LMFDB - Embedded newform 6039.2.a.m.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.52609 q^{2} +4.38115 q^{4} -1.93230 q^{5} +0.113912 q^{7} -6.01501 q^{8} +O(q^{10})$	\(q-2.52609 q^{2} +4.38115 q^{4} -1.93230 q^{5} +0.113912 q^{7} -6.01501 q^{8} +4.88117 q^{10} +1.00000 q^{11} +5.88237 q^{13} -0.287752 q^{14} +6.43219 q^{16} +1.78570 q^{17} +5.41761 q^{19} -8.46569 q^{20} -2.52609 q^{22} +4.94254 q^{23} -1.26622 q^{25} -14.8594 q^{26} +0.499065 q^{28} -8.48445 q^{29} +0.794958 q^{31} -4.21828 q^{32} -4.51084 q^{34} -0.220112 q^{35} -5.89557 q^{37} -13.6854 q^{38} +11.6228 q^{40} -6.11707 q^{41} -8.60678 q^{43} +4.38115 q^{44} -12.4853 q^{46} -12.9872 q^{47} -6.98702 q^{49} +3.19860 q^{50} +25.7715 q^{52} -3.32572 q^{53} -1.93230 q^{55} -0.685182 q^{56} +21.4325 q^{58} +5.19777 q^{59} +1.00000 q^{61} -2.00814 q^{62} -2.20859 q^{64} -11.3665 q^{65} -5.05734 q^{67} +7.82341 q^{68} +0.556023 q^{70} +11.7914 q^{71} +8.77470 q^{73} +14.8928 q^{74} +23.7354 q^{76} +0.113912 q^{77} -6.98919 q^{79} -12.4289 q^{80} +15.4523 q^{82} -13.1177 q^{83} -3.45050 q^{85} +21.7415 q^{86} -6.01501 q^{88} -7.93849 q^{89} +0.670072 q^{91} +21.6540 q^{92} +32.8070 q^{94} -10.4684 q^{95} +5.08118 q^{97} +17.6499 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.52609	−1.78622	−0.893109	−	0.449840i	$-0.851481\pi$
$2$	−2.52609	−1.78622	−0.893109	+	0.449840i	$0.851481\pi$
$3$	0	0
$3$	0	0
$4$	4.38115	2.19058
$5$	−1.93230	−0.864150	−0.432075	−	0.901838i	$-0.642218\pi$
$5$	−1.93230	−0.864150	−0.432075	+	0.901838i	$0.642218\pi$
$6$	0	0
$7$	0.113912	0.0430547	0.0215273	−	0.999768i	$-0.493147\pi$
$7$	0.113912	0.0430547	0.0215273	+	0.999768i	$0.493147\pi$
$8$	−6.01501	−2.12663
$9$	0	0
$10$	4.88117	1.54356
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.88237	1.63148	0.815738	−	0.578422i	$-0.196331\pi$
$13$	5.88237	1.63148	0.815738	+	0.578422i	$0.196331\pi$
$14$	−0.287752	−0.0769050
$15$	0	0
$16$	6.43219	1.60805
$17$	1.78570	0.433095	0.216547	−	0.976272i	$-0.430520\pi$
$17$	1.78570	0.433095	0.216547	+	0.976272i	$0.430520\pi$
$18$	0	0
$19$	5.41761	1.24288	0.621442	−	0.783460i	$-0.286547\pi$
$19$	5.41761	1.24288	0.621442	+	0.783460i	$0.286547\pi$
$20$	−8.46569	−1.89299
$21$	0	0
$22$	−2.52609	−0.538565
$23$	4.94254	1.03059	0.515295	−	0.857013i	$-0.327682\pi$
$23$	4.94254	1.03059	0.515295	+	0.857013i	$0.327682\pi$
$24$	0	0
$25$	−1.26622	−0.253245
$26$	−14.8594	−2.91417
$27$	0	0
$28$	0.499065	0.0943145
$29$	−8.48445	−1.57552	−0.787761	−	0.615981i	$-0.788760\pi$
$29$	−8.48445	−1.57552	−0.787761	+	0.615981i	$0.788760\pi$
$30$	0	0
$31$	0.794958	0.142779	0.0713893	−	0.997449i	$-0.477257\pi$
$31$	0.794958	0.142779	0.0713893	+	0.997449i	$0.477257\pi$
$32$	−4.21828	−0.745694
$33$	0	0
$34$	−4.51084	−0.773602
$35$	−0.220112	−0.0372057
$36$	0	0
$37$	−5.89557	−0.969226	−0.484613	−	0.874729i	$-0.661040\pi$
$37$	−5.89557	−0.969226	−0.484613	+	0.874729i	$0.661040\pi$
$38$	−13.6854	−2.22006
$39$	0	0
$40$	11.6228	1.83773
$41$	−6.11707	−0.955326	−0.477663	−	0.878543i	$-0.658516\pi$
$41$	−6.11707	−0.955326	−0.477663	+	0.878543i	$0.658516\pi$
$42$	0	0
$43$	−8.60678	−1.31252	−0.656261	−	0.754534i	$-0.727863\pi$
$43$	−8.60678	−1.31252	−0.656261	+	0.754534i	$0.727863\pi$
$44$	4.38115	0.660483
$45$	0	0
$46$	−12.4853	−1.84086
$47$	−12.9872	−1.89438	−0.947191	−	0.320670i	$-0.896092\pi$
$47$	−12.9872	−1.89438	−0.947191	+	0.320670i	$0.896092\pi$
$48$	0	0
$49$	−6.98702	−0.998146
$50$	3.19860	0.452350
$51$	0	0
$52$	25.7715	3.57387
$53$	−3.32572	−0.456823	−0.228411	−	0.973565i	$-0.573353\pi$
$53$	−3.32572	−0.456823	−0.228411	+	0.973565i	$0.573353\pi$
$54$	0	0
$55$	−1.93230	−0.260551
$56$	−0.685182	−0.0915613
$57$	0	0
$58$	21.4325	2.81423
$59$	5.19777	0.676692	0.338346	−	0.941022i	$-0.390133\pi$
$59$	5.19777	0.676692	0.338346	+	0.941022i	$0.390133\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−2.00814	−0.255034
$63$	0	0
$64$	−2.20859	−0.276074
$65$	−11.3665	−1.40984
$66$	0	0
$67$	−5.05734	−0.617853	−0.308926	−	0.951086i	$-0.599970\pi$
$67$	−5.05734	−0.617853	−0.308926	+	0.951086i	$0.599970\pi$
$68$	7.82341	0.948727
$69$	0	0
$70$	0.556023	0.0664575
$71$	11.7914	1.39939	0.699693	−	0.714444i	$-0.253320\pi$
$71$	11.7914	1.39939	0.699693	+	0.714444i	$0.253320\pi$
$72$	0	0
$73$	8.77470	1.02700	0.513500	−	0.858089i	$-0.328348\pi$
$73$	8.77470	1.02700	0.513500	+	0.858089i	$0.328348\pi$
$74$	14.8928	1.73125
$75$	0	0
$76$	23.7354	2.72263
$77$	0.113912	0.0129815
$78$	0	0
$79$	−6.98919	−0.786345	−0.393173	−	0.919465i	$-0.628623\pi$
$79$	−6.98919	−0.786345	−0.393173	+	0.919465i	$0.628623\pi$
$80$	−12.4289	−1.38959
$81$	0	0
$82$	15.4523	1.70642
$83$	−13.1177	−1.43986	−0.719928	−	0.694049i	$-0.755825\pi$
$83$	−13.1177	−1.43986	−0.719928	+	0.694049i	$0.755825\pi$
$84$	0	0
$85$	−3.45050	−0.374259
$86$	21.7415	2.34445
$87$	0	0
$88$	−6.01501	−0.641203
$89$	−7.93849	−0.841478	−0.420739	−	0.907182i	$-0.638229\pi$
$89$	−7.93849	−0.841478	−0.420739	+	0.907182i	$0.638229\pi$
$90$	0	0
$91$	0.670072	0.0702426
$92$	21.6540	2.25759
$93$	0	0
$94$	32.8070	3.38378
$95$	−10.4684	−1.07404
$96$	0	0
$97$	5.08118	0.515916	0.257958	−	0.966156i	$-0.416950\pi$
$97$	5.08118	0.515916	0.257958	+	0.966156i	$0.416950\pi$
$98$	17.6499	1.78291
$99$	0	0
$100$	−5.54752	−0.554752
$101$	16.4514	1.63697	0.818486	−	0.574526i	$-0.194814\pi$
$101$	16.4514	1.63697	0.818486	+	0.574526i	$0.194814\pi$
$102$	0	0
$103$	−1.89389	−0.186610	−0.0933051	−	0.995638i	$-0.529743\pi$
$103$	−1.89389	−0.186610	−0.0933051	+	0.995638i	$0.529743\pi$
$104$	−35.3825	−3.46954
$105$	0	0
$106$	8.40108	0.815985
$107$	5.25256	0.507785	0.253892	−	0.967232i	$-0.418289\pi$
$107$	5.25256	0.507785	0.253892	+	0.967232i	$0.418289\pi$
$108$	0	0
$109$	6.14259	0.588353	0.294177	−	0.955751i	$-0.404955\pi$
$109$	6.14259	0.588353	0.294177	+	0.955751i	$0.404955\pi$
$110$	4.88117	0.465401
$111$	0	0
$112$	0.732703	0.0692339
$113$	0.570143	0.0536345	0.0268173	−	0.999640i	$-0.491463\pi$
$113$	0.570143	0.0536345	0.0268173	+	0.999640i	$0.491463\pi$
$114$	0	0
$115$	−9.55045	−0.890584
$116$	−37.1716	−3.45130
$117$	0	0
$118$	−13.1301	−1.20872
$119$	0.203412	0.0186468
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.52609	−0.228702
$123$	0	0
$124$	3.48283	0.312768
$125$	12.1082	1.08299
$126$	0	0
$127$	4.53333	0.402268	0.201134	−	0.979564i	$-0.435537\pi$
$127$	4.53333	0.402268	0.201134	+	0.979564i	$0.435537\pi$
$128$	14.0157	1.23882
$129$	0	0
$130$	28.7128	2.51828
$131$	−14.7245	−1.28649	−0.643245	−	0.765661i	$-0.722412\pi$
$131$	−14.7245	−1.28649	−0.643245	+	0.765661i	$0.722412\pi$
$132$	0	0
$133$	0.617130	0.0535120
$134$	12.7753	1.10362
$135$	0	0
$136$	−10.7410	−0.921032
$137$	12.3369	1.05402	0.527008	−	0.849860i	$-0.323314\pi$
$137$	12.3369	1.05402	0.527008	+	0.849860i	$0.323314\pi$
$138$	0	0
$139$	1.82701	0.154965	0.0774823	−	0.996994i	$-0.475312\pi$
$139$	1.82701	0.154965	0.0774823	+	0.996994i	$0.475312\pi$
$140$	−0.964343	−0.0815019
$141$	0	0
$142$	−29.7863	−2.49961
$143$	5.88237	0.491908
$144$	0	0
$145$	16.3945	1.36149
$146$	−22.1657	−1.83445
$147$	0	0
$148$	−25.8294	−2.12316
$149$	−10.7611	−0.881584	−0.440792	−	0.897609i	$-0.645302\pi$
$149$	−10.7611	−0.881584	−0.440792	+	0.897609i	$0.645302\pi$
$150$	0	0
$151$	−14.6848	−1.19503	−0.597514	−	0.801858i	$-0.703845\pi$
$151$	−14.6848	−1.19503	−0.597514	+	0.801858i	$0.703845\pi$
$152$	−32.5870	−2.64315
$153$	0	0
$154$	−0.287752	−0.0231877
$155$	−1.53610	−0.123382
$156$	0	0
$157$	16.8329	1.34341	0.671705	−	0.740819i	$-0.265562\pi$
$157$	16.8329	1.34341	0.671705	+	0.740819i	$0.265562\pi$
$158$	17.6554	1.40458
$159$	0	0
$160$	8.15098	0.644392
$161$	0.563014	0.0443717
$162$	0	0
$163$	−6.93439	−0.543143	−0.271572	−	0.962418i	$-0.587543\pi$
$163$	−6.93439	−0.543143	−0.271572	+	0.962418i	$0.587543\pi$
$164$	−26.7998	−2.09271
$165$	0	0
$166$	33.1366	2.57190
$167$	−1.49920	−0.116012	−0.0580059	−	0.998316i	$-0.518474\pi$
$167$	−1.49920	−0.116012	−0.0580059	+	0.998316i	$0.518474\pi$
$168$	0	0
$169$	21.6022	1.66171
$170$	8.71628	0.668508
$171$	0	0
$172$	−37.7076	−2.87518
$173$	0.240929	0.0183175	0.00915873	−	0.999958i	$-0.497085\pi$
$173$	0.240929	0.0183175	0.00915873	+	0.999958i	$0.497085\pi$
$174$	0	0
$175$	−0.144238	−0.0109034
$176$	6.43219	0.484844
$177$	0	0
$178$	20.0534	1.50306
$179$	9.36366	0.699873	0.349937	−	0.936773i	$-0.386203\pi$
$179$	9.36366	0.699873	0.349937	+	0.936773i	$0.386203\pi$
$180$	0	0
$181$	5.25546	0.390635	0.195317	−	0.980740i	$-0.437426\pi$
$181$	5.25546	0.390635	0.195317	+	0.980740i	$0.437426\pi$
$182$	−1.69266	−0.125469
$183$	0	0
$184$	−29.7294	−2.19168
$185$	11.3920	0.837556
$186$	0	0
$187$	1.78570	0.130583
$188$	−56.8990	−4.14979
$189$	0	0
$190$	26.4443	1.91847
$191$	−1.29738	−0.0938749	−0.0469375	−	0.998898i	$-0.514946\pi$
$191$	−1.29738	−0.0938749	−0.0469375	+	0.998898i	$0.514946\pi$
$192$	0	0
$193$	−20.5045	−1.47594	−0.737972	−	0.674832i	$-0.764216\pi$
$193$	−20.5045	−1.47594	−0.737972	+	0.674832i	$0.764216\pi$
$194$	−12.8355	−0.921538
$195$	0	0
$196$	−30.6112	−2.18652
$197$	−16.8767	−1.20242	−0.601209	−	0.799092i	$-0.705314\pi$
$197$	−16.8767	−1.20242	−0.601209	+	0.799092i	$0.705314\pi$
$198$	0	0
$199$	−23.7790	−1.68565	−0.842824	−	0.538190i	$-0.819108\pi$
$199$	−23.7790	−1.68565	−0.842824	+	0.538190i	$0.819108\pi$
$200$	7.61635	0.538557
$201$	0	0
$202$	−41.5577	−2.92399
$203$	−0.966480	−0.0678336
$204$	0	0
$205$	11.8200	0.825545
$206$	4.78413	0.333326
$207$	0	0
$208$	37.8365	2.62349
$209$	5.41761	0.374744
$210$	0	0
$211$	2.20932	0.152096	0.0760480	−	0.997104i	$-0.475770\pi$
$211$	2.20932	0.152096	0.0760480	+	0.997104i	$0.475770\pi$
$212$	−14.5705	−1.00071
$213$	0	0
$214$	−13.2685	−0.907014
$215$	16.6309	1.13422
$216$	0	0
$217$	0.0905552	0.00614729
$218$	−15.5168	−1.05093
$219$	0	0
$220$	−8.46569	−0.570757
$221$	10.5041	0.706584
$222$	0	0
$223$	−11.3955	−0.763098	−0.381549	−	0.924349i	$-0.624609\pi$
$223$	−11.3955	−0.763098	−0.381549	+	0.924349i	$0.624609\pi$
$224$	−0.480513	−0.0321056
$225$	0	0
$226$	−1.44024	−0.0958030
$227$	−11.7724	−0.781365	−0.390682	−	0.920526i	$-0.627761\pi$
$227$	−11.7724	−0.781365	−0.390682	+	0.920526i	$0.627761\pi$
$228$	0	0
$229$	−25.5699	−1.68971	−0.844853	−	0.534998i	$-0.820312\pi$
$229$	−25.5699	−1.68971	−0.844853	+	0.534998i	$0.820312\pi$
$230$	24.1253	1.59078
$231$	0	0
$232$	51.0341	3.35055
$233$	20.1380	1.31928	0.659641	−	0.751581i	$-0.270708\pi$
$233$	20.1380	1.31928	0.659641	+	0.751581i	$0.270708\pi$
$234$	0	0
$235$	25.0952	1.63703
$236$	22.7722	1.48235
$237$	0	0
$238$	−0.513838	−0.0333072
$239$	3.94973	0.255487	0.127744	−	0.991807i	$-0.459227\pi$
$239$	3.94973	0.255487	0.127744	+	0.991807i	$0.459227\pi$
$240$	0	0
$241$	−18.8084	−1.21156	−0.605778	−	0.795634i	$-0.707138\pi$
$241$	−18.8084	−1.21156	−0.605778	+	0.795634i	$0.707138\pi$
$242$	−2.52609	−0.162383
$243$	0	0
$244$	4.38115	0.280475
$245$	13.5010	0.862548
$246$	0	0
$247$	31.8684	2.02774
$248$	−4.78168	−0.303637
$249$	0	0
$250$	−30.5865	−1.93446
$251$	−7.13194	−0.450164	−0.225082	−	0.974340i	$-0.572265\pi$
$251$	−7.13194	−0.450164	−0.225082	+	0.974340i	$0.572265\pi$
$252$	0	0
$253$	4.94254	0.310735
$254$	−11.4516	−0.718538
$255$	0	0
$256$	−30.9877	−1.93673
$257$	0.347296	0.0216637	0.0108319	−	0.999941i	$-0.496552\pi$
$257$	0.347296	0.0216637	0.0108319	+	0.999941i	$0.496552\pi$
$258$	0	0
$259$	−0.671576	−0.0417297
$260$	−49.7983	−3.08836
$261$	0	0
$262$	37.1956	2.29795
$263$	−23.5097	−1.44967	−0.724835	−	0.688922i	$-0.758084\pi$
$263$	−23.5097	−1.44967	−0.724835	+	0.688922i	$0.758084\pi$
$264$	0	0
$265$	6.42628	0.394764
$266$	−1.55893	−0.0955841
$267$	0	0
$268$	−22.1570	−1.35345
$269$	−14.6293	−0.891964	−0.445982	−	0.895042i	$-0.647145\pi$
$269$	−14.6293	−0.891964	−0.445982	+	0.895042i	$0.647145\pi$
$270$	0	0
$271$	5.11181	0.310520	0.155260	−	0.987874i	$-0.450378\pi$
$271$	5.11181	0.310520	0.155260	+	0.987874i	$0.450378\pi$
$272$	11.4859	0.696437
$273$	0	0
$274$	−31.1643	−1.88270
$275$	−1.26622	−0.0763561
$276$	0	0
$277$	18.5910	1.11702	0.558512	−	0.829497i	$-0.311373\pi$
$277$	18.5910	1.11702	0.558512	+	0.829497i	$0.311373\pi$
$278$	−4.61519	−0.276801
$279$	0	0
$280$	1.32398	0.0791227
$281$	−17.1458	−1.02283	−0.511417	−	0.859332i	$-0.670879\pi$
$281$	−17.1458	−1.02283	−0.511417	+	0.859332i	$0.670879\pi$
$282$	0	0
$283$	−28.5748	−1.69859	−0.849297	−	0.527915i	$-0.822974\pi$
$283$	−28.5748	−1.69859	−0.849297	+	0.527915i	$0.822974\pi$
$284$	51.6600	3.06546
$285$	0	0
$286$	−14.8594	−0.878656
$287$	−0.696808	−0.0411312
$288$	0	0
$289$	−13.8113	−0.812429
$290$	−41.4140	−2.43191
$291$	0	0
$292$	38.4433	2.24972
$293$	−18.0620	−1.05520	−0.527598	−	0.849494i	$-0.676907\pi$
$293$	−18.0620	−1.05520	−0.527598	+	0.849494i	$0.676907\pi$
$294$	0	0
$295$	−10.0436	−0.584764
$296$	35.4619	2.06118
$297$	0	0
$298$	27.1836	1.57470
$299$	29.0738	1.68138
$300$	0	0
$301$	−0.980415	−0.0565102
$302$	37.0951	2.13458
$303$	0	0
$304$	34.8471	1.99862
$305$	−1.93230	−0.110643
$306$	0	0
$307$	22.2069	1.26742	0.633708	−	0.773572i	$-0.281532\pi$
$307$	22.2069	1.26742	0.633708	+	0.773572i	$0.281532\pi$
$308$	0.499065	0.0284369
$309$	0	0
$310$	3.88032	0.220388
$311$	14.9639	0.848527	0.424264	−	0.905539i	$-0.360533\pi$
$311$	14.9639	0.848527	0.424264	+	0.905539i	$0.360533\pi$
$312$	0	0
$313$	21.5801	1.21978	0.609888	−	0.792487i	$-0.291214\pi$
$313$	21.5801	1.21978	0.609888	+	0.792487i	$0.291214\pi$
$314$	−42.5214	−2.39962
$315$	0	0
$316$	−30.6207	−1.72255
$317$	19.3449	1.08652	0.543259	−	0.839565i	$-0.317190\pi$
$317$	19.3449	1.08652	0.543259	+	0.839565i	$0.317190\pi$
$318$	0	0
$319$	−8.48445	−0.475038
$320$	4.26766	0.238569
$321$	0	0
$322$	−1.42223	−0.0792575
$323$	9.67420	0.538287
$324$	0	0
$325$	−7.44839	−0.413162
$326$	17.5169	0.970173
$327$	0	0
$328$	36.7943	2.03162
$329$	−1.47940	−0.0815620
$330$	0	0
$331$	−11.0474	−0.607221	−0.303611	−	0.952796i	$-0.598192\pi$
$331$	−11.0474	−0.607221	−0.303611	+	0.952796i	$0.598192\pi$
$332$	−57.4707	−3.15411
$333$	0	0
$334$	3.78713	0.207222
$335$	9.77230	0.533918
$336$	0	0
$337$	13.7114	0.746907	0.373454	−	0.927649i	$-0.378174\pi$
$337$	13.7114	0.746907	0.373454	+	0.927649i	$0.378174\pi$
$338$	−54.5693	−2.96818
$339$	0	0
$340$	−15.1172	−0.819843
$341$	0.794958	0.0430494
$342$	0	0
$343$	−1.59329	−0.0860295
$344$	51.7699	2.79125
$345$	0	0
$346$	−0.608608	−0.0327190
$347$	27.5074	1.47667	0.738337	−	0.674432i	$-0.235611\pi$
$347$	27.5074	1.47667	0.738337	+	0.674432i	$0.235611\pi$
$348$	0	0
$349$	−20.2655	−1.08479	−0.542395	−	0.840124i	$-0.682482\pi$
$349$	−20.2655	−1.08479	−0.542395	+	0.840124i	$0.682482\pi$
$350$	0.364359	0.0194758
$351$	0	0
$352$	−4.21828	−0.224835
$353$	28.4966	1.51672	0.758360	−	0.651836i	$-0.226001\pi$
$353$	28.4966	1.51672	0.758360	+	0.651836i	$0.226001\pi$
$354$	0	0
$355$	−22.7846	−1.20928
$356$	−34.7797	−1.84332
$357$	0	0
$358$	−23.6535	−1.25013
$359$	−19.0140	−1.00352	−0.501761	−	0.865006i	$-0.667314\pi$
$359$	−19.0140	−1.00352	−0.501761	+	0.865006i	$0.667314\pi$
$360$	0	0
$361$	10.3505	0.544762
$362$	−13.2758	−0.697759
$363$	0	0
$364$	2.93569	0.153872
$365$	−16.9553	−0.887483
$366$	0	0
$367$	−13.4763	−0.703459	−0.351729	−	0.936102i	$-0.614406\pi$
$367$	−13.4763	−0.703459	−0.351729	+	0.936102i	$0.614406\pi$
$368$	31.7913	1.65724
$369$	0	0
$370$	−28.7773	−1.49606
$371$	−0.378839	−0.0196684
$372$	0	0
$373$	−19.8448	−1.02752	−0.513761	−	0.857933i	$-0.671748\pi$
$373$	−19.8448	−1.02752	−0.513761	+	0.857933i	$0.671748\pi$
$374$	−4.51084	−0.233250
$375$	0	0
$376$	78.1184	4.02865
$377$	−49.9086	−2.57043
$378$	0	0
$379$	16.8845	0.867300	0.433650	−	0.901081i	$-0.357225\pi$
$379$	16.8845	0.867300	0.433650	+	0.901081i	$0.357225\pi$
$380$	−45.8638	−2.35276
$381$	0	0
$382$	3.27730	0.167681
$383$	34.0115	1.73790	0.868952	−	0.494896i	$-0.164794\pi$
$383$	34.0115	1.73790	0.868952	+	0.494896i	$0.164794\pi$
$384$	0	0
$385$	−0.220112	−0.0112179
$386$	51.7962	2.63636
$387$	0	0
$388$	22.2614	1.13015
$389$	−2.77557	−0.140727	−0.0703636	−	0.997521i	$-0.522416\pi$
$389$	−2.77557	−0.140727	−0.0703636	+	0.997521i	$0.522416\pi$
$390$	0	0
$391$	8.82587	0.446343
$392$	42.0270	2.12269
$393$	0	0
$394$	42.6322	2.14778
$395$	13.5052	0.679520
$396$	0	0
$397$	−25.5841	−1.28403	−0.642014	−	0.766693i	$-0.721901\pi$
$397$	−25.5841	−1.28403	−0.642014	+	0.766693i	$0.721901\pi$
$398$	60.0680	3.01093
$399$	0	0
$400$	−8.14459	−0.407229
$401$	25.5458	1.27569	0.637847	−	0.770163i	$-0.279825\pi$
$401$	25.5458	1.27569	0.637847	+	0.770163i	$0.279825\pi$
$402$	0	0
$403$	4.67623	0.232940
$404$	72.0759	3.58591
$405$	0	0
$406$	2.44142	0.121166
$407$	−5.89557	−0.292233
$408$	0	0
$409$	35.7912	1.76976	0.884880	−	0.465819i	$-0.154240\pi$
$409$	35.7912	1.76976	0.884880	+	0.465819i	$0.154240\pi$
$410$	−29.8585	−1.47460
$411$	0	0
$412$	−8.29740	−0.408784
$413$	0.592088	0.0291348
$414$	0	0
$415$	25.3473	1.24425
$416$	−24.8135	−1.21658
$417$	0	0
$418$	−13.6854	−0.669374
$419$	−33.6250	−1.64269	−0.821344	−	0.570434i	$-0.806775\pi$
$419$	−33.6250	−1.64269	−0.821344	+	0.570434i	$0.806775\pi$
$420$	0	0
$421$	20.1060	0.979908	0.489954	−	0.871748i	$-0.337014\pi$
$421$	20.1060	0.979908	0.489954	+	0.871748i	$0.337014\pi$
$422$	−5.58095	−0.271677
$423$	0	0
$424$	20.0043	0.971493
$425$	−2.26109	−0.109679
$426$	0	0
$427$	0.113912	0.00551258
$428$	23.0123	1.11234
$429$	0	0
$430$	−42.0111	−2.02596
$431$	5.81599	0.280146	0.140073	−	0.990141i	$-0.455266\pi$
$431$	5.81599	0.280146	0.140073	+	0.990141i	$0.455266\pi$
$432$	0	0
$433$	−15.9953	−0.768685	−0.384343	−	0.923191i	$-0.625572\pi$
$433$	−15.9953	−0.768685	−0.384343	+	0.923191i	$0.625572\pi$
$434$	−0.228751	−0.0109804
$435$	0	0
$436$	26.9116	1.28883
$437$	26.7767	1.28090
$438$	0	0
$439$	−19.1298	−0.913018	−0.456509	−	0.889719i	$-0.650900\pi$
$439$	−19.1298	−0.913018	−0.456509	+	0.889719i	$0.650900\pi$
$440$	11.6228	0.554095
$441$	0	0
$442$	−26.5344	−1.26211
$443$	39.2547	1.86505	0.932523	−	0.361110i	$-0.117602\pi$
$443$	39.2547	1.86505	0.932523	+	0.361110i	$0.117602\pi$
$444$	0	0
$445$	15.3395	0.727164
$446$	28.7861	1.36306
$447$	0	0
$448$	−0.251585	−0.0118863
$449$	−32.6933	−1.54289	−0.771446	−	0.636295i	$-0.780466\pi$
$449$	−32.6933	−1.54289	−0.771446	+	0.636295i	$0.780466\pi$
$450$	0	0
$451$	−6.11707	−0.288042
$452$	2.49788	0.117491
$453$	0	0
$454$	29.7383	1.39569
$455$	−1.29478	−0.0607002
$456$	0	0
$457$	−20.6606	−0.966461	−0.483230	−	0.875493i	$-0.660537\pi$
$457$	−20.6606	−0.966461	−0.483230	+	0.875493i	$0.660537\pi$
$458$	64.5920	3.01818
$459$	0	0
$460$	−41.8420	−1.95089
$461$	−28.9689	−1.34922	−0.674608	−	0.738176i	$-0.735688\pi$
$461$	−28.9689	−1.34922	−0.674608	+	0.738176i	$0.735688\pi$
$462$	0	0
$463$	13.0665	0.607251	0.303625	−	0.952792i	$-0.401803\pi$
$463$	13.0665	0.607251	0.303625	+	0.952792i	$0.401803\pi$
$464$	−54.5736	−2.53351
$465$	0	0
$466$	−50.8704	−2.35653
$467$	11.7648	0.544409	0.272205	−	0.962239i	$-0.412247\pi$
$467$	11.7648	0.544409	0.272205	+	0.962239i	$0.412247\pi$
$468$	0	0
$469$	−0.576092	−0.0266014
$470$	−63.3928	−2.92409
$471$	0	0
$472$	−31.2647	−1.43907
$473$	−8.60678	−0.395740
$474$	0	0
$475$	−6.85990	−0.314754
$476$	0.891179	0.0408471
$477$	0	0
$478$	−9.97740	−0.456356
$479$	−1.03072	−0.0470946	−0.0235473	−	0.999723i	$-0.507496\pi$
$479$	−1.03072	−0.0470946	−0.0235473	+	0.999723i	$0.507496\pi$
$480$	0	0
$481$	−34.6799	−1.58127
$482$	47.5118	2.16410
$483$	0	0
$484$	4.38115	0.199143
$485$	−9.81835	−0.445828
$486$	0	0
$487$	34.1680	1.54830	0.774151	−	0.633001i	$-0.218177\pi$
$487$	34.1680	1.54830	0.774151	+	0.633001i	$0.218177\pi$
$488$	−6.01501	−0.272287
$489$	0	0
$490$	−34.1048	−1.54070
$491$	−4.68285	−0.211334	−0.105667	−	0.994402i	$-0.533698\pi$
$491$	−4.68285	−0.211334	−0.105667	+	0.994402i	$0.533698\pi$
$492$	0	0
$493$	−15.1506	−0.682351
$494$	−80.5025	−3.62198
$495$	0	0
$496$	5.11332	0.229595
$497$	1.34318	0.0602501
$498$	0	0
$499$	−19.1658	−0.857981	−0.428990	−	0.903309i	$-0.641131\pi$
$499$	−19.1658	−0.857981	−0.428990	+	0.903309i	$0.641131\pi$
$500$	53.0479	2.37238
$501$	0	0
$502$	18.0160	0.804092
$503$	−43.3358	−1.93225	−0.966124	−	0.258080i	$-0.916910\pi$
$503$	−43.3358	−1.93225	−0.966124	+	0.258080i	$0.916910\pi$
$504$	0	0
$505$	−31.7890	−1.41459
$506$	−12.4853	−0.555040
$507$	0	0
$508$	19.8612	0.881198
$509$	−19.5101	−0.864771	−0.432385	−	0.901689i	$-0.642328\pi$
$509$	−19.5101	−0.864771	−0.432385	+	0.901689i	$0.642328\pi$
$510$	0	0
$511$	0.999543	0.0442172
$512$	50.2466	2.22061
$513$	0	0
$514$	−0.877302	−0.0386961
$515$	3.65955	0.161259
$516$	0	0
$517$	−12.9872	−0.571178
$518$	1.69646	0.0745383
$519$	0	0
$520$	68.3696	2.99820
$521$	−0.0566823	−0.00248330	−0.00124165	−	0.999999i	$-0.500395\pi$
$521$	−0.0566823	−0.00248330	−0.00124165	+	0.999999i	$0.500395\pi$
$522$	0	0
$523$	−12.8506	−0.561916	−0.280958	−	0.959720i	$-0.590652\pi$
$523$	−12.8506	−0.561916	−0.280958	+	0.959720i	$0.590652\pi$
$524$	−64.5105	−2.81815
$525$	0	0
$526$	59.3877	2.58943
$527$	1.41955	0.0618367
$528$	0	0
$529$	1.42866	0.0621156
$530$	−16.2334	−0.705134
$531$	0	0
$532$	2.70374	0.117222
$533$	−35.9829	−1.55859
$534$	0	0
$535$	−10.1495	−0.438802
$536$	30.4200	1.31394
$537$	0	0
$538$	36.9550	1.59324
$539$	−6.98702	−0.300952
$540$	0	0
$541$	2.68196	0.115307	0.0576533	−	0.998337i	$-0.481638\pi$
$541$	2.68196	0.115307	0.0576533	+	0.998337i	$0.481638\pi$
$542$	−12.9129	−0.554657
$543$	0	0
$544$	−7.53257	−0.322956
$545$	−11.8693	−0.508426
$546$	0	0
$547$	−32.5247	−1.39066	−0.695328	−	0.718693i	$-0.744741\pi$
$547$	−32.5247	−1.39066	−0.695328	+	0.718693i	$0.744741\pi$
$548$	54.0500	2.30890
$549$	0	0
$550$	3.19860	0.136389
$551$	−45.9654	−1.95819
$552$	0	0
$553$	−0.796152	−0.0338558
$554$	−46.9625	−1.99525
$555$	0	0
$556$	8.00439	0.339462
$557$	6.52321	0.276397	0.138199	−	0.990405i	$-0.455869\pi$
$557$	6.52321	0.276397	0.138199	+	0.990405i	$0.455869\pi$
$558$	0	0
$559$	−50.6282	−2.14135
$560$	−1.41580	−0.0598285
$561$	0	0
$562$	43.3120	1.82701
$563$	−10.5148	−0.443145	−0.221572	−	0.975144i	$-0.571119\pi$
$563$	−10.5148	−0.443145	−0.221572	+	0.975144i	$0.571119\pi$
$564$	0	0
$565$	−1.10169	−0.0463483
$566$	72.1826	3.03406
$567$	0	0
$568$	−70.9256	−2.97597
$569$	11.9224	0.499813	0.249907	−	0.968270i	$-0.419600\pi$
$569$	11.9224	0.499813	0.249907	+	0.968270i	$0.419600\pi$
$570$	0	0
$571$	6.87414	0.287674	0.143837	−	0.989601i	$-0.454056\pi$
$571$	6.87414	0.287674	0.143837	+	0.989601i	$0.454056\pi$
$572$	25.7715	1.07756
$573$	0	0
$574$	1.76020	0.0734694
$575$	−6.25835	−0.260991
$576$	0	0
$577$	−21.3629	−0.889348	−0.444674	−	0.895693i	$-0.646680\pi$
$577$	−21.3629	−0.889348	−0.444674	+	0.895693i	$0.646680\pi$
$578$	34.8886	1.45118
$579$	0	0
$580$	71.8267	2.98244
$581$	−1.49426	−0.0619925
$582$	0	0
$583$	−3.32572	−0.137737
$584$	−52.7799	−2.18405
$585$	0	0
$586$	45.6264	1.88481
$587$	45.4940	1.87774	0.938870	−	0.344271i	$-0.111874\pi$
$587$	45.4940	1.87774	0.938870	+	0.344271i	$0.111874\pi$
$588$	0	0
$589$	4.30677	0.177457
$590$	25.3712	1.04452
$591$	0	0
$592$	−37.9214	−1.55856
$593$	−20.3900	−0.837318	−0.418659	−	0.908144i	$-0.637500\pi$
$593$	−20.3900	−0.837318	−0.418659	+	0.908144i	$0.637500\pi$
$594$	0	0
$595$	−0.393053	−0.0161136
$596$	−47.1460	−1.93118
$597$	0	0
$598$	−73.4432	−3.00331
$599$	19.5418	0.798458	0.399229	−	0.916851i	$-0.369278\pi$
$599$	19.5418	0.798458	0.399229	+	0.916851i	$0.369278\pi$
$600$	0	0
$601$	−30.7158	−1.25292	−0.626461	−	0.779453i	$-0.715497\pi$
$601$	−30.7158	−1.25292	−0.626461	+	0.779453i	$0.715497\pi$
$602$	2.47662	0.100939
$603$	0	0
$604$	−64.3361	−2.61780
$605$	−1.93230	−0.0785591
$606$	0	0
$607$	40.2946	1.63551	0.817754	−	0.575568i	$-0.195219\pi$
$607$	40.2946	1.63551	0.817754	+	0.575568i	$0.195219\pi$
$608$	−22.8530	−0.926812
$609$	0	0
$610$	4.88117	0.197633
$611$	−76.3956	−3.09064
$612$	0	0
$613$	33.0708	1.33572	0.667858	−	0.744289i	$-0.267211\pi$
$613$	33.0708	1.33572	0.667858	+	0.744289i	$0.267211\pi$
$614$	−56.0968	−2.26388
$615$	0	0
$616$	−0.685182	−0.0276068
$617$	21.3239	0.858468	0.429234	−	0.903193i	$-0.358783\pi$
$617$	21.3239	0.858468	0.429234	+	0.903193i	$0.358783\pi$
$618$	0	0
$619$	−40.3497	−1.62179	−0.810896	−	0.585190i	$-0.801020\pi$
$619$	−40.3497	−1.62179	−0.810896	+	0.585190i	$0.801020\pi$
$620$	−6.72987	−0.270278
$621$	0	0
$622$	−37.8003	−1.51566
$623$	−0.904289	−0.0362296
$624$	0	0
$625$	−17.0656	−0.682622
$626$	−54.5132	−2.17879
$627$	0	0
$628$	73.7474	2.94284
$629$	−10.5277	−0.419767
$630$	0	0
$631$	−39.1524	−1.55863	−0.779317	−	0.626630i	$-0.784434\pi$
$631$	−39.1524	−1.55863	−0.779317	+	0.626630i	$0.784434\pi$
$632$	42.0401	1.67226
$633$	0	0
$634$	−48.8670	−1.94076
$635$	−8.75974	−0.347620
$636$	0	0
$637$	−41.1002	−1.62845
$638$	21.4325	0.848521
$639$	0	0
$640$	−27.0825	−1.07053
$641$	−16.0781	−0.635045	−0.317523	−	0.948251i	$-0.602851\pi$
$641$	−16.0781	−0.635045	−0.317523	+	0.948251i	$0.602851\pi$
$642$	0	0
$643$	−13.4326	−0.529730	−0.264865	−	0.964285i	$-0.585327\pi$
$643$	−13.4326	−0.529730	−0.264865	+	0.964285i	$0.585327\pi$
$644$	2.46665	0.0971996
$645$	0	0
$646$	−24.4379	−0.961498
$647$	44.2080	1.73799	0.868997	−	0.494817i	$-0.164765\pi$
$647$	44.2080	1.73799	0.868997	+	0.494817i	$0.164765\pi$
$648$	0	0
$649$	5.19777	0.204030
$650$	18.8153	0.737998
$651$	0	0
$652$	−30.3806	−1.18980
$653$	50.1092	1.96092	0.980462	−	0.196707i	$-0.0630248\pi$
$653$	50.1092	1.96092	0.980462	+	0.196707i	$0.0630248\pi$
$654$	0	0
$655$	28.4522	1.11172
$656$	−39.3462	−1.53621
$657$	0	0
$658$	3.73710	0.145688
$659$	−8.71389	−0.339445	−0.169722	−	0.985492i	$-0.554287\pi$
$659$	−8.71389	−0.339445	−0.169722	+	0.985492i	$0.554287\pi$
$660$	0	0
$661$	12.6647	0.492598	0.246299	−	0.969194i	$-0.420785\pi$
$661$	12.6647	0.492598	0.246299	+	0.969194i	$0.420785\pi$
$662$	27.9068	1.08463
$663$	0	0
$664$	78.9032	3.06204
$665$	−1.19248	−0.0462424
$666$	0	0
$667$	−41.9347	−1.62372
$668$	−6.56823	−0.254133
$669$	0	0
$670$	−24.6857	−0.953693
$671$	1.00000	0.0386046
$672$	0	0
$673$	−46.7415	−1.80175	−0.900876	−	0.434077i	$-0.857075\pi$
$673$	−46.7415	−1.80175	−0.900876	+	0.434077i	$0.857075\pi$
$674$	−34.6363	−1.33414
$675$	0	0
$676$	94.6427	3.64010
$677$	−29.8325	−1.14656	−0.573278	−	0.819361i	$-0.694328\pi$
$677$	−29.8325	−1.14656	−0.573278	+	0.819361i	$0.694328\pi$
$678$	0	0
$679$	0.578807	0.0222126
$680$	20.7548	0.795910
$681$	0	0
$682$	−2.00814	−0.0768956
$683$	−38.2200	−1.46245	−0.731224	−	0.682137i	$-0.761051\pi$
$683$	−38.2200	−1.46245	−0.731224	+	0.682137i	$0.761051\pi$
$684$	0	0
$685$	−23.8386	−0.910828
$686$	4.02480	0.153667
$687$	0	0
$688$	−55.3604	−2.11060
$689$	−19.5631	−0.745295
$690$	0	0
$691$	20.1391	0.766126	0.383063	−	0.923722i	$-0.374869\pi$
$691$	20.1391	0.766126	0.383063	+	0.923722i	$0.374869\pi$
$692$	1.05554	0.0401258
$693$	0	0
$694$	−69.4863	−2.63766
$695$	−3.53032	−0.133913
$696$	0	0
$697$	−10.9232	−0.413747
$698$	51.1927	1.93767
$699$	0	0
$700$	−0.631928	−0.0238846
$701$	−10.0684	−0.380279	−0.190140	−	0.981757i	$-0.560894\pi$
$701$	−10.0684	−0.380279	−0.190140	+	0.981757i	$0.560894\pi$
$702$	0	0
$703$	−31.9399	−1.20464
$704$	−2.20859	−0.0832394
$705$	0	0
$706$	−71.9850	−2.70919
$707$	1.87401	0.0704793
$708$	0	0
$709$	33.8351	1.27070	0.635352	−	0.772223i	$-0.280855\pi$
$709$	33.8351	1.27070	0.635352	+	0.772223i	$0.280855\pi$
$710$	57.5559	2.16004
$711$	0	0
$712$	47.7501	1.78951
$713$	3.92911	0.147146
$714$	0	0
$715$	−11.3665	−0.425083
$716$	41.0236	1.53313
$717$	0	0
$718$	48.0312	1.79251
$719$	−2.59000	−0.0965907	−0.0482954	−	0.998833i	$-0.515379\pi$
$719$	−2.59000	−0.0965907	−0.0482954	+	0.998833i	$0.515379\pi$
$720$	0	0
$721$	−0.215736	−0.00803444
$722$	−26.1463	−0.973065
$723$	0	0
$724$	23.0249	0.855715
$725$	10.7432	0.398993
$726$	0	0
$727$	35.8640	1.33012	0.665061	−	0.746789i	$-0.268406\pi$
$727$	35.8640	1.33012	0.665061	+	0.746789i	$0.268406\pi$
$728$	−4.03049	−0.149380
$729$	0	0
$730$	42.8308	1.58524
$731$	−15.3691	−0.568446
$732$	0	0
$733$	−26.9398	−0.995044	−0.497522	−	0.867451i	$-0.665757\pi$
$733$	−26.9398	−0.995044	−0.497522	+	0.867451i	$0.665757\pi$
$734$	34.0425	1.25653
$735$	0	0
$736$	−20.8490	−0.768505
$737$	−5.05734	−0.186290
$738$	0	0
$739$	4.20043	0.154515	0.0772577	−	0.997011i	$-0.475384\pi$
$739$	4.20043	0.154515	0.0772577	+	0.997011i	$0.475384\pi$
$740$	49.9101	1.83473
$741$	0	0
$742$	0.956984	0.0351320
$743$	20.8762	0.765874	0.382937	−	0.923774i	$-0.374913\pi$
$743$	20.8762	0.765874	0.382937	+	0.923774i	$0.374913\pi$
$744$	0	0
$745$	20.7937	0.761821
$746$	50.1297	1.83538
$747$	0	0
$748$	7.82341	0.286052
$749$	0.598330	0.0218625
$750$	0	0
$751$	−38.9121	−1.41992	−0.709961	−	0.704241i	$-0.751287\pi$
$751$	−38.9121	−1.41992	−0.709961	+	0.704241i	$0.751287\pi$
$752$	−83.5363	−3.04626
$753$	0	0
$754$	126.074	4.59134
$755$	28.3753	1.03268
$756$	0	0
$757$	40.0357	1.45512	0.727561	−	0.686043i	$-0.240654\pi$
$757$	40.0357	1.45512	0.727561	+	0.686043i	$0.240654\pi$
$758$	−42.6519	−1.54919
$759$	0	0
$760$	62.9678	2.28408
$761$	−45.0289	−1.63230	−0.816148	−	0.577842i	$-0.803895\pi$
$761$	−45.0289	−1.63230	−0.816148	+	0.577842i	$0.803895\pi$
$762$	0	0
$763$	0.699714	0.0253314
$764$	−5.68401	−0.205640
$765$	0	0
$766$	−85.9162	−3.10428
$767$	30.5752	1.10401
$768$	0	0
$769$	14.5387	0.524279	0.262140	−	0.965030i	$-0.415572\pi$
$769$	14.5387	0.524279	0.262140	+	0.965030i	$0.415572\pi$
$770$	0.556023	0.0200377
$771$	0	0
$772$	−89.8332	−3.23317
$773$	−33.5429	−1.20645	−0.603227	−	0.797569i	$-0.706119\pi$
$773$	−33.5429	−1.20645	−0.603227	+	0.797569i	$0.706119\pi$
$774$	0	0
$775$	−1.00659	−0.0361579
$776$	−30.5634	−1.09716
$777$	0	0
$778$	7.01136	0.251369
$779$	−33.1399	−1.18736
$780$	0	0
$781$	11.7914	0.421931
$782$	−22.2950	−0.797267
$783$	0	0
$784$	−44.9419	−1.60507
$785$	−32.5261	−1.16091
$786$	0	0
$787$	19.0854	0.680323	0.340161	−	0.940367i	$-0.389518\pi$
$787$	19.0854	0.680323	0.340161	+	0.940367i	$0.389518\pi$
$788$	−73.9396	−2.63399
$789$	0	0
$790$	−34.1154	−1.21377
$791$	0.0649461	0.00230922
$792$	0	0
$793$	5.88237	0.208889
$794$	64.6277	2.29355
$795$	0	0
$796$	−104.179	−3.69254
$797$	−19.2914	−0.683337	−0.341669	−	0.939820i	$-0.610992\pi$
$797$	−19.2914	−0.683337	−0.341669	+	0.939820i	$0.610992\pi$
$798$	0	0
$799$	−23.1912	−0.820447
$800$	5.34129	0.188843
$801$	0	0
$802$	−64.5310	−2.27867
$803$	8.77470	0.309652
$804$	0	0
$805$	−1.08791	−0.0383438
$806$	−11.8126	−0.416081
$807$	0	0
$808$	−98.9552	−3.48123
$809$	21.0014	0.738370	0.369185	−	0.929356i	$-0.379637\pi$
$809$	21.0014	0.738370	0.369185	+	0.929356i	$0.379637\pi$
$810$	0	0
$811$	−7.94660	−0.279043	−0.139521	−	0.990219i	$-0.544556\pi$
$811$	−7.94660	−0.279043	−0.139521	+	0.990219i	$0.544556\pi$
$812$	−4.23429	−0.148595
$813$	0	0
$814$	14.8928	0.521991
$815$	13.3993	0.469357
$816$	0	0
$817$	−46.6282	−1.63131
$818$	−90.4119	−3.16118
$819$	0	0
$820$	51.7853	1.80842
$821$	−41.6733	−1.45441	−0.727204	−	0.686421i	$-0.759181\pi$
$821$	−41.6733	−1.45441	−0.727204	+	0.686421i	$0.759181\pi$
$822$	0	0
$823$	−45.5881	−1.58910	−0.794550	−	0.607199i	$-0.792293\pi$
$823$	−45.5881	−1.58910	−0.794550	+	0.607199i	$0.792293\pi$
$824$	11.3918	0.396850
$825$	0	0
$826$	−1.49567	−0.0520411
$827$	10.7820	0.374928	0.187464	−	0.982271i	$-0.439973\pi$
$827$	10.7820	0.374928	0.187464	+	0.982271i	$0.439973\pi$
$828$	0	0
$829$	17.5248	0.608662	0.304331	−	0.952566i	$-0.401567\pi$
$829$	17.5248	0.608662	0.304331	+	0.952566i	$0.401567\pi$
$830$	−64.0297	−2.22250
$831$	0	0
$832$	−12.9917	−0.450408
$833$	−12.4767	−0.432292
$834$	0	0
$835$	2.89691	0.100252
$836$	23.7354	0.820905
$837$	0	0
$838$	84.9398	2.93420
$839$	−0.132237	−0.00456533	−0.00228266	−	0.999997i	$-0.500727\pi$
$839$	−0.132237	−0.00456533	−0.00228266	+	0.999997i	$0.500727\pi$
$840$	0	0
$841$	42.9858	1.48227
$842$	−50.7897	−1.75033
$843$	0	0
$844$	9.67937	0.333178
$845$	−41.7420	−1.43597
$846$	0	0
$847$	0.113912	0.00391406
$848$	−21.3917	−0.734593
$849$	0	0
$850$	5.71173	0.195911
$851$	−29.1391	−0.998874
$852$	0	0
$853$	−8.37840	−0.286871	−0.143435	−	0.989660i	$-0.545815\pi$
$853$	−8.37840	−0.286871	−0.143435	+	0.989660i	$0.545815\pi$
$854$	−0.287752	−0.00984668
$855$	0	0
$856$	−31.5943	−1.07987
$857$	7.02518	0.239976	0.119988	−	0.992775i	$-0.461714\pi$
$857$	7.02518	0.239976	0.119988	+	0.992775i	$0.461714\pi$
$858$	0	0
$859$	41.2585	1.40772	0.703860	−	0.710338i	$-0.251458\pi$
$859$	41.2585	1.40772	0.703860	+	0.710338i	$0.251458\pi$
$860$	72.8623	2.48458
$861$	0	0
$862$	−14.6917	−0.500402
$863$	−4.80155	−0.163447	−0.0817233	−	0.996655i	$-0.526042\pi$
$863$	−4.80155	−0.163447	−0.0817233	+	0.996655i	$0.526042\pi$
$864$	0	0
$865$	−0.465546	−0.0158290
$866$	40.4056	1.37304
$867$	0	0
$868$	0.396736	0.0134661
$869$	−6.98919	−0.237092
$870$	0	0
$871$	−29.7492	−1.00801
$872$	−36.9478	−1.25121
$873$	0	0
$874$	−67.6405	−2.28798
$875$	1.37927	0.0466278
$876$	0	0
$877$	52.6925	1.77930	0.889650	−	0.456642i	$-0.150948\pi$
$877$	52.6925	1.77930	0.889650	+	0.456642i	$0.150948\pi$
$878$	48.3238	1.63085
$879$	0	0
$880$	−12.4289	−0.418978
$881$	25.0082	0.842547	0.421273	−	0.906934i	$-0.361583\pi$
$881$	25.0082	0.842547	0.421273	+	0.906934i	$0.361583\pi$
$882$	0	0
$883$	19.7149	0.663459	0.331730	−	0.943374i	$-0.392368\pi$
$883$	19.7149	0.663459	0.331730	+	0.943374i	$0.392368\pi$
$884$	46.0201	1.54782
$885$	0	0
$886$	−99.1610	−3.33138
$887$	24.4373	0.820523	0.410261	−	0.911968i	$-0.365438\pi$
$887$	24.4373	0.820523	0.410261	+	0.911968i	$0.365438\pi$
$888$	0	0
$889$	0.516400	0.0173195
$890$	−38.7491	−1.29887
$891$	0	0
$892$	−49.9254	−1.67163
$893$	−70.3597	−2.35450
$894$	0	0
$895$	−18.0934	−0.604795
$896$	1.59655	0.0533371
$897$	0	0
$898$	82.5863	2.75594
$899$	−6.74478	−0.224951
$900$	0	0
$901$	−5.93873	−0.197848
$902$	15.4523	0.514505
$903$	0	0
$904$	−3.42942	−0.114061
$905$	−10.1551	−0.337567
$906$	0	0
$907$	23.4467	0.778534	0.389267	−	0.921125i	$-0.372728\pi$
$907$	23.4467	0.778534	0.389267	+	0.921125i	$0.372728\pi$
$908$	−51.5769	−1.71164
$909$	0	0
$910$	3.27073	0.108424
$911$	15.7042	0.520304	0.260152	−	0.965568i	$-0.416227\pi$
$911$	15.7042	0.520304	0.260152	+	0.965568i	$0.416227\pi$
$912$	0	0
$913$	−13.1177	−0.434133
$914$	52.1905	1.72631
$915$	0	0
$916$	−112.026	−3.70143
$917$	−1.67730	−0.0553894
$918$	0	0
$919$	−0.889911	−0.0293555	−0.0146777	−	0.999892i	$-0.504672\pi$
$919$	−0.889911	−0.0293555	−0.0146777	+	0.999892i	$0.504672\pi$
$920$	57.4461	1.89394
$921$	0	0
$922$	73.1782	2.41000
$923$	69.3615	2.28306
$924$	0	0
$925$	7.46511	0.245451
$926$	−33.0072	−1.08468
$927$	0	0
$928$	35.7898	1.17486
$929$	20.0785	0.658754	0.329377	−	0.944198i	$-0.393161\pi$
$929$	20.0785	0.658754	0.329377	+	0.944198i	$0.393161\pi$
$930$	0	0
$931$	−37.8530	−1.24058
$932$	88.2275	2.88999
$933$	0	0
$934$	−29.7189	−0.972434
$935$	−3.45050	−0.112843
$936$	0	0
$937$	−17.0528	−0.557090	−0.278545	−	0.960423i	$-0.589852\pi$
$937$	−17.0528	−0.557090	−0.278545	+	0.960423i	$0.589852\pi$
$938$	1.45526	0.0475160
$939$	0	0
$940$	109.946	3.58604
$941$	−41.6953	−1.35923	−0.679613	−	0.733571i	$-0.737852\pi$
$941$	−41.6953	−1.35923	−0.679613	+	0.733571i	$0.737852\pi$
$942$	0	0
$943$	−30.2338	−0.984550
$944$	33.4331	1.08815
$945$	0	0
$946$	21.7415	0.706878
$947$	10.7242	0.348491	0.174245	−	0.984702i	$-0.444251\pi$
$947$	10.7242	0.348491	0.174245	+	0.984702i	$0.444251\pi$
$948$	0	0
$949$	51.6160	1.67553
$950$	17.3288	0.562219
$951$	0	0
$952$	−1.22353	−0.0396547
$953$	−18.7031	−0.605853	−0.302927	−	0.953014i	$-0.597964\pi$
$953$	−18.7031	−0.605853	−0.302927	+	0.953014i	$0.597964\pi$
$954$	0	0
$955$	2.50692	0.0811220
$956$	17.3044	0.559664
$957$	0	0
$958$	2.60368	0.0841212
$959$	1.40532	0.0453803
$960$	0	0
$961$	−30.3680	−0.979614
$962$	87.6047	2.82449
$963$	0	0
$964$	−82.4024	−2.65400
$965$	39.6207	1.27544
$966$	0	0
$967$	14.4362	0.464236	0.232118	−	0.972688i	$-0.425434\pi$
$967$	14.4362	0.464236	0.232118	+	0.972688i	$0.425434\pi$
$968$	−6.01501	−0.193330
$969$	0	0
$970$	24.8021	0.796347
$971$	−21.3589	−0.685439	−0.342720	−	0.939438i	$-0.611348\pi$
$971$	−21.3589	−0.685439	−0.342720	+	0.939438i	$0.611348\pi$
$972$	0	0
$973$	0.208118	0.00667195
$974$	−86.3117	−2.76560
$975$	0	0
$976$	6.43219	0.205889
$977$	17.5438	0.561276	0.280638	−	0.959814i	$-0.409454\pi$
$977$	17.5438	0.561276	0.280638	+	0.959814i	$0.409454\pi$
$978$	0	0
$979$	−7.93849	−0.253715
$980$	59.1500	1.88948
$981$	0	0
$982$	11.8293	0.377489
$983$	−29.2091	−0.931625	−0.465812	−	0.884884i	$-0.654238\pi$
$983$	−29.2091	−0.931625	−0.465812	+	0.884884i	$0.654238\pi$
$984$	0	0
$985$	32.6109	1.03907
$986$	38.2720	1.21883
$987$	0	0
$988$	139.620	4.44191
$989$	−42.5393	−1.35267
$990$	0	0
$991$	−27.3370	−0.868389	−0.434194	−	0.900819i	$-0.642967\pi$
$991$	−27.3370	−0.868389	−0.434194	+	0.900819i	$0.642967\pi$
$992$	−3.35336	−0.106469
$993$	0	0
$994$	−3.39301	−0.107620
$995$	45.9481	1.45665
$996$	0	0
$997$	21.9937	0.696548	0.348274	−	0.937393i	$-0.386768\pi$
$997$	21.9937	0.696548	0.348274	+	0.937393i	$0.386768\pi$
$998$	48.4147	1.53254
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.3	✓	25
3.2	odd	2		6039.2.a.p.1.23	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.3	✓	25	1.1	even	1	trivial
6039.2.a.p.1.23	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)

\(f(q)\)	\(=\)	\(q-2.52609 q^{2} +4.38115 q^{4} -1.93230 q^{5} +0.113912 q^{7} -6.01501 q^{8} +O(q^{10})\)	\(q-2.52609 q^{2} +4.38115 q^{4} -1.93230 q^{5} +0.113912 q^{7} -6.01501 q^{8} +4.88117 q^{10} +1.00000 q^{11} +5.88237 q^{13} -0.287752 q^{14} +6.43219 q^{16} +1.78570 q^{17} +5.41761 q^{19} -8.46569 q^{20} -2.52609 q^{22} +4.94254 q^{23} -1.26622 q^{25} -14.8594 q^{26} +0.499065 q^{28} -8.48445 q^{29} +0.794958 q^{31} -4.21828 q^{32} -4.51084 q^{34} -0.220112 q^{35} -5.89557 q^{37} -13.6854 q^{38} +11.6228 q^{40} -6.11707 q^{41} -8.60678 q^{43} +4.38115 q^{44} -12.4853 q^{46} -12.9872 q^{47} -6.98702 q^{49} +3.19860 q^{50} +25.7715 q^{52} -3.32572 q^{53} -1.93230 q^{55} -0.685182 q^{56} +21.4325 q^{58} +5.19777 q^{59} +1.00000 q^{61} -2.00814 q^{62} -2.20859 q^{64} -11.3665 q^{65} -5.05734 q^{67} +7.82341 q^{68} +0.556023 q^{70} +11.7914 q^{71} +8.77470 q^{73} +14.8928 q^{74} +23.7354 q^{76} +0.113912 q^{77} -6.98919 q^{79} -12.4289 q^{80} +15.4523 q^{82} -13.1177 q^{83} -3.45050 q^{85} +21.7415 q^{86} -6.01501 q^{88} -7.93849 q^{89} +0.670072 q^{91} +21.6540 q^{92} +32.8070 q^{94} -10.4684 q^{95} +5.08118 q^{97} +17.6499 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Introduction

L-functions

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