$	0.298839	0.0334112
$81$	0	0
$82$	−0.405315	−0.0447595
$83$	−4.90970	−0.538909	−0.269455	−	0.963013i	$-0.586843\pi$
$83$	−4.90970	−0.538909	−0.269455	+	0.963013i	$0.586843\pi$
$84$	0	0
$85$	−0.330309	−0.0358270
$86$	0.396534	0.0427593
$87$	0	0
$88$	−0.441081	−0.0470194
$89$	−6.47445	−0.686290	−0.343145	−	0.939282i	$-0.611492\pi$
$89$	−6.47445	−0.686290	−0.343145	+	0.939282i	$0.611492\pi$
$90$	0	0
$91$	10.8791	1.14044
$92$	17.5447	1.82917
$93$	0	0
$94$	0.771469	0.0795710
$95$	0.607445	0.0623226
$96$	0	0
$97$	7.14406	0.725370	0.362685	−	0.931912i	$-0.381860\pi$
$97$	7.14406	0.725370	0.362685	+	0.931912i	$0.381860\pi$
$98$	−0.819969	−0.0828294
$99$	0	0
$100$	9.92732	0.992732
$101$	9.53683	0.948951	0.474475	−	0.880269i	$-0.342638\pi$
$101$	9.53683	0.948951	0.474475	+	0.880269i	$0.342638\pi$
$102$	0	0
$103$	10.6635	1.05070	0.525351	−	0.850886i	$-0.323934\pi$
$103$	10.6635	1.05070	0.525351	+	0.850886i	$0.323934\pi$
$104$	1.26395	0.123940
$105$	0	0
$106$	−1.05804	−0.102766
$107$	5.71072	0.552076	0.276038	−	0.961147i	$-0.410978\pi$
$107$	5.71072	0.552076	0.276038	+	0.961147i	$0.410978\pi$
$108$	0	0
$109$	−8.11069	−0.776864	−0.388432	−	0.921477i	$-0.626983\pi$
$109$	−8.11069	−0.776864	−0.388432	+	0.921477i	$0.626983\pi$
$110$	0.00841771	0.000802597 0
$111$	0	0
$112$	14.9078	1.40865
$113$	7.42753	0.698723	0.349362	−	0.936988i	$-0.386398\pi$
$113$	7.42753	0.698723	0.349362	+	0.936988i	$0.386398\pi$
$114$	0	0
$115$	−0.671717	−0.0626380
$116$	17.4329	1.61860
$117$	0	0
$118$	−0.505773	−0.0465602
$119$	−16.4777	−1.51051
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.110609	0.0100140
$123$	0	0
$124$	−4.27633	−0.384025
$125$	−0.760595	−0.0680297
$126$	0	0
$127$	−12.5087	−1.10997	−0.554983	−	0.831862i	$-0.687275\pi$
$127$	−12.5087	−1.10997	−0.554983	+	0.831862i	$0.687275\pi$
$128$	3.48555	0.308082
$129$	0	0
$130$	−0.0241215	−0.00211559
$131$	−17.7841	−1.55380	−0.776900	−	0.629624i	$-0.783209\pi$
$131$	−17.7841	−1.55380	−0.776900	+	0.629624i	$0.783209\pi$
$132$	0	0
$133$	30.3028	2.62759
$134$	0.725594	0.0626818
$135$	0	0
$136$	−1.91440	−0.164159
$137$	21.6229	1.84737	0.923685	−	0.383153i	$-0.125162\pi$
$137$	21.6229	1.84737	0.923685	+	0.383153i	$0.125162\pi$
$138$	0	0
$139$	0.746613	0.0633269	0.0316635	−	0.999499i	$-0.489920\pi$
$139$	0.746613	0.0633269	0.0316635	+	0.999499i	$0.489920\pi$
$140$	−0.574316	−0.0485386
$141$	0	0
$142$	0.786702	0.0660186
$143$	−2.86557	−0.239631
$144$	0	0
$145$	−0.667436	−0.0554275
$146$	0.563614	0.0466450
$147$	0	0
$148$	18.4606	1.51745
$149$	−5.42693	−0.444591	−0.222296	−	0.974979i	$-0.571355\pi$
$149$	−5.42693	−0.444591	−0.222296	+	0.974979i	$0.571355\pi$
$150$	0	0
$151$	−2.58101	−0.210040	−0.105020	−	0.994470i	$-0.533491\pi$
$151$	−2.58101	−0.210040	−0.105020	+	0.994470i	$0.533491\pi$
$152$	3.52063	0.285561
$153$	0	0
$154$	0.419923	0.0338384
$155$	0.163723	0.0131506
$156$	0	0
$157$	−18.9785	−1.51465	−0.757325	−	0.653038i	$-0.773494\pi$
$157$	−18.9785	−1.51465	−0.757325	+	0.653038i	$0.773494\pi$
$158$	−0.616139	−0.0490174
$159$	0	0
$160$	−0.100190	−0.00792071
$161$	−33.5091	−2.64089
$162$	0	0
$163$	−11.3676	−0.890379	−0.445189	−	0.895436i	$-0.646864\pi$
$163$	−11.3676	−0.890379	−0.445189	+	0.895436i	$0.646864\pi$
$164$	−7.28398	−0.568783
$165$	0	0
$166$	0.543055	0.0421492
$167$	−11.8499	−0.916969	−0.458485	−	0.888702i	$-0.651608\pi$
$167$	−11.8499	−0.916969	−0.458485	+	0.888702i	$0.651608\pi$
$168$	0	0
$169$	−4.78854	−0.368349
$170$	0.0365350	0.00280210
$171$	0	0
$172$	7.12617	0.543366
$173$	−9.34584	−0.710551	−0.355276	−	0.934762i	$-0.615613\pi$
$173$	−9.34584	−0.710551	−0.355276	+	0.934762i	$0.615613\pi$
$174$	0	0
$175$	−18.9604	−1.43327
$176$	−3.92674	−0.295989
$177$	0	0
$178$	0.716130	0.0536762
$179$	−17.0201	−1.27215	−0.636073	−	0.771629i	$-0.719442\pi$
$179$	−17.0201	−1.27215	−0.636073	+	0.771629i	$0.719442\pi$
$180$	0	0
$181$	22.6884	1.68641	0.843206	−	0.537590i	$-0.180665\pi$
$181$	22.6884	1.68641	0.843206	+	0.537590i	$0.180665\pi$
$182$	−1.20332	−0.0891958
$183$	0	0
$184$	−3.89314	−0.287006
$185$	−0.706780	−0.0519635
$186$	0	0
$187$	4.34025	0.317391
$188$	13.8642	1.01115
$189$	0	0
$190$	−0.0671887	−0.00487438
$191$	−19.8345	−1.43517	−0.717586	−	0.696470i	$-0.754753\pi$
$191$	−19.8345	−1.43517	−0.717586	+	0.696470i	$0.754753\pi$
$192$	0	0
$193$	−6.95311	−0.500496	−0.250248	−	0.968182i	$-0.580512\pi$
$193$	−6.95311	−0.500496	−0.250248	+	0.968182i	$0.580512\pi$
$194$	−0.790195	−0.0567327
$195$	0	0
$196$	−14.7358	−1.05256
$197$	3.09263	0.220341	0.110170	−	0.993913i	$-0.464860\pi$
$197$	3.09263	0.220341	0.110170	+	0.993913i	$0.464860\pi$
$198$	0	0
$199$	5.62476	0.398729	0.199364	−	0.979925i	$-0.436112\pi$
$199$	5.62476	0.398729	0.199364	+	0.979925i	$0.436112\pi$
$200$	−2.20285	−0.155765
$201$	0	0
$202$	−1.05486	−0.0742194
$203$	−33.2955	−2.33689
$204$	0	0
$205$	0.278874	0.0194774
$206$	−1.17947	−0.0821775
$207$	0	0
$208$	11.2523	0.780210
$209$	−7.98182	−0.552114
$210$	0	0
$211$	−7.98551	−0.549745	−0.274873	−	0.961481i	$-0.588636\pi$
$211$	−7.98551	−0.549745	−0.274873	+	0.961481i	$0.588636\pi$
$212$	−19.0142	−1.30590
$213$	0	0
$214$	−0.631654	−0.0431790
$215$	−0.272833	−0.0186070
$216$	0	0
$217$	8.16745	0.554443
$218$	0.897113	0.0607601
$219$	0	0
$220$	0.151276	0.0101990
$221$	−12.4373	−0.836622
$222$	0	0
$223$	21.7057	1.45352	0.726759	−	0.686892i	$-0.241025\pi$
$223$	21.7057	1.45352	0.726759	+	0.686892i	$0.241025\pi$
$224$	−4.99804	−0.333946
$225$	0	0
$226$	−0.821549	−0.0546486
$227$	23.0947	1.53285	0.766423	−	0.642336i	$-0.222035\pi$
$227$	23.0947	1.53285	0.766423	+	0.642336i	$0.222035\pi$
$228$	0	0
$229$	−21.0725	−1.39251	−0.696257	−	0.717793i	$-0.745152\pi$
$229$	−21.0725	−1.39251	−0.696257	+	0.717793i	$0.745152\pi$
$230$	0.0742977	0.00489905
$231$	0	0
$232$	−3.86833	−0.253968
$233$	19.5223	1.27895	0.639475	−	0.768812i	$-0.279152\pi$
$233$	19.5223	1.27895	0.639475	+	0.768812i	$0.279152\pi$
$234$	0	0
$235$	−0.530805	−0.0346259
$236$	−9.08934	−0.591665
$237$	0	0
$238$	1.82257	0.118140
$239$	8.05007	0.520716	0.260358	−	0.965512i	$-0.416159\pi$
$239$	8.05007	0.520716	0.260358	+	0.965512i	$0.416159\pi$
$240$	0	0
$241$	−16.0330	−1.03278	−0.516388	−	0.856355i	$-0.672724\pi$
$241$	−16.0330	−1.03278	−0.516388	+	0.856355i	$0.672724\pi$
$242$	−0.110609	−0.00711019
$243$	0	0
$244$	1.98777	0.127254
$245$	0.564175	0.0360438
$246$	0	0
$247$	22.8724	1.45534
$248$	0.948908	0.0602557
$249$	0	0
$250$	0.0841283	0.00532074
$251$	−18.4562	−1.16494	−0.582471	−	0.812851i	$-0.697914\pi$
$251$	−18.4562	−1.16494	−0.582471	+	0.812851i	$0.697914\pi$
$252$	0	0
$253$	8.82636	0.554908
$254$	1.38357	0.0868128
$255$	0	0
$256$	15.0302	0.939388
$257$	−4.27534	−0.266689	−0.133344	−	0.991070i	$-0.542572\pi$
$257$	−4.27534	−0.266689	−0.133344	+	0.991070i	$0.542572\pi$
$258$	0	0
$259$	−35.2582	−2.19084
$260$	−0.433491	−0.0268840
$261$	0	0
$262$	1.96707	0.121526
$263$	26.3258	1.62332	0.811659	−	0.584132i	$-0.198565\pi$
$263$	26.3258	1.62332	0.811659	+	0.584132i	$0.198565\pi$
$264$	0	0
$265$	0.727977	0.0447193
$266$	−3.35175	−0.205509
$267$	0	0
$268$	13.0398	0.796530
$269$	23.7635	1.44889	0.724444	−	0.689333i	$-0.242097\pi$
$269$	23.7635	1.44889	0.724444	+	0.689333i	$0.242097\pi$
$270$	0	0
$271$	−5.94283	−0.361001	−0.180501	−	0.983575i	$-0.557772\pi$
$271$	−5.94283	−0.361001	−0.180501	+	0.983575i	$0.557772\pi$
$272$	−17.0431	−1.03339
$273$	0	0
$274$	−2.39168	−0.144487
$275$	4.99421	0.301162
$276$	0	0
$277$	−2.28627	−0.137369	−0.0686844	−	0.997638i	$-0.521880\pi$
$277$	−2.28627	−0.137369	−0.0686844	+	0.997638i	$0.521880\pi$
$278$	−0.0825818	−0.00495293
$279$	0	0
$280$	0.127440	0.00761597
$281$	−11.7431	−0.700536	−0.350268	−	0.936650i	$-0.613909\pi$
$281$	−11.7431	−0.700536	−0.350268	+	0.936650i	$0.613909\pi$
$282$	0	0
$283$	−23.9783	−1.42536	−0.712682	−	0.701487i	$-0.752520\pi$
$283$	−23.9783	−1.42536	−0.712682	+	0.701487i	$0.752520\pi$
$284$	14.1380	0.838934
$285$	0	0
$286$	0.316956	0.0187420
$287$	13.9118	0.821190
$288$	0	0
$289$	1.83778	0.108105
$290$	0.0738241	0.00433510
$291$	0	0
$292$	10.1288	0.592743
$293$	−29.2544	−1.70906	−0.854531	−	0.519401i	$-0.826155\pi$
$293$	−29.2544	−1.70906	−0.854531	+	0.519401i	$0.826155\pi$
$294$	0	0
$295$	0.347994	0.0202610
$296$	−4.09636	−0.238096
$297$	0	0
$298$	0.600265	0.0347724
$299$	−25.2925	−1.46270
$300$	0	0
$301$	−13.6104	−0.784493
$302$	0.285482	0.0164277
$303$	0	0
$304$	31.3426	1.79762
$305$	−0.0761036	−0.00435768
$306$	0	0
$307$	−11.5984	−0.661954	−0.330977	−	0.943639i	$-0.607378\pi$
$307$	−11.5984	−0.661954	−0.330977	+	0.943639i	$0.607378\pi$
$308$	7.54651	0.430002
$309$	0	0
$310$	−0.0181092	−0.00102853
$311$	−25.0665	−1.42139	−0.710695	−	0.703501i	$-0.751619\pi$
$311$	−25.0665	−1.42139	−0.710695	+	0.703501i	$0.751619\pi$
$312$	0	0
$313$	−22.6390	−1.27963	−0.639816	−	0.768528i	$-0.720989\pi$
$313$	−22.6390	−1.27963	−0.639816	+	0.768528i	$0.720989\pi$
$314$	2.09919	0.118464
$315$	0	0
$316$	−11.0727	−0.622890
$317$	20.0501	1.12613	0.563063	−	0.826414i	$-0.309623\pi$
$317$	20.0501	1.12613	0.563063	+	0.826414i	$0.309623\pi$
$318$	0	0
$319$	8.77010	0.491031
$320$	−0.586597	−0.0327917
$321$	0	0
$322$	3.70639	0.206549
$323$	−34.6431	−1.92759
$324$	0	0
$325$	−14.3112	−0.793844
$326$	1.25735	0.0696384
$327$	0	0
$328$	1.61630	0.0892453
$329$	−26.4796	−1.45987
$330$	0	0
$331$	−14.3521	−0.788865	−0.394433	−	0.918925i	$-0.629059\pi$
$331$	−14.3521	−0.788865	−0.394433	+	0.918925i	$0.629059\pi$
$332$	9.75933	0.535613
$333$	0	0
$334$	1.31070	0.0717181
$335$	−0.499240	−0.0272764
$336$	0	0
$337$	20.9937	1.14360	0.571800	−	0.820393i	$-0.306245\pi$
$337$	20.9937	1.14360	0.571800	+	0.820393i	$0.306245\pi$
$338$	0.529653	0.0288093
$339$	0	0
$340$	0.656576	0.0356078
$341$	−2.15132	−0.116501
$342$	0	0
$343$	1.56890	0.0847127
$344$	−1.58128	−0.0852571
$345$	0	0
$346$	1.03373	0.0555737
$347$	−28.1758	−1.51256	−0.756278	−	0.654250i	$-0.772984\pi$
$347$	−28.1758	−1.51256	−0.756278	+	0.654250i	$0.772984\pi$
$348$	0	0
$349$	8.23859	0.441002	0.220501	−	0.975387i	$-0.429231\pi$
$349$	8.23859	0.441002	0.220501	+	0.975387i	$0.429231\pi$
$350$	2.09718	0.112099
$351$	0	0
$352$	1.31649	0.0701694
$353$	11.3361	0.603359	0.301680	−	0.953409i	$-0.402453\pi$
$353$	11.3361	0.603359	0.301680	+	0.953409i	$0.402453\pi$
$354$	0	0
$355$	−0.541286	−0.0287285
$356$	12.8697	0.682092
$357$	0	0
$358$	1.88257	0.0994972
$359$	18.9216	0.998643	0.499322	−	0.866417i	$-0.333583\pi$
$359$	18.9216	0.998643	0.499322	+	0.866417i	$0.333583\pi$
$360$	0	0
$361$	44.7095	2.35313
$362$	−2.50953	−0.131898
$363$	0	0
$364$	−21.6250	−1.13346
$365$	−0.387791	−0.0202979
$366$	0	0
$367$	7.65543	0.399610	0.199805	−	0.979836i	$-0.435969\pi$
$367$	7.65543	0.399610	0.199805	+	0.979836i	$0.435969\pi$
$368$	−34.6589	−1.80672
$369$	0	0
$370$	0.0781760	0.00406418
$371$	36.3156	1.88541
$372$	0	0
$373$	−0.0460594	−0.00238487	−0.00119243	−	0.999999i	$-0.500380\pi$
$373$	−0.0460594	−0.00238487	−0.00119243	+	0.999999i	$0.500380\pi$
$374$	−0.480069	−0.0248238
$375$	0	0
$376$	−3.07644	−0.158655
$377$	−25.1313	−1.29433
$378$	0	0
$379$	6.07669	0.312139	0.156069	−	0.987746i	$-0.450118\pi$
$379$	6.07669	0.312139	0.156069	+	0.987746i	$0.450118\pi$
$380$	−1.20746	−0.0619413
$381$	0	0
$382$	2.19386	0.112248
$383$	−19.1399	−0.978002	−0.489001	−	0.872283i	$-0.662639\pi$
$383$	−19.1399	−0.978002	−0.489001	+	0.872283i	$0.662639\pi$
$384$	0	0
$385$	−0.288926	−0.0147250
$386$	0.769074	0.0391449
$387$	0	0
$388$	−14.2007	−0.720932
$389$	30.0214	1.52215	0.761073	−	0.648666i	$-0.224673\pi$
$389$	30.0214	1.52215	0.761073	+	0.648666i	$0.224673\pi$
$390$	0	0
$391$	38.3086	1.93735
$392$	3.26985	0.165152
$393$	0	0
$394$	−0.342071	−0.0172333
$395$	0.423930	0.0213303
$396$	0	0
$397$	23.6843	1.18868	0.594342	−	0.804213i	$-0.297413\pi$
$397$	23.6843	1.18868	0.594342	+	0.804213i	$0.297413\pi$
$398$	−0.622147	−0.0311854
$399$	0	0
$400$	−19.6110	−0.980549
$401$	−17.4907	−0.873443	−0.436722	−	0.899597i	$-0.643860\pi$
$401$	−17.4907	−0.873443	−0.436722	+	0.899597i	$0.643860\pi$
$402$	0	0
$403$	6.16476	0.307088
$404$	−18.9570	−0.943146
$405$	0	0
$406$	3.68277	0.182773
$407$	9.28709	0.460344
$408$	0	0
$409$	−12.0343	−0.595058	−0.297529	−	0.954713i	$-0.596162\pi$
$409$	−12.0343	−0.595058	−0.297529	+	0.954713i	$0.596162\pi$
$410$	−0.0308459	−0.00152337
$411$	0	0
$412$	−21.1964	−1.04427
$413$	17.3599	0.854226
$414$	0	0
$415$	−0.373645	−0.0183415
$416$	−3.77250	−0.184962
$417$	0	0
$418$	0.882858	0.0431820
$419$	−4.71859	−0.230518	−0.115259	−	0.993335i	$-0.536770\pi$
$419$	−4.71859	−0.230518	−0.115259	+	0.993335i	$0.536770\pi$
$420$	0	0
$421$	−3.31420	−0.161524	−0.0807622	−	0.996733i	$-0.525735\pi$
$421$	−3.31420	−0.161524	−0.0807622	+	0.996733i	$0.525735\pi$
$422$	0.883266	0.0429967
$423$	0	0
$424$	4.21921	0.204903
$425$	21.6761	1.05145
$426$	0	0
$427$	−3.79648	−0.183724
$428$	−11.3516	−0.548699
$429$	0	0
$430$	0.0301776	0.00145529
$431$	−27.1154	−1.30610	−0.653051	−	0.757314i	$-0.726511\pi$
$431$	−27.1154	−1.30610	−0.653051	+	0.757314i	$0.726511\pi$
$432$	0	0
$433$	−22.4342	−1.07812	−0.539059	−	0.842268i	$-0.681220\pi$
$433$	−22.4342	−1.07812	−0.539059	+	0.842268i	$0.681220\pi$
$434$	−0.903390	−0.0433641
$435$	0	0
$436$	16.1222	0.772111
$437$	−70.4504	−3.37010
$438$	0	0
$439$	9.76170	0.465900	0.232950	−	0.972489i	$-0.425162\pi$
$439$	9.76170	0.465900	0.232950	+	0.972489i	$0.425162\pi$
$440$	−0.0335679	−0.00160028
$441$	0	0
$442$	1.37567	0.0654339
$443$	22.8703	1.08660	0.543300	−	0.839539i	$-0.317175\pi$
$443$	22.8703	1.08660	0.543300	+	0.839539i	$0.317175\pi$
$444$	0	0
$445$	−0.492729	−0.0233576
$446$	−2.40083	−0.113683
$447$	0	0
$448$	−29.2628	−1.38254
$449$	−11.6616	−0.550344	−0.275172	−	0.961395i	$-0.588735\pi$
$449$	−11.6616	−0.550344	−0.275172	+	0.961395i	$0.588735\pi$
$450$	0	0
$451$	−3.66441	−0.172550
$452$	−14.7642	−0.694449
$453$	0	0
$454$	−2.55447	−0.119887
$455$	0.827935	0.0388142
$456$	0	0
$457$	−25.9533	−1.21405	−0.607023	−	0.794684i	$-0.707636\pi$
$457$	−25.9533	−1.21405	−0.607023	+	0.794684i	$0.707636\pi$
$458$	2.33081	0.108911
$459$	0	0
$460$	1.33522	0.0622548
$461$	−19.5400	−0.910066	−0.455033	−	0.890474i	$-0.650373\pi$
$461$	−19.5400	−0.910066	−0.455033	+	0.890474i	$0.650373\pi$
$462$	0	0
$463$	15.5942	0.724724	0.362362	−	0.932037i	$-0.381970\pi$
$463$	15.5942	0.724724	0.362362	+	0.932037i	$0.381970\pi$
$464$	−34.4379	−1.59874
$465$	0	0
$466$	−2.15934	−0.100029
$467$	−2.26539	−0.104830	−0.0524149	−	0.998625i	$-0.516692\pi$
$467$	−2.26539	−0.104830	−0.0524149	+	0.998625i	$0.516692\pi$
$468$	0	0
$469$	−24.9049	−1.15000
$470$	0.0587116	0.00270816
$471$	0	0
$472$	2.01691	0.0928356
$473$	3.58502	0.164839
$474$	0	0
$475$	−39.8629	−1.82903
$476$	32.7538	1.50127
$477$	0	0
$478$	−0.890407	−0.0407263
$479$	−42.1326	−1.92509	−0.962545	−	0.271123i	$-0.912605\pi$
$479$	−42.1326	−1.92509	−0.962545	+	0.271123i	$0.912605\pi$
$480$	0	0
$481$	−26.6128	−1.21344
$482$	1.77339	0.0807756
$483$	0	0
$484$	−1.98777	−0.0903530
$485$	0.543689	0.0246876
$486$	0	0
$487$	16.6975	0.756638	0.378319	−	0.925675i	$-0.376502\pi$
$487$	16.6975	0.756638	0.378319	+	0.925675i	$0.376502\pi$
$488$	−0.441081	−0.0199668
$489$	0	0
$490$	−0.0624026	−0.00281906
$491$	−12.2889	−0.554590	−0.277295	−	0.960785i	$-0.589438\pi$
$491$	−12.2889	−0.554590	−0.277295	+	0.960785i	$0.589438\pi$
$492$	0	0
$493$	38.0644	1.71434
$494$	−2.52989	−0.113825
$495$	0	0
$496$	8.44769	0.379313
$497$	−27.0024	−1.21122
$498$	0	0
$499$	15.4398	0.691181	0.345591	−	0.938385i	$-0.387679\pi$
$499$	15.4398	0.691181	0.345591	+	0.938385i	$0.387679\pi$
$500$	1.51188	0.0676135
$501$	0	0
$502$	2.04141	0.0911126
$503$	6.34296	0.282818	0.141409	−	0.989951i	$-0.454837\pi$
$503$	6.34296	0.282818	0.141409	+	0.989951i	$0.454837\pi$
$504$	0	0
$505$	0.725787	0.0322971
$506$	−0.976271	−0.0434006
$507$	0	0
$508$	24.8643	1.10318
$509$	−10.7065	−0.474559	−0.237279	−	0.971441i	$-0.576256\pi$
$509$	−10.7065	−0.474559	−0.237279	+	0.971441i	$0.576256\pi$
$510$	0	0
$511$	−19.3452	−0.855783
$512$	−8.63356	−0.381553
$513$	0	0
$514$	0.472890	0.0208583
$515$	0.811527	0.0357601
$516$	0	0
$517$	6.97477	0.306750
$518$	3.89986	0.171350
$519$	0	0
$520$	0.0961909	0.00421825
$521$	23.3915	1.02480	0.512401	−	0.858746i	$-0.328756\pi$
$521$	23.3915	1.02480	0.512401	+	0.858746i	$0.328756\pi$
$522$	0	0
$523$	0.600364	0.0262521	0.0131260	−	0.999914i	$-0.495822\pi$
$523$	0.600364	0.0262521	0.0131260	+	0.999914i	$0.495822\pi$
$524$	35.3505	1.54430
$525$	0	0
$526$	−2.91186	−0.126963
$527$	−9.33728	−0.406738
$528$	0	0
$529$	54.9046	2.38716
$530$	−0.0805205	−0.00349759
$531$	0	0
$532$	−60.2349	−2.61151
$533$	10.5006	0.454831
$534$	0	0
$535$	0.434606	0.0187896
$536$	−2.89350	−0.124980
$537$	0	0
$538$	−2.62845	−0.113321
$539$	−7.41325	−0.319311
$540$	0	0
$541$	−11.4295	−0.491393	−0.245697	−	0.969347i	$-0.579017\pi$
$541$	−11.4295	−0.491393	−0.245697	+	0.969347i	$0.579017\pi$
$542$	0.657328	0.0282347
$543$	0	0
$544$	5.71392	0.244982
$545$	−0.617253	−0.0264402
$546$	0	0
$547$	12.3464	0.527894	0.263947	−	0.964537i	$-0.414976\pi$
$547$	12.3464	0.527894	0.263947	+	0.964537i	$0.414976\pi$
$548$	−42.9813	−1.83607
$549$	0	0
$550$	−0.552402	−0.0235545
$551$	−70.0014	−2.98216
$552$	0	0
$553$	21.1481	0.899307
$554$	0.252882	0.0107439
$555$	0	0
$556$	−1.48409	−0.0629395
$557$	−0.0358158	−0.00151757	−0.000758783	−	1.00000i	$-0.500242\pi$
$557$	−0.0358158	−0.00151757	−0.000758783		1.00000i	$0.500242\pi$
$558$	0	0
$559$	−10.2731	−0.434506
$560$	1.13454	0.0479429
$561$	0	0
$562$	1.29889	0.0547904
$563$	34.8146	1.46726	0.733631	−	0.679548i	$-0.237824\pi$
$563$	34.8146	1.46726	0.733631	+	0.679548i	$0.237824\pi$
$564$	0	0
$565$	0.565261	0.0237807
$566$	2.65221	0.111481
$567$	0	0
$568$	−3.13719	−0.131633
$569$	1.86930	0.0783650	0.0391825	−	0.999232i	$-0.487525\pi$
$569$	1.86930	0.0783650	0.0391825	+	0.999232i	$0.487525\pi$
$570$	0	0
$571$	−21.4826	−0.899019	−0.449510	−	0.893275i	$-0.648401\pi$
$571$	−21.4826	−0.899019	−0.449510	+	0.893275i	$0.648401\pi$
$572$	5.69607	0.238165
$573$	0	0
$574$	−1.53877	−0.0642270
$575$	44.0807	1.83829
$576$	0	0
$577$	−9.99358	−0.416038	−0.208019	−	0.978125i	$-0.566702\pi$
$577$	−9.99358	−0.416038	−0.208019	+	0.978125i	$0.566702\pi$
$578$	−0.203274	−0.00845509
$579$	0	0
$580$	1.32671	0.0550885
$581$	−18.6396	−0.773299
$582$	0	0
$583$	−9.56561	−0.396167
$584$	−2.24756	−0.0930047
$585$	0	0
$586$	3.23579	0.133669
$587$	−38.0086	−1.56878	−0.784391	−	0.620267i	$-0.787024\pi$
$587$	−38.0086	−1.56878	−0.784391	+	0.620267i	$0.787024\pi$
$588$	0	0
$589$	17.1715	0.707538
$590$	−0.0384911	−0.00158466
$591$	0	0
$592$	−36.4680	−1.49883
$593$	−2.69125	−0.110516	−0.0552582	−	0.998472i	$-0.517598\pi$
$593$	−2.69125	−0.110516	−0.0552582	+	0.998472i	$0.517598\pi$
$594$	0	0
$595$	−1.25401	−0.0514094
$596$	10.7875	0.441872
$597$	0	0
$598$	2.79757	0.114401
$599$	38.5122	1.57357	0.786784	−	0.617229i	$-0.211745\pi$
$599$	38.5122	1.57357	0.786784	+	0.617229i	$0.211745\pi$
$600$	0	0
$601$	−6.39375	−0.260806	−0.130403	−	0.991461i	$-0.541627\pi$
$601$	−6.39375	−0.260806	−0.130403	+	0.991461i	$0.541627\pi$
$602$	1.50543	0.0613568
$603$	0	0
$604$	5.13045	0.208755
$605$	0.0761036	0.00309405
$606$	0	0
$607$	−21.9533	−0.891056	−0.445528	−	0.895268i	$-0.646984\pi$
$607$	−21.9533	−0.891056	−0.445528	+	0.895268i	$0.646984\pi$
$608$	−10.5080	−0.426157
$609$	0	0
$610$	0.00841771	0.000340823 0
$611$	−19.9867	−0.808573
$612$	0	0
$613$	−13.0981	−0.529028	−0.264514	−	0.964382i	$-0.585212\pi$
$613$	−13.0981	−0.529028	−0.264514	+	0.964382i	$0.585212\pi$
$614$	1.28288	0.0517728
$615$	0	0
$616$	−1.67456	−0.0674698
$617$	−5.20052	−0.209365	−0.104683	−	0.994506i	$-0.533383\pi$
$617$	−5.20052	−0.209365	−0.104683	+	0.994506i	$0.533383\pi$
$618$	0	0
$619$	41.8564	1.68235	0.841175	−	0.540762i	$-0.181864\pi$
$619$	41.8564	1.68235	0.841175	+	0.540762i	$0.181864\pi$
$620$	−0.325444	−0.0130701
$621$	0	0
$622$	2.77257	0.111170
$623$	−24.5801	−0.984781
$624$	0	0
$625$	24.9132	0.996526
$626$	2.50407	0.100083
$627$	0	0
$628$	37.7249	1.50539
$629$	40.3083	1.60720
$630$	0	0
$631$	−30.4353	−1.21161	−0.605805	−	0.795613i	$-0.707149\pi$
$631$	−30.4353	−1.21161	−0.605805	+	0.795613i	$0.707149\pi$
$632$	2.45702	0.0977349
$633$	0	0
$634$	−2.21771	−0.0880767
$635$	−0.951955	−0.0377772
$636$	0	0
$637$	21.2432	0.841685
$638$	−0.970048	−0.0384046
$639$	0	0
$640$	0.265262	0.0104854
$641$	−5.45615	−0.215505	−0.107752	−	0.994178i	$-0.534365\pi$
$641$	−5.45615	−0.215505	−0.107752	+	0.994178i	$0.534365\pi$
$642$	0	0
$643$	13.9108	0.548587	0.274294	−	0.961646i	$-0.411556\pi$
$643$	13.9108	0.548587	0.274294	+	0.961646i	$0.411556\pi$
$644$	66.6082	2.62473
$645$	0	0
$646$	3.83183	0.150761
$647$	−34.4350	−1.35378	−0.676891	−	0.736084i	$-0.736673\pi$
$647$	−34.4350	−1.35378	−0.676891	+	0.736084i	$0.736673\pi$
$648$	0	0
$649$	−4.57264	−0.179492
$650$	1.58295	0.0620882
$651$	0	0
$652$	22.5961	0.884932
$653$	−18.1858	−0.711665	−0.355833	−	0.934550i	$-0.615803\pi$
$653$	−18.1858	−0.711665	−0.355833	+	0.934550i	$0.615803\pi$
$654$	0	0
$655$	−1.35343	−0.0528829
$656$	14.3892	0.561803
$657$	0	0
$658$	2.92887	0.114179
$659$	40.3831	1.57310	0.786551	−	0.617525i	$-0.211865\pi$
$659$	40.3831	1.57310	0.786551	+	0.617525i	$0.211865\pi$
$660$	0	0
$661$	−22.0696	−0.858408	−0.429204	−	0.903208i	$-0.641206\pi$
$661$	−22.0696	−0.858408	−0.429204	+	0.903208i	$0.641206\pi$
$662$	1.58747	0.0616988
$663$	0	0
$664$	−2.16558	−0.0840406
$665$	2.30615	0.0894288
$666$	0	0
$667$	77.4080	2.99725
$668$	23.5547	0.911360
$669$	0	0
$670$	0.0552203	0.00213334
$671$	1.00000	0.0386046
$672$	0	0
$673$	−25.2391	−0.972897	−0.486448	−	0.873709i	$-0.661708\pi$
$673$	−25.2391	−0.972897	−0.486448	+	0.873709i	$0.661708\pi$
$674$	−2.32209	−0.0894434
$675$	0	0
$676$	9.51849	0.366096
$677$	29.9067	1.14941	0.574704	−	0.818362i	$-0.305117\pi$
$677$	29.9067	1.14941	0.574704	+	0.818362i	$0.305117\pi$
$678$	0	0
$679$	27.1223	1.04086
$680$	−0.145693	−0.00558707
$681$	0	0
$682$	0.237955	0.00911176
$683$	36.2768	1.38809	0.694046	−	0.719931i	$-0.255826\pi$
$683$	36.2768	1.38809	0.694046	+	0.719931i	$0.255826\pi$
$684$	0	0
$685$	1.64558	0.0628744
$686$	−0.173534	−0.00662556
$687$	0	0
$688$	−14.0774	−0.536697
$689$	27.4109	1.04427
$690$	0	0
$691$	−37.1636	−1.41377	−0.706885	−	0.707328i	$-0.749900\pi$
$691$	−37.1636	−1.41377	−0.706885	+	0.707328i	$0.749900\pi$
$692$	18.5773	0.706205
$693$	0	0
$694$	3.11649	0.118300
$695$	0.0568199	0.00215530
$696$	0	0
$697$	−15.9044	−0.602424
$698$	−0.911259	−0.0344917
$699$	0	0
$700$	37.6888	1.42450
$701$	22.1251	0.835654	0.417827	−	0.908527i	$-0.362792\pi$
$701$	22.1251	0.835654	0.417827	+	0.908527i	$0.362792\pi$
$702$	0	0
$703$	−74.1279	−2.79579
$704$	7.70787	0.290501
$705$	0	0
$706$	−1.25387	−0.0471900
$707$	36.2064	1.36168
$708$	0	0
$709$	−3.10803	−0.116724	−0.0583622	−	0.998295i	$-0.518588\pi$
$709$	−3.10803	−0.116724	−0.0583622	+	0.998295i	$0.518588\pi$
$710$	0.0598709	0.00224691
$711$	0	0
$712$	−2.85576	−0.107024
$713$	−18.9883	−0.711119
$714$	0	0
$715$	−0.218080	−0.00815572
$716$	33.8321	1.26436
$717$	0	0
$718$	−2.09289	−0.0781060
$719$	−31.2247	−1.16448	−0.582242	−	0.813016i	$-0.697824\pi$
$719$	−31.2247	−1.16448	−0.582242	+	0.813016i	$0.697824\pi$
$720$	0	0
$721$	40.4836	1.50769
$722$	−4.94526	−0.184043
$723$	0	0
$724$	−45.0992	−1.67610
$725$	43.7997	1.62668
$726$	0	0
$727$	38.9601	1.44495	0.722475	−	0.691397i	$-0.243005\pi$
$727$	38.9601	1.44495	0.722475	+	0.691397i	$0.243005\pi$
$728$	4.79855	0.177846
$729$	0	0
$730$	0.0428930	0.00158754
$731$	15.5599	0.575503
$732$	0	0
$733$	0.872224	0.0322163	0.0161082	−	0.999870i	$-0.494872\pi$
$733$	0.872224	0.0322163	0.0161082	+	0.999870i	$0.494872\pi$
$734$	−0.846757	−0.0312544
$735$	0	0
$736$	11.6199	0.428313
$737$	6.56001	0.241641
$738$	0	0
$739$	−44.8816	−1.65100	−0.825499	−	0.564404i	$-0.809106\pi$
$739$	−44.8816	−1.65100	−0.825499	+	0.564404i	$0.809106\pi$
$740$	1.40491	0.0516457
$741$	0	0
$742$	−4.01682	−0.147462
$743$	17.1705	0.629924	0.314962	−	0.949104i	$-0.398008\pi$
$743$	17.1705	0.629924	0.314962	+	0.949104i	$0.398008\pi$
$744$	0	0
$745$	−0.413009	−0.0151315
$746$	0.00509457	0.000186525 0
$747$	0	0
$748$	−8.62740	−0.315449
$749$	21.6806	0.792192
$750$	0	0
$751$	−8.10694	−0.295827	−0.147913	−	0.989000i	$-0.547256\pi$
$751$	−8.10694	−0.295827	−0.147913	+	0.989000i	$0.547256\pi$
$752$	−27.3881	−0.998742
$753$	0	0
$754$	2.77974	0.101232
$755$	−0.196424	−0.00714861
$756$	0	0
$757$	−43.1699	−1.56904	−0.784519	−	0.620105i	$-0.787090\pi$
$757$	−43.1699	−1.56904	−0.784519	+	0.620105i	$0.787090\pi$
$758$	−0.672134	−0.0244130
$759$	0	0
$760$	0.267933	0.00971894
$761$	−5.86825	−0.212724	−0.106362	−	0.994327i	$-0.533920\pi$
$761$	−5.86825	−0.212724	−0.106362	+	0.994327i	$0.533920\pi$
$762$	0	0
$763$	−30.7921	−1.11475
$764$	39.4263	1.42639
$765$	0	0
$766$	2.11704	0.0764916
$767$	13.1032	0.473129
$768$	0	0
$769$	11.4647	0.413426	0.206713	−	0.978402i	$-0.433723\pi$
$769$	11.4647	0.413426	0.206713	+	0.978402i	$0.433723\pi$
$770$	0.0319577	0.00115167
$771$	0	0
$772$	13.8212	0.497434
$773$	9.19356	0.330669	0.165335	−	0.986238i	$-0.447130\pi$
$773$	9.19356	0.330669	0.165335	+	0.986238i	$0.447130\pi$
$774$	0	0
$775$	−10.7442	−0.385941
$776$	3.15111	0.113118
$777$	0	0
$778$	−3.32063	−0.119050
$779$	29.2486	1.04794
$780$	0	0
$781$	7.11249	0.254505
$782$	−4.23726	−0.151524
$783$	0	0
$784$	29.1099	1.03964
$785$	−1.44433	−0.0515505
$786$	0	0
$787$	42.3426	1.50935	0.754675	−	0.656098i	$-0.227794\pi$
$787$	42.3426	1.50935	0.754675	+	0.656098i	$0.227794\pi$
$788$	−6.14742	−0.218993
$789$	0	0
$790$	−0.0468904	−0.00166828
$791$	28.1985	1.00262
$792$	0	0
$793$	−2.86557	−0.101759
$794$	−2.61969	−0.0929694
$795$	0	0
$796$	−11.1807	−0.396290
$797$	−33.9733	−1.20340	−0.601698	−	0.798723i	$-0.705509\pi$
$797$	−33.9733	−1.20340	−0.601698	+	0.798723i	$0.705509\pi$
$798$	0	0
$799$	30.2722	1.07096
$800$	6.57485	0.232456
$801$	0	0
$802$	1.93462	0.0683138
$803$	5.09557	0.179819
$804$	0	0
$805$	−2.55016	−0.0898814
$806$	−0.681875	−0.0240180
$807$	0	0
$808$	4.20652	0.147985
$809$	−30.1801	−1.06108	−0.530538	−	0.847661i	$-0.678010\pi$
$809$	−30.1801	−1.06108	−0.530538	+	0.847661i	$0.678010\pi$
$810$	0	0
$811$	−18.6481	−0.654823	−0.327411	−	0.944882i	$-0.606176\pi$
$811$	−18.6481	−0.654823	−0.327411	+	0.944882i	$0.606176\pi$
$812$	66.1836	2.32259
$813$	0	0
$814$	−1.02723	−0.0360045
$815$	−0.865114	−0.0303036
$816$	0	0
$817$	−28.6150	−1.00111
$818$	1.33110	0.0465407
$819$	0	0
$820$	−0.554337	−0.0193583
$821$	9.96643	0.347831	0.173915	−	0.984761i	$-0.444358\pi$
$821$	9.96643	0.347831	0.173915	+	0.984761i	$0.444358\pi$
$822$	0	0
$823$	33.7617	1.17686	0.588430	−	0.808548i	$-0.299746\pi$
$823$	33.7617	1.17686	0.588430	+	0.808548i	$0.299746\pi$
$824$	4.70345	0.163852
$825$	0	0
$826$	−1.92016	−0.0668108
$827$	−4.59052	−0.159628	−0.0798141	−	0.996810i	$-0.525433\pi$
$827$	−4.59052	−0.159628	−0.0798141	+	0.996810i	$0.525433\pi$
$828$	0	0
$829$	−2.64187	−0.0917560	−0.0458780	−	0.998947i	$-0.514609\pi$
$829$	−2.64187	−0.0917560	−0.0458780	+	0.998947i	$0.514609\pi$
$830$	0.0413284	0.00143453
$831$	0	0
$832$	−22.0874	−0.765743
$833$	−32.1754	−1.11481
$834$	0	0
$835$	−0.901816	−0.0312086
$836$	15.8660	0.548737
$837$	0	0
$838$	0.521917	0.0180293
$839$	36.6518	1.26536	0.632681	−	0.774413i	$-0.281955\pi$
$839$	36.6518	1.26536	0.632681	+	0.774413i	$0.281955\pi$
$840$	0	0
$841$	47.9146	1.65223
$842$	0.366579	0.0126332
$843$	0	0
$844$	15.8733	0.546383
$845$	−0.364425	−0.0125366
$846$	0	0
$847$	3.79648	0.130449
$848$	37.5617	1.28987
$849$	0	0
$850$	−2.39757	−0.0822358
$851$	81.9712	2.80994
$852$	0	0
$853$	23.0204	0.788204	0.394102	−	0.919067i	$-0.371056\pi$
$853$	23.0204	0.788204	0.394102	+	0.919067i	$0.371056\pi$
$854$	0.419923	0.0143695
$855$	0	0
$856$	2.51889	0.0860939
$857$	6.71504	0.229382	0.114691	−	0.993401i	$-0.463412\pi$
$857$	6.71504	0.229382	0.114691	+	0.993401i	$0.463412\pi$
$858$	0	0
$859$	−55.6212	−1.89777	−0.948886	−	0.315620i	$-0.897787\pi$
$859$	−55.6212	−1.89777	−0.948886	+	0.315620i	$0.897787\pi$
$860$	0.542327	0.0184932
$861$	0	0
$862$	2.99920	0.102153
$863$	10.1564	0.345726	0.172863	−	0.984946i	$-0.444698\pi$
$863$	10.1564	0.345726	0.172863	+	0.984946i	$0.444698\pi$
$864$	0	0
$865$	−0.711252	−0.0241833
$866$	2.48141	0.0843219
$867$	0	0
$868$	−16.2350	−0.551051
$869$	−5.57044	−0.188964
$870$	0	0
$871$	−18.7981	−0.636951
$872$	−3.57748	−0.121149
$873$	0	0
$874$	7.79243	0.263583
$875$	−2.88758	−0.0976181
$876$	0	0
$877$	−42.2582	−1.42696	−0.713480	−	0.700676i	$-0.752882\pi$
$877$	−42.2582	−1.42696	−0.713480	+	0.700676i	$0.752882\pi$
$878$	−1.07973	−0.0364390
$879$	0	0
$880$	−0.298839	−0.0100739
$881$	12.3079	0.414664	0.207332	−	0.978271i	$-0.433522\pi$
$881$	12.3079	0.414664	0.207332	+	0.978271i	$0.433522\pi$
$882$	0	0
$883$	28.4863	0.958639	0.479320	−	0.877640i	$-0.340883\pi$
$883$	28.4863	0.958639	0.479320	+	0.877640i	$0.340883\pi$
$884$	24.7224	0.831504
$885$	0	0
$886$	−2.52965	−0.0849852
$887$	−28.3316	−0.951284	−0.475642	−	0.879639i	$-0.657784\pi$
$887$	−28.3316	−0.951284	−0.475642	+	0.879639i	$0.657784\pi$
$888$	0	0
$889$	−47.4889	−1.59273
$890$	0.0545000	0.00182685
$891$	0	0
$892$	−43.1458	−1.44463
$893$	−55.6714	−1.86297
$894$	0	0
$895$	−1.29529	−0.0432969
$896$	13.2328	0.442077
$897$	0	0
$898$	1.28987	0.0430436
$899$	−18.8673	−0.629260
$900$	0	0
$901$	−41.5172	−1.38314
$902$	0.405315	0.0134955
$903$	0	0
$904$	3.27614	0.108963
$905$	1.72667	0.0573963
$906$	0	0
$907$	17.6810	0.587087	0.293544	−	0.955946i	$-0.405165\pi$
$907$	17.6810	0.587087	0.293544	+	0.955946i	$0.405165\pi$
$908$	−45.9068	−1.52347
$909$	0	0
$910$	−0.0915767	−0.00303574
$911$	58.8525	1.94987	0.974934	−	0.222493i	$-0.0714194\pi$
$911$	58.8525	1.94987	0.974934	+	0.222493i	$0.0714194\pi$
$912$	0	0
$913$	4.90970	0.162487
$914$	2.87066	0.0949531
$915$	0	0
$916$	41.8873	1.38400
$917$	−67.5168	−2.22960
$918$	0	0
$919$	38.4730	1.26911	0.634554	−	0.772878i	$-0.281184\pi$
$919$	38.4730	1.26911	0.634554	+	0.772878i	$0.281184\pi$
$920$	−0.296282	−0.00976813
$921$	0	0
$922$	2.16129	0.0711782
$923$	−20.3813	−0.670859
$924$	0	0
$925$	46.3816	1.52502
$926$	−1.72485	−0.0566822
$927$	0	0
$928$	11.5458	0.379009
$929$	34.7384	1.13973	0.569865	−	0.821738i	$-0.306995\pi$
$929$	34.7384	1.13973	0.569865	+	0.821738i	$0.306995\pi$
$930$	0	0
$931$	59.1713	1.93926
$932$	−38.8058	−1.27113
$933$	0	0
$934$	0.250572	0.00819896
$935$	0.330309	0.0108022
$936$	0	0
$937$	4.40555	0.143923	0.0719616	−	0.997407i	$-0.477074\pi$
$937$	4.40555	0.143923	0.0719616	+	0.997407i	$0.477074\pi$
$938$	2.75470	0.0899442
$939$	0	0
$940$	1.05512	0.0344141
$941$	−52.8759	−1.72370	−0.861852	−	0.507160i	$-0.830695\pi$
$941$	−52.8759	−1.72370	−0.861852	+	0.507160i	$0.830695\pi$
$942$	0	0
$943$	−32.3434	−1.05324
$944$	17.9556	0.584405
$945$	0	0
$946$	−0.396534	−0.0128924
$947$	−19.7608	−0.642138	−0.321069	−	0.947056i	$-0.604042\pi$
$947$	−19.7608	−0.642138	−0.321069	+	0.947056i	$0.604042\pi$
$948$	0	0
$949$	−14.6017	−0.473991
$950$	4.40918	0.143053
$951$	0	0
$952$	−7.26799	−0.235557
$953$	21.8127	0.706583	0.353291	−	0.935513i	$-0.385062\pi$
$953$	21.8127	0.706583	0.353291	+	0.935513i	$0.385062\pi$
$954$	0	0
$955$	−1.50947	−0.0488454
$956$	−16.0017	−0.517531
$957$	0	0
$958$	4.66023	0.150565
$959$	82.0909	2.65085
$960$	0	0
$961$	−26.3718	−0.850704
$962$	2.94360	0.0949055
$963$	0	0
$964$	31.8698	1.02646
$965$	−0.529157	−0.0170342
$966$	0	0
$967$	14.5254	0.467107	0.233553	−	0.972344i	$-0.424965\pi$
$967$	14.5254	0.467107	0.233553	+	0.972344i	$0.424965\pi$
$968$	0.441081	0.0141769
$969$	0	0
$970$	−0.0601366	−0.00193087
$971$	−21.6701	−0.695426	−0.347713	−	0.937601i	$-0.613042\pi$
$971$	−21.6701	−0.695426	−0.347713	+	0.937601i	$0.613042\pi$
$972$	0	0
$973$	2.83450	0.0908699
$974$	−1.84689	−0.0591783
$975$	0	0
$976$	−3.92674	−0.125692
$977$	−41.3933	−1.32429	−0.662145	−	0.749376i	$-0.730354\pi$
$977$	−41.3933	−1.32429	−0.662145	+	0.749376i	$0.730354\pi$
$978$	0	0
$979$	6.47445	0.206924
$980$	−1.12145	−0.0358233
$981$	0	0
$982$	1.35926	0.0433757
$983$	42.7148	1.36239	0.681196	−	0.732101i	$-0.261460\pi$
$983$	42.7148	1.36239	0.681196	+	0.732101i	$0.261460\pi$
$984$	0	0
$985$	0.235360	0.00749919
$986$	−4.21025	−0.134082
$987$	0	0
$988$	−45.4650	−1.44644
$989$	31.6426	1.00618
$990$	0	0
$991$	−1.59075	−0.0505317	−0.0252659	−	0.999681i	$-0.508043\pi$
$991$	−1.59075	−0.0505317	−0.0252659	+	0.999681i	$0.508043\pi$
$992$	−2.83220	−0.0899226
$993$	0	0
$994$	2.98670	0.0947323
$995$	0.428064	0.0135706
$996$	0	0
$997$	3.25667	0.103140	0.0515699	−	0.998669i	$-0.483578\pi$
$997$	3.25667	0.103140	0.0515699	+	0.998669i	$0.483578\pi$
$998$	−1.70778	−0.0540587
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.n.1.13	✓	25
3.2	odd	2		6039.2.a.o.1.13	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.13	✓	25	1.1	even	1	trivial
6039.2.a.o.1.13	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)

\(f(q)\)	\(=\)	\(q-0.110609 q^{2} -1.98777 q^{4} +0.0761036 q^{5} +3.79648 q^{7} +0.441081 q^{8} +O(q^{10})\)	\(q-0.110609 q^{2} -1.98777 q^{4} +0.0761036 q^{5} +3.79648 q^{7} +0.441081 q^{8} -0.00841771 q^{10} -1.00000 q^{11} +2.86557 q^{13} -0.419923 q^{14} +3.92674 q^{16} -4.34025 q^{17} +7.98182 q^{19} -0.151276 q^{20} +0.110609 q^{22} -8.82636 q^{23} -4.99421 q^{25} -0.316956 q^{26} -7.54651 q^{28} -8.77010 q^{29} +2.15132 q^{31} -1.31649 q^{32} +0.480069 q^{34} +0.288926 q^{35} -9.28709 q^{37} -0.882858 q^{38} +0.0335679 q^{40} +3.66441 q^{41} -3.58502 q^{43} +1.98777 q^{44} +0.976271 q^{46} -6.97477 q^{47} +7.41325 q^{49} +0.552402 q^{50} -5.69607 q^{52} +9.56561 q^{53} -0.0761036 q^{55} +1.67456 q^{56} +0.970048 q^{58} +4.57264 q^{59} -1.00000 q^{61} -0.237955 q^{62} -7.70787 q^{64} +0.218080 q^{65} -6.56001 q^{67} +8.62740 q^{68} -0.0319577 q^{70} -7.11249 q^{71} -5.09557 q^{73} +1.02723 q^{74} -15.8660 q^{76} -3.79648 q^{77} +5.57044 q^{79} +0.298839 q^{80} -0.405315 q^{82} -4.90970 q^{83} -0.330309 q^{85} +0.396534 q^{86} -0.441081 q^{88} -6.47445 q^{89} +10.8791 q^{91} +17.5447 q^{92} +0.771469 q^{94} +0.607445 q^{95} +7.14406 q^{97} -0.819969 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) 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

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Database

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