LMFDB - Embedded newform 6039.2.a.m.1.7

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			1.7
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)

$f(q)$	$=$	$q-1.84847 q^{2} +1.41684 q^{4} +3.30338 q^{5} -3.95713 q^{7} +1.07795 q^{8} +O(q^{10})$	\(q-1.84847 q^{2} +1.41684 q^{4} +3.30338 q^{5} -3.95713 q^{7} +1.07795 q^{8} -6.10619 q^{10} +1.00000 q^{11} -2.98905 q^{13} +7.31464 q^{14} -4.82624 q^{16} +2.21325 q^{17} +3.56465 q^{19} +4.68036 q^{20} -1.84847 q^{22} +4.72434 q^{23} +5.91229 q^{25} +5.52517 q^{26} -5.60662 q^{28} -9.75580 q^{29} -4.49208 q^{31} +6.76526 q^{32} -4.09113 q^{34} -13.0719 q^{35} -5.68381 q^{37} -6.58915 q^{38} +3.56088 q^{40} +1.33980 q^{41} -1.84241 q^{43} +1.41684 q^{44} -8.73279 q^{46} +3.92388 q^{47} +8.65888 q^{49} -10.9287 q^{50} -4.23501 q^{52} +8.83416 q^{53} +3.30338 q^{55} -4.26560 q^{56} +18.0333 q^{58} -14.2933 q^{59} +1.00000 q^{61} +8.30347 q^{62} -2.85289 q^{64} -9.87396 q^{65} +11.8404 q^{67} +3.13582 q^{68} +24.1630 q^{70} +10.1261 q^{71} -7.19928 q^{73} +10.5064 q^{74} +5.05054 q^{76} -3.95713 q^{77} +7.49052 q^{79} -15.9429 q^{80} -2.47659 q^{82} -15.2439 q^{83} +7.31120 q^{85} +3.40564 q^{86} +1.07795 q^{88} +4.27782 q^{89} +11.8281 q^{91} +6.69363 q^{92} -7.25317 q^{94} +11.7754 q^{95} -8.72931 q^{97} -16.0057 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.84847	−1.30707	−0.653533	−	0.756898i	$-0.726714\pi$
$2$	−1.84847	−1.30707	−0.653533	+	0.756898i	$0.726714\pi$
$3$	0	0
$3$	0	0
$4$	1.41684	0.708420
$5$	3.30338	1.47731	0.738657	−	0.674081i	$-0.235460\pi$
$5$	3.30338	1.47731	0.738657	+	0.674081i	$0.235460\pi$
$6$	0	0
$7$	−3.95713	−1.49565	−0.747827	−	0.663893i	$-0.768903\pi$
$7$	−3.95713	−1.49565	−0.747827	+	0.663893i	$0.768903\pi$
$8$	1.07795	0.381114
$9$	0	0
$10$	−6.10619	−1.93095
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.98905	−0.829013	−0.414507	−	0.910046i	$-0.636046\pi$
$13$	−2.98905	−0.829013	−0.414507	+	0.910046i	$0.636046\pi$
$14$	7.31464	1.95492
$15$	0	0
$16$	−4.82624	−1.20656
$17$	2.21325	0.536792	0.268396	−	0.963309i	$-0.413506\pi$
$17$	2.21325	0.536792	0.268396	+	0.963309i	$0.413506\pi$
$18$	0	0
$19$	3.56465	0.817787	0.408894	−	0.912582i	$-0.365915\pi$
$19$	3.56465	0.817787	0.408894	+	0.912582i	$0.365915\pi$
$20$	4.68036	1.04656
$21$	0	0
$22$	−1.84847	−0.394095
$23$	4.72434	0.985092	0.492546	−	0.870286i	$-0.336066\pi$
$23$	4.72434	0.985092	0.492546	+	0.870286i	$0.336066\pi$
$24$	0	0
$25$	5.91229	1.18246
$26$	5.52517	1.08357
$27$	0	0
$28$	−5.60662	−1.05955
$29$	−9.75580	−1.81161	−0.905803	−	0.423699i	$-0.860732\pi$
$29$	−9.75580	−1.81161	−0.905803	+	0.423699i	$0.860732\pi$
$30$	0	0
$31$	−4.49208	−0.806801	−0.403400	−	0.915024i	$-0.632172\pi$
$31$	−4.49208	−0.806801	−0.403400	+	0.915024i	$0.632172\pi$
$32$	6.76526	1.19594
$33$	0	0
$34$	−4.09113	−0.701623
$35$	−13.0719	−2.20955
$36$	0	0
$37$	−5.68381	−0.934413	−0.467207	−	0.884148i	$-0.654740\pi$
$37$	−5.68381	−0.934413	−0.467207	+	0.884148i	$0.654740\pi$
$38$	−6.58915	−1.06890
$39$	0	0
$40$	3.56088	0.563025
$41$	1.33980	0.209242	0.104621	−	0.994512i	$-0.466637\pi$
$41$	1.33980	0.209242	0.104621	+	0.994512i	$0.466637\pi$
$42$	0	0
$43$	−1.84241	−0.280965	−0.140482	−	0.990083i	$-0.544865\pi$
$43$	−1.84241	−0.280965	−0.140482	+	0.990083i	$0.544865\pi$
$44$	1.41684	0.213597
$45$	0	0
$46$	−8.73279	−1.28758
$47$	3.92388	0.572357	0.286178	−	0.958176i	$-0.407615\pi$
$47$	3.92388	0.572357	0.286178	+	0.958176i	$0.407615\pi$
$48$	0	0
$49$	8.65888	1.23698
$50$	−10.9287	−1.54555
$51$	0	0
$52$	−4.23501	−0.587290
$53$	8.83416	1.21347	0.606733	−	0.794906i	$-0.292480\pi$
$53$	8.83416	1.21347	0.606733	+	0.794906i	$0.292480\pi$
$54$	0	0
$55$	3.30338	0.445427
$56$	−4.26560	−0.570015
$57$	0	0
$58$	18.0333	2.36789
$59$	−14.2933	−1.86083	−0.930415	−	0.366508i	$-0.880553\pi$
$59$	−14.2933	−1.86083	−0.930415	+	0.366508i	$0.880553\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	8.30347	1.05454
$63$	0	0
$64$	−2.85289	−0.356612
$65$	−9.87396	−1.22471
$66$	0	0
$67$	11.8404	1.44653	0.723267	−	0.690568i	$-0.242640\pi$
$67$	11.8404	1.44653	0.723267	+	0.690568i	$0.242640\pi$
$68$	3.13582	0.380275
$69$	0	0
$70$	24.1630	2.88803
$71$	10.1261	1.20175	0.600875	−	0.799343i	$-0.294819\pi$
$71$	10.1261	1.20175	0.600875	+	0.799343i	$0.294819\pi$
$72$	0	0
$73$	−7.19928	−0.842612	−0.421306	−	0.906918i	$-0.638428\pi$
$73$	−7.19928	−0.842612	−0.421306	+	0.906918i	$0.638428\pi$
$74$	10.5064	1.22134
$75$	0	0
$76$	5.05054	0.579337
$77$	−3.95713	−0.450957
$78$	0	0
$79$	7.49052	0.842750	0.421375	−	0.906887i	$-0.361548\pi$
$79$	7.49052	0.842750	0.421375	+	0.906887i	$0.361548\pi$
$80$	−15.9429	−1.78247
$81$	0	0
$82$	−2.47659	−0.273493
$83$	−15.2439	−1.67323	−0.836616	−	0.547789i	$-0.815470\pi$
$83$	−15.2439	−1.67323	−0.836616	+	0.547789i	$0.815470\pi$
$84$	0	0
$85$	7.31120	0.793011
$86$	3.40564	0.367239
$87$	0	0
$88$	1.07795	0.114910
$89$	4.27782	0.453448	0.226724	−	0.973959i	$-0.427198\pi$
$89$	4.27782	0.453448	0.226724	+	0.973959i	$0.427198\pi$
$90$	0	0
$91$	11.8281	1.23992
$92$	6.69363	0.697859
$93$	0	0
$94$	−7.25317	−0.748108
$95$	11.7754	1.20813
$96$	0	0
$97$	−8.72931	−0.886327	−0.443164	−	0.896441i	$-0.646144\pi$
$97$	−8.72931	−0.886327	−0.443164	+	0.896441i	$0.646144\pi$
$98$	−16.0057	−1.61682
$99$	0	0
$100$	8.37678	0.837678
$101$	−8.41434	−0.837258	−0.418629	−	0.908157i	$-0.637489\pi$
$101$	−8.41434	−0.837258	−0.418629	+	0.908157i	$0.637489\pi$
$102$	0	0
$103$	−12.0092	−1.18330	−0.591652	−	0.806194i	$-0.701524\pi$
$103$	−12.0092	−1.18330	−0.591652	+	0.806194i	$0.701524\pi$
$104$	−3.22205	−0.315948
$105$	0	0
$106$	−16.3297	−1.58608
$107$	−4.38059	−0.423488	−0.211744	−	0.977325i	$-0.567914\pi$
$107$	−4.38059	−0.423488	−0.211744	+	0.977325i	$0.567914\pi$
$108$	0	0
$109$	4.23951	0.406072	0.203036	−	0.979171i	$-0.434919\pi$
$109$	4.23951	0.406072	0.203036	+	0.979171i	$0.434919\pi$
$110$	−6.10619	−0.582202
$111$	0	0
$112$	19.0981	1.80460
$113$	4.63806	0.436312	0.218156	−	0.975914i	$-0.429996\pi$
$113$	4.63806	0.436312	0.218156	+	0.975914i	$0.429996\pi$
$114$	0	0
$115$	15.6063	1.45529
$116$	−13.8224	−1.28338
$117$	0	0
$118$	26.4207	2.43223
$119$	−8.75812	−0.802856
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.84847	−0.167353
$123$	0	0
$124$	−6.36456	−0.571554
$125$	3.01365	0.269549
$126$	0	0
$127$	−4.86801	−0.431966	−0.215983	−	0.976397i	$-0.569296\pi$
$127$	−4.86801	−0.431966	−0.215983	+	0.976397i	$0.569296\pi$
$128$	−8.25704	−0.729826
$129$	0	0
$130$	18.2517	1.60078
$131$	−15.3923	−1.34483	−0.672414	−	0.740175i	$-0.734743\pi$
$131$	−15.3923	−1.34483	−0.672414	+	0.740175i	$0.734743\pi$
$132$	0	0
$133$	−14.1058	−1.22313
$134$	−21.8866	−1.89071
$135$	0	0
$136$	2.38578	0.204579
$137$	−10.4869	−0.895953	−0.447976	−	0.894045i	$-0.647855\pi$
$137$	−10.4869	−0.895953	−0.447976	+	0.894045i	$0.647855\pi$
$138$	0	0
$139$	15.8531	1.34464	0.672321	−	0.740259i	$-0.265297\pi$
$139$	15.8531	1.34464	0.672321	+	0.740259i	$0.265297\pi$
$140$	−18.5208	−1.56529
$141$	0	0
$142$	−18.7178	−1.57077
$143$	−2.98905	−0.249957
$144$	0	0
$145$	−32.2271	−2.67631
$146$	13.3077	1.10135
$147$	0	0
$148$	−8.05306	−0.661957
$149$	−23.5778	−1.93157	−0.965783	−	0.259352i	$-0.916491\pi$
$149$	−23.5778	−1.93157	−0.965783	+	0.259352i	$0.916491\pi$
$150$	0	0
$151$	−7.83173	−0.637337	−0.318668	−	0.947866i	$-0.603236\pi$
$151$	−7.83173	−0.637337	−0.318668	+	0.947866i	$0.603236\pi$
$152$	3.84253	0.311670
$153$	0	0
$154$	7.31464	0.589430
$155$	−14.8390	−1.19190
$156$	0	0
$157$	3.23571	0.258238	0.129119	−	0.991629i	$-0.458785\pi$
$157$	3.23571	0.258238	0.129119	+	0.991629i	$0.458785\pi$
$158$	−13.8460	−1.10153
$159$	0	0
$160$	22.3482	1.76678
$161$	−18.6948	−1.47336
$162$	0	0
$163$	−10.2249	−0.800875	−0.400437	−	0.916324i	$-0.631142\pi$
$163$	−10.2249	−0.800875	−0.400437	+	0.916324i	$0.631142\pi$
$164$	1.89829	0.148231
$165$	0	0
$166$	28.1778	2.18703
$167$	16.3966	1.26881	0.634405	−	0.773000i	$-0.281245\pi$
$167$	16.3966	1.26881	0.634405	+	0.773000i	$0.281245\pi$
$168$	0	0
$169$	−4.06558	−0.312737
$170$	−13.5145	−1.03652
$171$	0	0
$172$	−2.61040	−0.199041
$173$	5.35272	0.406960	0.203480	−	0.979079i	$-0.434775\pi$
$173$	5.35272	0.406960	0.203480	+	0.979079i	$0.434775\pi$
$174$	0	0
$175$	−23.3957	−1.76855
$176$	−4.82624	−0.363792
$177$	0	0
$178$	−7.90743	−0.592687
$179$	−0.134569	−0.0100582	−0.00502909	−	0.999987i	$-0.501601\pi$
$179$	−0.134569	−0.0100582	−0.00502909	+	0.999987i	$0.501601\pi$
$180$	0	0
$181$	1.95147	0.145051	0.0725257	−	0.997367i	$-0.476894\pi$
$181$	1.95147	0.145051	0.0725257	+	0.997367i	$0.476894\pi$
$182$	−21.8638	−1.62065
$183$	0	0
$184$	5.09261	0.375432
$185$	−18.7758	−1.38042
$186$	0	0
$187$	2.21325	0.161849
$188$	5.55951	0.405469
$189$	0	0
$190$	−21.7664	−1.57910
$191$	23.0282	1.66626	0.833130	−	0.553078i	$-0.186547\pi$
$191$	23.0282	1.66626	0.833130	+	0.553078i	$0.186547\pi$
$192$	0	0
$193$	19.6738	1.41615	0.708076	−	0.706136i	$-0.249563\pi$
$193$	19.6738	1.41615	0.708076	+	0.706136i	$0.249563\pi$
$194$	16.1359	1.15849
$195$	0	0
$196$	12.2683	0.876304
$197$	−4.30898	−0.307002	−0.153501	−	0.988149i	$-0.549055\pi$
$197$	−4.30898	−0.307002	−0.153501	+	0.988149i	$0.549055\pi$
$198$	0	0
$199$	7.33897	0.520246	0.260123	−	0.965576i	$-0.416237\pi$
$199$	7.33897	0.520246	0.260123	+	0.965576i	$0.416237\pi$
$200$	6.37317	0.450651
$201$	0	0
$202$	15.5537	1.09435
$203$	38.6050	2.70954
$204$	0	0
$205$	4.42587	0.309116
$206$	22.1987	1.54665
$207$	0	0
$208$	14.4259	1.00026
$209$	3.56465	0.246572
$210$	0	0
$211$	−17.5685	−1.20947	−0.604733	−	0.796428i	$-0.706720\pi$
$211$	−17.5685	−1.20947	−0.604733	+	0.796428i	$0.706720\pi$
$212$	12.5166	0.859644
$213$	0	0
$214$	8.09740	0.553527
$215$	−6.08617	−0.415073
$216$	0	0
$217$	17.7757	1.20670
$218$	−7.83661	−0.530762
$219$	0	0
$220$	4.68036	0.315550
$221$	−6.61552	−0.445008
$222$	0	0
$223$	−19.0149	−1.27333	−0.636666	−	0.771140i	$-0.719687\pi$
$223$	−19.0149	−1.27333	−0.636666	+	0.771140i	$0.719687\pi$
$224$	−26.7710	−1.78871
$225$	0	0
$226$	−8.57331	−0.570288
$227$	21.0709	1.39852	0.699262	−	0.714866i	$-0.253512\pi$
$227$	21.0709	1.39852	0.699262	+	0.714866i	$0.253512\pi$
$228$	0	0
$229$	−6.19206	−0.409183	−0.204592	−	0.978847i	$-0.565587\pi$
$229$	−6.19206	−0.409183	−0.204592	+	0.978847i	$0.565587\pi$
$230$	−28.8477	−1.90216
$231$	0	0
$232$	−10.5163	−0.690428
$233$	−10.4925	−0.687387	−0.343694	−	0.939082i	$-0.611678\pi$
$233$	−10.4925	−0.687387	−0.343694	+	0.939082i	$0.611678\pi$
$234$	0	0
$235$	12.9621	0.845551
$236$	−20.2513	−1.31825
$237$	0	0
$238$	16.1891	1.04939
$239$	−7.59425	−0.491231	−0.245616	−	0.969367i	$-0.578990\pi$
$239$	−7.59425	−0.491231	−0.245616	+	0.969367i	$0.578990\pi$
$240$	0	0
$241$	−16.8949	−1.08829	−0.544147	−	0.838990i	$-0.683147\pi$
$241$	−16.8949	−1.08829	−0.544147	+	0.838990i	$0.683147\pi$
$242$	−1.84847	−0.118824
$243$	0	0
$244$	1.41684	0.0907039
$245$	28.6035	1.82741
$246$	0	0
$247$	−10.6549	−0.677957
$248$	−4.84225	−0.307483
$249$	0	0
$250$	−5.57065	−0.352319
$251$	11.9875	0.756646	0.378323	−	0.925674i	$-0.376501\pi$
$251$	11.9875	0.756646	0.378323	+	0.925674i	$0.376501\pi$
$252$	0	0
$253$	4.72434	0.297016
$254$	8.99837	0.564608
$255$	0	0
$256$	20.9687	1.31054
$257$	1.46850	0.0916027	0.0458014	−	0.998951i	$-0.485416\pi$
$257$	1.46850	0.0916027	0.0458014	+	0.998951i	$0.485416\pi$
$258$	0	0
$259$	22.4916	1.39756
$260$	−13.9898	−0.867612
$261$	0	0
$262$	28.4521	1.75778
$263$	−5.52056	−0.340412	−0.170206	−	0.985408i	$-0.554443\pi$
$263$	−5.52056	−0.340412	−0.170206	+	0.985408i	$0.554443\pi$
$264$	0	0
$265$	29.1826	1.79267
$266$	26.0741	1.59871
$267$	0	0
$268$	16.7759	1.02475
$269$	17.4972	1.06682	0.533412	−	0.845856i	$-0.320910\pi$
$269$	17.4972	1.06682	0.533412	+	0.845856i	$0.320910\pi$
$270$	0	0
$271$	−7.52293	−0.456985	−0.228493	−	0.973546i	$-0.573380\pi$
$271$	−7.52293	−0.456985	−0.228493	+	0.973546i	$0.573380\pi$
$272$	−10.6817	−0.647673
$273$	0	0
$274$	19.3846	1.17107
$275$	5.91229	0.356525
$276$	0	0
$277$	−7.90278	−0.474832	−0.237416	−	0.971408i	$-0.576300\pi$
$277$	−7.90278	−0.474832	−0.237416	+	0.971408i	$0.576300\pi$
$278$	−29.3040	−1.75754
$279$	0	0
$280$	−14.0909	−0.842091
$281$	0.766106	0.0457020	0.0228510	−	0.999739i	$-0.492726\pi$
$281$	0.766106	0.0457020	0.0228510	+	0.999739i	$0.492726\pi$
$282$	0	0
$283$	14.2211	0.845357	0.422678	−	0.906280i	$-0.361090\pi$
$283$	14.2211	0.845357	0.422678	+	0.906280i	$0.361090\pi$
$284$	14.3471	0.851344
$285$	0	0
$286$	5.52517	0.326710
$287$	−5.30178	−0.312954
$288$	0	0
$289$	−12.1015	−0.711854
$290$	59.5708	3.49812
$291$	0	0
$292$	−10.2002	−0.596924
$293$	−3.44663	−0.201354	−0.100677	−	0.994919i	$-0.532101\pi$
$293$	−3.44663	−0.201354	−0.100677	+	0.994919i	$0.532101\pi$
$294$	0	0
$295$	−47.2162	−2.74903
$296$	−6.12688	−0.356118
$297$	0	0
$298$	43.5828	2.52468
$299$	−14.1213	−0.816655
$300$	0	0
$301$	7.29065	0.420226
$302$	14.4767	0.833041
$303$	0	0
$304$	−17.2039	−0.986710
$305$	3.30338	0.189151
$306$	0	0
$307$	−11.4971	−0.656171	−0.328086	−	0.944648i	$-0.606403\pi$
$307$	−11.4971	−0.656171	−0.328086	+	0.944648i	$0.606403\pi$
$308$	−5.60662	−0.319467
$309$	0	0
$310$	27.4295	1.55789
$311$	14.3564	0.814075	0.407038	−	0.913411i	$-0.366562\pi$
$311$	14.3564	0.814075	0.407038	+	0.913411i	$0.366562\pi$
$312$	0	0
$313$	−17.6147	−0.995644	−0.497822	−	0.867279i	$-0.665867\pi$
$313$	−17.6147	−0.995644	−0.497822	+	0.867279i	$0.665867\pi$
$314$	−5.98112	−0.337534
$315$	0	0
$316$	10.6129	0.597021
$317$	−13.7541	−0.772506	−0.386253	−	0.922393i	$-0.626231\pi$
$317$	−13.7541	−0.772506	−0.386253	+	0.922393i	$0.626231\pi$
$318$	0	0
$319$	−9.75580	−0.546220
$320$	−9.42418	−0.526828
$321$	0	0
$322$	34.5568	1.92577
$323$	7.88947	0.438982
$324$	0	0
$325$	−17.6721	−0.980274
$326$	18.9004	1.04680
$327$	0	0
$328$	1.44424	0.0797450
$329$	−15.5273	−0.856048
$330$	0	0
$331$	29.7124	1.63314	0.816569	−	0.577247i	$-0.195873\pi$
$331$	29.7124	1.63314	0.816569	+	0.577247i	$0.195873\pi$
$332$	−21.5981	−1.18535
$333$	0	0
$334$	−30.3087	−1.65842
$335$	39.1133	2.13699
$336$	0	0
$337$	−12.1870	−0.663867	−0.331933	−	0.943303i	$-0.607701\pi$
$337$	−12.1870	−0.663867	−0.331933	+	0.943303i	$0.607701\pi$
$338$	7.51510	0.408767
$339$	0	0
$340$	10.3588	0.561785
$341$	−4.49208	−0.243260
$342$	0	0
$343$	−6.56440	−0.354445
$344$	−1.98603	−0.107080
$345$	0	0
$346$	−9.89434	−0.531923
$347$	28.8989	1.55137	0.775686	−	0.631119i	$-0.217404\pi$
$347$	28.8989	1.55137	0.775686	+	0.631119i	$0.217404\pi$
$348$	0	0
$349$	−9.51210	−0.509171	−0.254585	−	0.967050i	$-0.581939\pi$
$349$	−9.51210	−0.509171	−0.254585	+	0.967050i	$0.581939\pi$
$350$	43.2463	2.31161
$351$	0	0
$352$	6.76526	0.360590
$353$	−4.38429	−0.233352	−0.116676	−	0.993170i	$-0.537224\pi$
$353$	−4.38429	−0.233352	−0.116676	+	0.993170i	$0.537224\pi$
$354$	0	0
$355$	33.4504	1.77536
$356$	6.06099	0.321232
$357$	0	0
$358$	0.248747	0.0131467
$359$	−34.8151	−1.83747	−0.918735	−	0.394874i	$-0.870788\pi$
$359$	−34.8151	−1.83747	−0.918735	+	0.394874i	$0.870788\pi$
$360$	0	0
$361$	−6.29326	−0.331224
$362$	−3.60723	−0.189592
$363$	0	0
$364$	16.7585	0.878383
$365$	−23.7819	−1.24480
$366$	0	0
$367$	0.614056	0.0320535	0.0160267	−	0.999872i	$-0.494898\pi$
$367$	0.614056	0.0320535	0.0160267	+	0.999872i	$0.494898\pi$
$368$	−22.8008	−1.18857
$369$	0	0
$370$	34.7065	1.80430
$371$	−34.9579	−1.81493
$372$	0	0
$373$	−5.38763	−0.278961	−0.139481	−	0.990225i	$-0.544543\pi$
$373$	−5.38763	−0.278961	−0.139481	+	0.990225i	$0.544543\pi$
$374$	−4.09113	−0.211547
$375$	0	0
$376$	4.22976	0.218133
$377$	29.1606	1.50185
$378$	0	0
$379$	−18.5494	−0.952819	−0.476410	−	0.879223i	$-0.658062\pi$
$379$	−18.5494	−0.952819	−0.476410	+	0.879223i	$0.658062\pi$
$380$	16.6838	0.855863
$381$	0	0
$382$	−42.5669	−2.17791
$383$	−8.41132	−0.429798	−0.214899	−	0.976636i	$-0.568942\pi$
$383$	−8.41132	−0.429798	−0.214899	+	0.976636i	$0.568942\pi$
$384$	0	0
$385$	−13.0719	−0.666205
$386$	−36.3664	−1.85100
$387$	0	0
$388$	−12.3680	−0.627892
$389$	−20.5368	−1.04126	−0.520628	−	0.853784i	$-0.674302\pi$
$389$	−20.5368	−1.04126	−0.520628	+	0.853784i	$0.674302\pi$
$390$	0	0
$391$	10.4561	0.528790
$392$	9.33386	0.471431
$393$	0	0
$394$	7.96501	0.401271
$395$	24.7440	1.24501
$396$	0	0
$397$	25.7098	1.29034	0.645169	−	0.764040i	$-0.276787\pi$
$397$	25.7098	1.29034	0.645169	+	0.764040i	$0.276787\pi$
$398$	−13.5659	−0.679995
$399$	0	0
$400$	−28.5342	−1.42671
$401$	−26.9601	−1.34633	−0.673163	−	0.739494i	$-0.735065\pi$
$401$	−26.9601	−1.34633	−0.673163	+	0.739494i	$0.735065\pi$
$402$	0	0
$403$	13.4270	0.668849
$404$	−11.9218	−0.593131
$405$	0	0
$406$	−71.3601	−3.54154
$407$	−5.68381	−0.281736
$408$	0	0
$409$	−11.6253	−0.574833	−0.287416	−	0.957806i	$-0.592796\pi$
$409$	−11.6253	−0.574833	−0.287416	+	0.957806i	$0.592796\pi$
$410$	−8.18109	−0.404035
$411$	0	0
$412$	−17.0151	−0.838276
$413$	56.5605	2.78316
$414$	0	0
$415$	−50.3563	−2.47189
$416$	−20.2217	−0.991451
$417$	0	0
$418$	−6.58915	−0.322286
$419$	−34.0716	−1.66451	−0.832254	−	0.554394i	$-0.812950\pi$
$419$	−34.0716	−1.66451	−0.832254	+	0.554394i	$0.812950\pi$
$420$	0	0
$421$	13.6562	0.665563	0.332781	−	0.943004i	$-0.392013\pi$
$421$	13.6562	0.665563	0.332781	+	0.943004i	$0.392013\pi$
$422$	32.4749	1.58085
$423$	0	0
$424$	9.52281	0.462468
$425$	13.0854	0.634735
$426$	0	0
$427$	−3.95713	−0.191499
$428$	−6.20660	−0.300008
$429$	0	0
$430$	11.2501	0.542528
$431$	−27.0917	−1.30496	−0.652481	−	0.757805i	$-0.726272\pi$
$431$	−27.0917	−1.30496	−0.652481	+	0.757805i	$0.726272\pi$
$432$	0	0
$433$	−4.42284	−0.212548	−0.106274	−	0.994337i	$-0.533892\pi$
$433$	−4.42284	−0.212548	−0.106274	+	0.994337i	$0.533892\pi$
$434$	−32.8579	−1.57723
$435$	0	0
$436$	6.00672	0.287670
$437$	16.8406	0.805596
$438$	0	0
$439$	−24.4951	−1.16909	−0.584544	−	0.811362i	$-0.698726\pi$
$439$	−24.4951	−1.16909	−0.584544	+	0.811362i	$0.698726\pi$
$440$	3.56088	0.169758
$441$	0	0
$442$	12.2286	0.581655
$443$	−14.2630	−0.677655	−0.338827	−	0.940849i	$-0.610030\pi$
$443$	−14.2630	−0.677655	−0.338827	+	0.940849i	$0.610030\pi$
$444$	0	0
$445$	14.1313	0.669886
$446$	35.1484	1.66433
$447$	0	0
$448$	11.2893	0.533368
$449$	0.585554	0.0276340	0.0138170	−	0.999905i	$-0.495602\pi$
$449$	0.585554	0.0276340	0.0138170	+	0.999905i	$0.495602\pi$
$450$	0	0
$451$	1.33980	0.0630889
$452$	6.57139	0.309092
$453$	0	0
$454$	−38.9489	−1.82796
$455$	39.0725	1.83175
$456$	0	0
$457$	17.4744	0.817416	0.408708	−	0.912665i	$-0.365979\pi$
$457$	17.4744	0.817416	0.408708	+	0.912665i	$0.365979\pi$
$458$	11.4458	0.534829
$459$	0	0
$460$	22.1116	1.03096
$461$	20.5326	0.956300	0.478150	−	0.878278i	$-0.341308\pi$
$461$	20.5326	0.956300	0.478150	+	0.878278i	$0.341308\pi$
$462$	0	0
$463$	8.88087	0.412729	0.206365	−	0.978475i	$-0.433837\pi$
$463$	8.88087	0.412729	0.206365	+	0.978475i	$0.433837\pi$
$464$	47.0839	2.18581
$465$	0	0
$466$	19.3951	0.898460
$467$	3.46933	0.160541	0.0802707	−	0.996773i	$-0.474422\pi$
$467$	3.46933	0.160541	0.0802707	+	0.996773i	$0.474422\pi$
$468$	0	0
$469$	−46.8540	−2.16352
$470$	−23.9600	−1.10519
$471$	0	0
$472$	−15.4075	−0.709188
$473$	−1.84241	−0.0847141
$474$	0	0
$475$	21.0753	0.967000
$476$	−12.4089	−0.568759
$477$	0	0
$478$	14.0377	0.642072
$479$	−16.0150	−0.731745	−0.365873	−	0.930665i	$-0.619229\pi$
$479$	−16.0150	−0.731745	−0.365873	+	0.930665i	$0.619229\pi$
$480$	0	0
$481$	16.9892	0.774641
$482$	31.2296	1.42247
$483$	0	0
$484$	1.41684	0.0644018
$485$	−28.8362	−1.30938
$486$	0	0
$487$	7.12455	0.322844	0.161422	−	0.986885i	$-0.448392\pi$
$487$	7.12455	0.322844	0.161422	+	0.986885i	$0.448392\pi$
$488$	1.07795	0.0487966
$489$	0	0
$490$	−52.8728	−2.38855
$491$	−24.6208	−1.11112	−0.555561	−	0.831476i	$-0.687496\pi$
$491$	−24.6208	−1.11112	−0.555561	+	0.831476i	$0.687496\pi$
$492$	0	0
$493$	−21.5920	−0.972456
$494$	19.6953	0.886134
$495$	0	0
$496$	21.6799	0.973454
$497$	−40.0704	−1.79740
$498$	0	0
$499$	35.6515	1.59598	0.797991	−	0.602669i	$-0.205896\pi$
$499$	35.6515	1.59598	0.797991	+	0.602669i	$0.205896\pi$
$500$	4.26987	0.190954
$501$	0	0
$502$	−22.1586	−0.988986
$503$	34.2193	1.52576	0.762881	−	0.646539i	$-0.223784\pi$
$503$	34.2193	1.52576	0.762881	+	0.646539i	$0.223784\pi$
$504$	0	0
$505$	−27.7957	−1.23689
$506$	−8.73279	−0.388220
$507$	0	0
$508$	−6.89719	−0.306013
$509$	−4.88930	−0.216714	−0.108357	−	0.994112i	$-0.534559\pi$
$509$	−4.88930	−0.216714	−0.108357	+	0.994112i	$0.534559\pi$
$510$	0	0
$511$	28.4885	1.26026
$512$	−22.2459	−0.983138
$513$	0	0
$514$	−2.71448	−0.119731
$515$	−39.6710	−1.74811
$516$	0	0
$517$	3.92388	0.172572
$518$	−41.5750	−1.82670
$519$	0	0
$520$	−10.6437	−0.466755
$521$	−0.638526	−0.0279743	−0.0139872	−	0.999902i	$-0.504452\pi$
$521$	−0.638526	−0.0279743	−0.0139872	+	0.999902i	$0.504452\pi$
$522$	0	0
$523$	−4.03487	−0.176433	−0.0882163	−	0.996101i	$-0.528117\pi$
$523$	−4.03487	−0.176433	−0.0882163	+	0.996101i	$0.528117\pi$
$524$	−21.8084	−0.952704
$525$	0	0
$526$	10.2046	0.444941
$527$	−9.94210	−0.433084
$528$	0	0
$529$	−0.680653	−0.0295936
$530$	−53.9431	−2.34314
$531$	0	0
$532$	−19.9857	−0.866488
$533$	−4.00474	−0.173464
$534$	0	0
$535$	−14.4708	−0.625625
$536$	12.7634	0.551294
$537$	0	0
$538$	−32.3430	−1.39441
$539$	8.65888	0.372964
$540$	0	0
$541$	−1.47979	−0.0636213	−0.0318107	−	0.999494i	$-0.510127\pi$
$541$	−1.47979	−0.0636213	−0.0318107	+	0.999494i	$0.510127\pi$
$542$	13.9059	0.597310
$543$	0	0
$544$	14.9732	0.641972
$545$	14.0047	0.599896
$546$	0	0
$547$	3.12677	0.133691	0.0668454	−	0.997763i	$-0.478707\pi$
$547$	3.12677	0.133691	0.0668454	+	0.997763i	$0.478707\pi$
$548$	−14.8582	−0.634711
$549$	0	0
$550$	−10.9287	−0.466001
$551$	−34.7760	−1.48151
$552$	0	0
$553$	−29.6410	−1.26046
$554$	14.6081	0.620637
$555$	0	0
$556$	22.4613	0.952572
$557$	23.5146	0.996347	0.498174	−	0.867077i	$-0.334004\pi$
$557$	23.5146	0.996347	0.498174	+	0.867077i	$0.334004\pi$
$558$	0	0
$559$	5.50705	0.232924
$560$	63.0881	2.66596
$561$	0	0
$562$	−1.41612	−0.0597356
$563$	22.9262	0.966226	0.483113	−	0.875558i	$-0.339506\pi$
$563$	22.9262	0.966226	0.483113	+	0.875558i	$0.339506\pi$
$564$	0	0
$565$	15.3213	0.644570
$566$	−26.2873	−1.10494
$567$	0	0
$568$	10.9155	0.458003
$569$	−33.6211	−1.40947	−0.704734	−	0.709471i	$-0.748934\pi$
$569$	−33.6211	−1.40947	−0.704734	+	0.709471i	$0.748934\pi$
$570$	0	0
$571$	−26.9729	−1.12878	−0.564391	−	0.825507i	$-0.690889\pi$
$571$	−26.9729	−1.12878	−0.564391	+	0.825507i	$0.690889\pi$
$572$	−4.23501	−0.177075
$573$	0	0
$574$	9.80017	0.409051
$575$	27.9317	1.16483
$576$	0	0
$577$	3.22915	0.134431	0.0672157	−	0.997738i	$-0.478588\pi$
$577$	3.22915	0.134431	0.0672157	+	0.997738i	$0.478588\pi$
$578$	22.3693	0.930440
$579$	0	0
$580$	−45.6606	−1.89595
$581$	60.3220	2.50258
$582$	0	0
$583$	8.83416	0.365874
$584$	−7.76049	−0.321131
$585$	0	0
$586$	6.37099	0.263183
$587$	1.21224	0.0500345	0.0250173	−	0.999687i	$-0.492036\pi$
$587$	1.21224	0.0500345	0.0250173	+	0.999687i	$0.492036\pi$
$588$	0	0
$589$	−16.0127	−0.659791
$590$	87.2777	3.59316
$591$	0	0
$592$	27.4315	1.12743
$593$	−20.2271	−0.830627	−0.415314	−	0.909678i	$-0.636328\pi$
$593$	−20.2271	−0.830627	−0.415314	+	0.909678i	$0.636328\pi$
$594$	0	0
$595$	−28.9314	−1.18607
$596$	−33.4059	−1.36836
$597$	0	0
$598$	26.1028	1.06742
$599$	35.5417	1.45219	0.726097	−	0.687593i	$-0.241332\pi$
$599$	35.5417	1.45219	0.726097	+	0.687593i	$0.241332\pi$
$600$	0	0
$601$	−31.4502	−1.28288	−0.641440	−	0.767173i	$-0.721663\pi$
$601$	−31.4502	−1.28288	−0.641440	+	0.767173i	$0.721663\pi$
$602$	−13.4766	−0.549263
$603$	0	0
$604$	−11.0963	−0.451502
$605$	3.30338	0.134301
$606$	0	0
$607$	40.2027	1.63178	0.815890	−	0.578208i	$-0.196248\pi$
$607$	40.2027	1.63178	0.815890	+	0.578208i	$0.196248\pi$
$608$	24.1158	0.978025
$609$	0	0
$610$	−6.10619	−0.247232
$611$	−11.7287	−0.474492
$612$	0	0
$613$	−24.3320	−0.982761	−0.491381	−	0.870945i	$-0.663508\pi$
$613$	−24.3320	−0.982761	−0.491381	+	0.870945i	$0.663508\pi$
$614$	21.2520	0.857659
$615$	0	0
$616$	−4.26560	−0.171866
$617$	−41.3610	−1.66513	−0.832565	−	0.553928i	$-0.813128\pi$
$617$	−41.3610	−1.66513	−0.832565	+	0.553928i	$0.813128\pi$
$618$	0	0
$619$	13.0276	0.523625	0.261813	−	0.965119i	$-0.415680\pi$
$619$	13.0276	0.523625	0.261813	+	0.965119i	$0.415680\pi$
$620$	−21.0245	−0.844365
$621$	0	0
$622$	−26.5373	−1.06405
$623$	−16.9279	−0.678202
$624$	0	0
$625$	−19.6062	−0.784250
$626$	32.5603	1.30137
$627$	0	0
$628$	4.58449	0.182941
$629$	−12.5797	−0.501586
$630$	0	0
$631$	−3.09168	−0.123078	−0.0615388	−	0.998105i	$-0.519601\pi$
$631$	−3.09168	−0.123078	−0.0615388	+	0.998105i	$0.519601\pi$
$632$	8.07443	0.321183
$633$	0	0
$634$	25.4240	1.00972
$635$	−16.0809	−0.638150
$636$	0	0
$637$	−25.8818	−1.02548
$638$	18.0333	0.713945
$639$	0	0
$640$	−27.2761	−1.07818
$641$	−31.6850	−1.25148	−0.625740	−	0.780031i	$-0.715203\pi$
$641$	−31.6850	−1.25148	−0.625740	+	0.780031i	$0.715203\pi$
$642$	0	0
$643$	−41.9634	−1.65488	−0.827438	−	0.561557i	$-0.810203\pi$
$643$	−41.9634	−1.65488	−0.827438	+	0.561557i	$0.810203\pi$
$644$	−26.4876	−1.04376
$645$	0	0
$646$	−14.5834	−0.573778
$647$	6.08892	0.239380	0.119690	−	0.992811i	$-0.461810\pi$
$647$	6.08892	0.239380	0.119690	+	0.992811i	$0.461810\pi$
$648$	0	0
$649$	−14.2933	−0.561061
$650$	32.6664	1.28128
$651$	0	0
$652$	−14.4870	−0.567356
$653$	−3.83537	−0.150090	−0.0750449	−	0.997180i	$-0.523910\pi$
$653$	−3.83537	−0.150090	−0.0750449	+	0.997180i	$0.523910\pi$
$654$	0	0
$655$	−50.8465	−1.98674
$656$	−6.46622	−0.252463
$657$	0	0
$658$	28.7018	1.11891
$659$	−42.8074	−1.66754	−0.833769	−	0.552113i	$-0.813822\pi$
$659$	−42.8074	−1.66754	−0.833769	+	0.552113i	$0.813822\pi$
$660$	0	0
$661$	3.43048	0.133430	0.0667151	−	0.997772i	$-0.478748\pi$
$661$	3.43048	0.133430	0.0667151	+	0.997772i	$0.478748\pi$
$662$	−54.9224	−2.13462
$663$	0	0
$664$	−16.4322	−0.637692
$665$	−46.5967	−1.80694
$666$	0	0
$667$	−46.0897	−1.78460
$668$	23.2314	0.898851
$669$	0	0
$670$	−72.2997	−2.79318
$671$	1.00000	0.0386046
$672$	0	0
$673$	−43.2194	−1.66599	−0.832993	−	0.553283i	$-0.813375\pi$
$673$	−43.2194	−1.66599	−0.832993	+	0.553283i	$0.813375\pi$
$674$	22.5273	0.867718
$675$	0	0
$676$	−5.76027	−0.221549
$677$	19.5997	0.753276	0.376638	−	0.926360i	$-0.377080\pi$
$677$	19.5997	0.753276	0.376638	+	0.926360i	$0.377080\pi$
$678$	0	0
$679$	34.5430	1.32564
$680$	7.88113	0.302228
$681$	0	0
$682$	8.30347	0.317956
$683$	−5.19077	−0.198619	−0.0993096	−	0.995057i	$-0.531663\pi$
$683$	−5.19077	−0.198619	−0.0993096	+	0.995057i	$0.531663\pi$
$684$	0	0
$685$	−34.6420	−1.32360
$686$	12.1341	0.463282
$687$	0	0
$688$	8.89192	0.339001
$689$	−26.4058	−1.00598
$690$	0	0
$691$	−17.2822	−0.657446	−0.328723	−	0.944426i	$-0.606618\pi$
$691$	−17.2822	−0.657446	−0.328723	+	0.944426i	$0.606618\pi$
$692$	7.58395	0.288298
$693$	0	0
$694$	−53.4187	−2.02775
$695$	52.3688	1.98646
$696$	0	0
$697$	2.96532	0.112320
$698$	17.5828	0.665520
$699$	0	0
$700$	−33.1480	−1.25288
$701$	−49.5798	−1.87260	−0.936301	−	0.351199i	$-0.885774\pi$
$701$	−49.5798	−1.87260	−0.936301	+	0.351199i	$0.885774\pi$
$702$	0	0
$703$	−20.2608	−0.764151
$704$	−2.85289	−0.107522
$705$	0	0
$706$	8.10423	0.305007
$707$	33.2966	1.25225
$708$	0	0
$709$	−11.6865	−0.438895	−0.219448	−	0.975624i	$-0.570425\pi$
$709$	−11.6865	−0.438895	−0.219448	+	0.975624i	$0.570425\pi$
$710$	−61.8320	−2.32052
$711$	0	0
$712$	4.61129	0.172815
$713$	−21.2221	−0.794773
$714$	0	0
$715$	−9.87396	−0.369265
$716$	−0.190663	−0.00712542
$717$	0	0
$718$	64.3547	2.40169
$719$	−36.7278	−1.36971	−0.684857	−	0.728677i	$-0.740136\pi$
$719$	−36.7278	−1.36971	−0.684857	+	0.728677i	$0.740136\pi$
$720$	0	0
$721$	47.5220	1.76981
$722$	11.6329	0.432932
$723$	0	0
$724$	2.76492	0.102757
$725$	−57.6792	−2.14215
$726$	0	0
$727$	40.1648	1.48963	0.744815	−	0.667270i	$-0.232537\pi$
$727$	40.1648	1.48963	0.744815	+	0.667270i	$0.232537\pi$
$728$	12.7501	0.472550
$729$	0	0
$730$	43.9602	1.62704
$731$	−4.07771	−0.150820
$732$	0	0
$733$	34.6513	1.27988	0.639938	−	0.768427i	$-0.278960\pi$
$733$	34.6513	1.27988	0.639938	+	0.768427i	$0.278960\pi$
$734$	−1.13506	−0.0418960
$735$	0	0
$736$	31.9614	1.17811
$737$	11.8404	0.436146
$738$	0	0
$739$	−45.3526	−1.66832	−0.834161	−	0.551520i	$-0.814048\pi$
$739$	−45.3526	−1.66832	−0.834161	+	0.551520i	$0.814048\pi$
$740$	−26.6023	−0.977919
$741$	0	0
$742$	64.6187	2.37223
$743$	−5.06913	−0.185968	−0.0929842	−	0.995668i	$-0.529641\pi$
$743$	−5.06913	−0.185968	−0.0929842	+	0.995668i	$0.529641\pi$
$744$	0	0
$745$	−77.8862	−2.85353
$746$	9.95888	0.364620
$747$	0	0
$748$	3.13582	0.114657
$749$	17.3346	0.633392
$750$	0	0
$751$	−29.1802	−1.06480	−0.532400	−	0.846493i	$-0.678710\pi$
$751$	−29.1802	−1.06480	−0.532400	+	0.846493i	$0.678710\pi$
$752$	−18.9376	−0.690583
$753$	0	0
$754$	−53.9024	−1.96301
$755$	−25.8711	−0.941547
$756$	0	0
$757$	37.6485	1.36836	0.684179	−	0.729314i	$-0.260161\pi$
$757$	37.6485	1.36836	0.684179	+	0.729314i	$0.260161\pi$
$758$	34.2880	1.24540
$759$	0	0
$760$	12.6933	0.460435
$761$	7.57930	0.274749	0.137375	−	0.990519i	$-0.456134\pi$
$761$	7.57930	0.274749	0.137375	+	0.990519i	$0.456134\pi$
$762$	0	0
$763$	−16.7763	−0.607343
$764$	32.6272	1.18041
$765$	0	0
$766$	15.5481	0.561774
$767$	42.7234	1.54265
$768$	0	0
$769$	−1.43737	−0.0518330	−0.0259165	−	0.999664i	$-0.508250\pi$
$769$	−1.43737	−0.0518330	−0.0259165	+	0.999664i	$0.508250\pi$
$770$	24.1630	0.870774
$771$	0	0
$772$	27.8747	1.00323
$773$	−25.8996	−0.931544	−0.465772	−	0.884905i	$-0.654223\pi$
$773$	−25.8996	−0.931544	−0.465772	+	0.884905i	$0.654223\pi$
$774$	0	0
$775$	−26.5585	−0.954009
$776$	−9.40978	−0.337791
$777$	0	0
$778$	37.9616	1.36099
$779$	4.77593	0.171115
$780$	0	0
$781$	10.1261	0.362341
$782$	−19.3279	−0.691163
$783$	0	0
$784$	−41.7899	−1.49250
$785$	10.6888	0.381499
$786$	0	0
$787$	47.0848	1.67839	0.839196	−	0.543830i	$-0.183026\pi$
$787$	47.0848	1.67839	0.839196	+	0.543830i	$0.183026\pi$
$788$	−6.10513	−0.217486
$789$	0	0
$790$	−45.7386	−1.62730
$791$	−18.3534	−0.652572
$792$	0	0
$793$	−2.98905	−0.106144
$794$	−47.5238	−1.68656
$795$	0	0
$796$	10.3981	0.368553
$797$	−18.7516	−0.664217	−0.332109	−	0.943241i	$-0.607760\pi$
$797$	−18.7516	−0.664217	−0.332109	+	0.943241i	$0.607760\pi$
$798$	0	0
$799$	8.68454	0.307237
$800$	39.9982	1.41415
$801$	0	0
$802$	49.8350	1.75974
$803$	−7.19928	−0.254057
$804$	0	0
$805$	−61.7560	−2.17661
$806$	−24.8195	−0.874229
$807$	0	0
$808$	−9.07026	−0.319091
$809$	−32.1859	−1.13160	−0.565799	−	0.824543i	$-0.691432\pi$
$809$	−32.1859	−1.13160	−0.565799	+	0.824543i	$0.691432\pi$
$810$	0	0
$811$	−50.3819	−1.76915	−0.884574	−	0.466401i	$-0.845551\pi$
$811$	−50.3819	−1.76915	−0.884574	+	0.466401i	$0.845551\pi$
$812$	54.6971	1.91949
$813$	0	0
$814$	10.5064	0.368248
$815$	−33.7766	−1.18314
$816$	0	0
$817$	−6.56755	−0.229769
$818$	21.4890	0.751344
$819$	0	0
$820$	6.27076	0.218984
$821$	42.9668	1.49955	0.749776	−	0.661692i	$-0.230161\pi$
$821$	42.9668	1.49955	0.749776	+	0.661692i	$0.230161\pi$
$822$	0	0
$823$	52.8913	1.84367	0.921837	−	0.387578i	$-0.126688\pi$
$823$	52.8913	1.84367	0.921837	+	0.387578i	$0.126688\pi$
$824$	−12.9454	−0.450973
$825$	0	0
$826$	−104.550	−3.63777
$827$	5.30584	0.184502	0.0922510	−	0.995736i	$-0.470594\pi$
$827$	5.30584	0.184502	0.0922510	+	0.995736i	$0.470594\pi$
$828$	0	0
$829$	−25.9232	−0.900349	−0.450174	−	0.892941i	$-0.648638\pi$
$829$	−25.9232	−0.900349	−0.450174	+	0.892941i	$0.648638\pi$
$830$	93.0820	3.23092
$831$	0	0
$832$	8.52744	0.295636
$833$	19.1643	0.664003
$834$	0	0
$835$	54.1643	1.87443
$836$	5.05054	0.174677
$837$	0	0
$838$	62.9804	2.17562
$839$	−11.8004	−0.407394	−0.203697	−	0.979034i	$-0.565296\pi$
$839$	−11.8004	−0.407394	−0.203697	+	0.979034i	$0.565296\pi$
$840$	0	0
$841$	66.1756	2.28192
$842$	−25.2431	−0.869934
$843$	0	0
$844$	−24.8918	−0.856810
$845$	−13.4301	−0.462011
$846$	0	0
$847$	−3.95713	−0.135969
$848$	−42.6358	−1.46412
$849$	0	0
$850$	−24.1880	−0.829640
$851$	−26.8522	−0.920483
$852$	0	0
$853$	−23.0587	−0.789514	−0.394757	−	0.918785i	$-0.629171\pi$
$853$	−23.0587	−0.789514	−0.394757	+	0.918785i	$0.629171\pi$
$854$	7.31464	0.250302
$855$	0	0
$856$	−4.72207	−0.161397
$857$	−39.0031	−1.33232	−0.666161	−	0.745808i	$-0.732064\pi$
$857$	−39.0031	−1.33232	−0.666161	+	0.745808i	$0.732064\pi$
$858$	0	0
$859$	0.832111	0.0283913	0.0141956	−	0.999899i	$-0.495481\pi$
$859$	0.832111	0.0283913	0.0141956	+	0.999899i	$0.495481\pi$
$860$	−8.62313	−0.294046
$861$	0	0
$862$	50.0783	1.70567
$863$	−10.3690	−0.352965	−0.176482	−	0.984304i	$-0.556472\pi$
$863$	−10.3690	−0.352965	−0.176482	+	0.984304i	$0.556472\pi$
$864$	0	0
$865$	17.6820	0.601207
$866$	8.17548	0.277814
$867$	0	0
$868$	25.1854	0.854847
$869$	7.49052	0.254099
$870$	0	0
$871$	−35.3915	−1.19920
$872$	4.57000	0.154760
$873$	0	0
$874$	−31.1294	−1.05297
$875$	−11.9254	−0.403153
$876$	0	0
$877$	50.6809	1.71137	0.855686	−	0.517496i	$-0.173136\pi$
$877$	50.6809	1.71137	0.855686	+	0.517496i	$0.173136\pi$
$878$	45.2785	1.52807
$879$	0	0
$880$	−15.9429	−0.537435
$881$	−16.3346	−0.550327	−0.275164	−	0.961397i	$-0.588732\pi$
$881$	−16.3346	−0.550327	−0.275164	+	0.961397i	$0.588732\pi$
$882$	0	0
$883$	23.1354	0.778569	0.389285	−	0.921118i	$-0.372722\pi$
$883$	23.1354	0.778569	0.389285	+	0.921118i	$0.372722\pi$
$884$	−9.37314	−0.315253
$885$	0	0
$886$	26.3647	0.885739
$887$	−29.4067	−0.987379	−0.493689	−	0.869638i	$-0.664352\pi$
$887$	−29.4067	−0.987379	−0.493689	+	0.869638i	$0.664352\pi$
$888$	0	0
$889$	19.2633	0.646072
$890$	−26.1212	−0.875585
$891$	0	0
$892$	−26.9411	−0.902054
$893$	13.9873	0.468066
$894$	0	0
$895$	−0.444533	−0.0148591
$896$	32.6742	1.09157
$897$	0	0
$898$	−1.08238	−0.0361194
$899$	43.8238	1.46161
$900$	0	0
$901$	19.5522	0.651379
$902$	−2.47659	−0.0824613
$903$	0	0
$904$	4.99961	0.166284
$905$	6.44643	0.214287
$906$	0	0
$907$	−42.3225	−1.40530	−0.702648	−	0.711537i	$-0.747999\pi$
$907$	−42.3225	−1.40530	−0.702648	+	0.711537i	$0.747999\pi$
$908$	29.8541	0.990742
$909$	0	0
$910$	−72.2244	−2.39422
$911$	26.5480	0.879575	0.439788	−	0.898102i	$-0.355054\pi$
$911$	26.5480	0.879575	0.439788	+	0.898102i	$0.355054\pi$
$912$	0	0
$913$	−15.2439	−0.504499
$914$	−32.3008	−1.06842
$915$	0	0
$916$	−8.77317	−0.289874
$917$	60.9092	2.01140
$918$	0	0
$919$	−20.5096	−0.676550	−0.338275	−	0.941047i	$-0.609843\pi$
$919$	−20.5096	−0.676550	−0.338275	+	0.941047i	$0.609843\pi$
$920$	16.8228	0.554632
$921$	0	0
$922$	−37.9540	−1.24995
$923$	−30.2675	−0.996267
$924$	0	0
$925$	−33.6044	−1.10491
$926$	−16.4160	−0.539464
$927$	0	0
$928$	−66.0005	−2.16657
$929$	−56.6319	−1.85803	−0.929016	−	0.370039i	$-0.879344\pi$
$929$	−56.6319	−1.85803	−0.929016	+	0.370039i	$0.879344\pi$
$930$	0	0
$931$	30.8659	1.01159
$932$	−14.8662	−0.486959
$933$	0	0
$934$	−6.41295	−0.209838
$935$	7.31120	0.239102
$936$	0	0
$937$	21.0886	0.688934	0.344467	−	0.938798i	$-0.388060\pi$
$937$	21.0886	0.688934	0.344467	+	0.938798i	$0.388060\pi$
$938$	86.6081	2.82786
$939$	0	0
$940$	18.3652	0.599006
$941$	−16.2132	−0.528536	−0.264268	−	0.964449i	$-0.585130\pi$
$941$	−16.2132	−0.528536	−0.264268	+	0.964449i	$0.585130\pi$
$942$	0	0
$943$	6.32968	0.206123
$944$	68.9830	2.24520
$945$	0	0
$946$	3.40564	0.110727
$947$	7.65428	0.248731	0.124365	−	0.992236i	$-0.460311\pi$
$947$	7.65428	0.248731	0.124365	+	0.992236i	$0.460311\pi$
$948$	0	0
$949$	21.5190	0.698537
$950$	−38.9570	−1.26393
$951$	0	0
$952$	−9.44084	−0.305980
$953$	−19.9857	−0.647400	−0.323700	−	0.946160i	$-0.604927\pi$
$953$	−19.9857	−0.647400	−0.323700	+	0.946160i	$0.604927\pi$
$954$	0	0
$955$	76.0707	2.46159
$956$	−10.7598	−0.347998
$957$	0	0
$958$	29.6033	0.956439
$959$	41.4979	1.34004
$960$	0	0
$961$	−10.8212	−0.349073
$962$	−31.4040	−1.01251
$963$	0	0
$964$	−23.9373	−0.770970
$965$	64.9900	2.09210
$966$	0	0
$967$	32.9320	1.05902	0.529511	−	0.848303i	$-0.322376\pi$
$967$	32.9320	1.05902	0.529511	+	0.848303i	$0.322376\pi$
$968$	1.07795	0.0346467
$969$	0	0
$970$	53.3028	1.71145
$971$	28.1643	0.903834	0.451917	−	0.892060i	$-0.350740\pi$
$971$	28.1643	0.903834	0.451917	+	0.892060i	$0.350740\pi$
$972$	0	0
$973$	−62.7328	−2.01112
$974$	−13.1695	−0.421978
$975$	0	0
$976$	−4.82624	−0.154484
$977$	−33.0812	−1.05836	−0.529180	−	0.848510i	$-0.677500\pi$
$977$	−33.0812	−1.05836	−0.529180	+	0.848510i	$0.677500\pi$
$978$	0	0
$979$	4.27782	0.136720
$980$	40.5267	1.29458
$981$	0	0
$982$	45.5108	1.45231
$983$	−51.5783	−1.64509	−0.822547	−	0.568698i	$-0.807447\pi$
$983$	−51.5783	−1.64509	−0.822547	+	0.568698i	$0.807447\pi$
$984$	0	0
$985$	−14.2342	−0.453538
$986$	39.9122	1.27106
$987$	0	0
$988$	−15.0963	−0.480278
$989$	−8.70416	−0.276776
$990$	0	0
$991$	44.8494	1.42469	0.712345	−	0.701830i	$-0.247633\pi$
$991$	44.8494	1.42469	0.712345	+	0.701830i	$0.247633\pi$
$992$	−30.3901	−0.964886
$993$	0	0
$994$	74.0689	2.34932
$995$	24.2434	0.768567
$996$	0	0
$997$	−26.9286	−0.852838	−0.426419	−	0.904526i	$-0.640225\pi$
$997$	−26.9286	−0.852838	−0.426419	+	0.904526i	$0.640225\pi$
$998$	−65.9008	−2.08605
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.7	✓	25
3.2	odd	2		6039.2.a.p.1.19	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.7	✓	25	1.1	even	1	trivial
6039.2.a.p.1.19	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)

\(f(q)\)	\(=\)	\(q-1.84847 q^{2} +1.41684 q^{4} +3.30338 q^{5} -3.95713 q^{7} +1.07795 q^{8} +O(q^{10})\)	\(q-1.84847 q^{2} +1.41684 q^{4} +3.30338 q^{5} -3.95713 q^{7} +1.07795 q^{8} -6.10619 q^{10} +1.00000 q^{11} -2.98905 q^{13} +7.31464 q^{14} -4.82624 q^{16} +2.21325 q^{17} +3.56465 q^{19} +4.68036 q^{20} -1.84847 q^{22} +4.72434 q^{23} +5.91229 q^{25} +5.52517 q^{26} -5.60662 q^{28} -9.75580 q^{29} -4.49208 q^{31} +6.76526 q^{32} -4.09113 q^{34} -13.0719 q^{35} -5.68381 q^{37} -6.58915 q^{38} +3.56088 q^{40} +1.33980 q^{41} -1.84241 q^{43} +1.41684 q^{44} -8.73279 q^{46} +3.92388 q^{47} +8.65888 q^{49} -10.9287 q^{50} -4.23501 q^{52} +8.83416 q^{53} +3.30338 q^{55} -4.26560 q^{56} +18.0333 q^{58} -14.2933 q^{59} +1.00000 q^{61} +8.30347 q^{62} -2.85289 q^{64} -9.87396 q^{65} +11.8404 q^{67} +3.13582 q^{68} +24.1630 q^{70} +10.1261 q^{71} -7.19928 q^{73} +10.5064 q^{74} +5.05054 q^{76} -3.95713 q^{77} +7.49052 q^{79} -15.9429 q^{80} -2.47659 q^{82} -15.2439 q^{83} +7.31120 q^{85} +3.40564 q^{86} +1.07795 q^{88} +4.27782 q^{89} +11.8281 q^{91} +6.69363 q^{92} -7.25317 q^{94} +11.7754 q^{95} -8.72931 q^{97} -16.0057 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 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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