LMFDB - Embedded newform 6039.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57687 q^{2} +4.64028 q^{4} +2.12696 q^{5} -3.45468 q^{7} -6.80367 q^{8} +O(q^{10})$	\(q-2.57687 q^{2} +4.64028 q^{4} +2.12696 q^{5} -3.45468 q^{7} -6.80367 q^{8} -5.48091 q^{10} -1.00000 q^{11} -2.79669 q^{13} +8.90228 q^{14} +8.25164 q^{16} -6.16482 q^{17} -2.59340 q^{19} +9.86969 q^{20} +2.57687 q^{22} +0.281731 q^{23} -0.476043 q^{25} +7.20672 q^{26} -16.0307 q^{28} -0.776820 q^{29} +3.56496 q^{31} -7.65609 q^{32} +15.8860 q^{34} -7.34797 q^{35} -11.5199 q^{37} +6.68285 q^{38} -14.4711 q^{40} +10.0908 q^{41} -4.63845 q^{43} -4.64028 q^{44} -0.725985 q^{46} -0.572400 q^{47} +4.93483 q^{49} +1.22670 q^{50} -12.9774 q^{52} -1.09426 q^{53} -2.12696 q^{55} +23.5045 q^{56} +2.00177 q^{58} +10.9151 q^{59} +1.00000 q^{61} -9.18644 q^{62} +3.22551 q^{64} -5.94845 q^{65} -11.7199 q^{67} -28.6065 q^{68} +18.9348 q^{70} +0.0926970 q^{71} -3.82910 q^{73} +29.6853 q^{74} -12.0341 q^{76} +3.45468 q^{77} -8.59873 q^{79} +17.5509 q^{80} -26.0028 q^{82} -2.92218 q^{83} -13.1123 q^{85} +11.9527 q^{86} +6.80367 q^{88} -5.34082 q^{89} +9.66168 q^{91} +1.30731 q^{92} +1.47500 q^{94} -5.51605 q^{95} +4.05883 q^{97} -12.7164 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.57687	−1.82213	−0.911063	−	0.412268i	$-0.864737\pi$
$2$	−2.57687	−1.82213	−0.911063	+	0.412268i	$0.864737\pi$
$3$	0	0
$3$	0	0
$4$	4.64028	2.32014
$5$	2.12696	0.951205	0.475603	−	0.879660i	$-0.342230\pi$
$5$	2.12696	0.951205	0.475603	+	0.879660i	$0.342230\pi$
$6$	0	0
$7$	−3.45468	−1.30575	−0.652874	−	0.757467i	$-0.726437\pi$
$7$	−3.45468	−1.30575	−0.652874	+	0.757467i	$0.726437\pi$
$8$	−6.80367	−2.40546
$9$	0	0
$10$	−5.48091	−1.73321
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.79669	−0.775663	−0.387831	−	0.921730i	$-0.626776\pi$
$13$	−2.79669	−0.775663	−0.387831	+	0.921730i	$0.626776\pi$
$14$	8.90228	2.37923
$15$	0	0
$16$	8.25164	2.06291
$17$	−6.16482	−1.49519	−0.747594	−	0.664156i	$-0.768791\pi$
$17$	−6.16482	−1.49519	−0.747594	+	0.664156i	$0.768791\pi$
$18$	0	0
$19$	−2.59340	−0.594966	−0.297483	−	0.954727i	$-0.596147\pi$
$19$	−2.59340	−0.594966	−0.297483	+	0.954727i	$0.596147\pi$
$20$	9.86969	2.20693
$21$	0	0
$22$	2.57687	0.549391
$23$	0.281731	0.0587450	0.0293725	−	0.999569i	$-0.490649\pi$
$23$	0.281731	0.0587450	0.0293725	+	0.999569i	$0.490649\pi$
$24$	0	0
$25$	−0.476043	−0.0952086
$26$	7.20672	1.41335
$27$	0	0
$28$	−16.0307	−3.02952
$29$	−0.776820	−0.144252	−0.0721260	−	0.997396i	$-0.522978\pi$
$29$	−0.776820	−0.144252	−0.0721260	+	0.997396i	$0.522978\pi$
$30$	0	0
$31$	3.56496	0.640285	0.320143	−	0.947369i	$-0.396269\pi$
$31$	3.56496	0.640285	0.320143	+	0.947369i	$0.396269\pi$
$32$	−7.65609	−1.35342
$33$	0	0
$34$	15.8860	2.72442
$35$	−7.34797	−1.24203
$36$	0	0
$37$	−11.5199	−1.89386	−0.946929	−	0.321444i	$-0.895832\pi$
$37$	−11.5199	−1.89386	−0.946929	+	0.321444i	$0.895832\pi$
$38$	6.68285	1.08410
$39$	0	0
$40$	−14.4711	−2.28809
$41$	10.0908	1.57593	0.787963	−	0.615723i	$-0.211136\pi$
$41$	10.0908	1.57593	0.787963	+	0.615723i	$0.211136\pi$
$42$	0	0
$43$	−4.63845	−0.707358	−0.353679	−	0.935367i	$-0.615069\pi$
$43$	−4.63845	−0.707358	−0.353679	+	0.935367i	$0.615069\pi$
$44$	−4.64028	−0.699548
$45$	0	0
$46$	−0.725985	−0.107041
$47$	−0.572400	−0.0834931	−0.0417465	−	0.999128i	$-0.513292\pi$
$47$	−0.572400	−0.0834931	−0.0417465	+	0.999128i	$0.513292\pi$
$48$	0	0
$49$	4.93483	0.704975
$50$	1.22670	0.173482
$51$	0	0
$52$	−12.9774	−1.79965
$53$	−1.09426	−0.150308	−0.0751541	−	0.997172i	$-0.523945\pi$
$53$	−1.09426	−0.150308	−0.0751541	+	0.997172i	$0.523945\pi$
$54$	0	0
$55$	−2.12696	−0.286799
$56$	23.5045	3.14092
$57$	0	0
$58$	2.00177	0.262845
$59$	10.9151	1.42103	0.710516	−	0.703681i	$-0.248462\pi$
$59$	10.9151	1.42103	0.710516	+	0.703681i	$0.248462\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−9.18644	−1.16668
$63$	0	0
$64$	3.22551	0.403189
$65$	−5.94845	−0.737814
$66$	0	0
$67$	−11.7199	−1.43181	−0.715907	−	0.698195i	$-0.753987\pi$
$67$	−11.7199	−1.43181	−0.715907	+	0.698195i	$0.753987\pi$
$68$	−28.6065	−3.46904
$69$	0	0
$70$	18.9348	2.26314
$71$	0.0926970	0.0110011	0.00550056	−	0.999985i	$-0.498249\pi$
$71$	0.0926970	0.0110011	0.00550056	+	0.999985i	$0.498249\pi$
$72$	0	0
$73$	−3.82910	−0.448162	−0.224081	−	0.974571i	$-0.571938\pi$
$73$	−3.82910	−0.448162	−0.224081	+	0.974571i	$0.571938\pi$
$74$	29.6853	3.45084
$75$	0	0
$76$	−12.0341	−1.38040
$77$	3.45468	0.393698
$78$	0	0
$79$	−8.59873	−0.967432	−0.483716	−	0.875225i	$-0.660713\pi$
$79$	−8.59873	−0.967432	−0.483716	+	0.875225i	$0.660713\pi$
$80$	17.5509	1.96225
$81$	0	0
$82$	−26.0028	−2.87153
$83$	−2.92218	−0.320751	−0.160375	−	0.987056i	$-0.551270\pi$
$83$	−2.92218	−0.320751	−0.160375	+	0.987056i	$0.551270\pi$
$84$	0	0
$85$	−13.1123	−1.42223
$86$	11.9527	1.28889
$87$	0	0
$88$	6.80367	0.725273
$89$	−5.34082	−0.566126	−0.283063	−	0.959101i	$-0.591351\pi$
$89$	−5.34082	−0.566126	−0.283063	+	0.959101i	$0.591351\pi$
$90$	0	0
$91$	9.66168	1.01282
$92$	1.30731	0.136297
$93$	0	0
$94$	1.47500	0.152135
$95$	−5.51605	−0.565935
$96$	0	0
$97$	4.05883	0.412112	0.206056	−	0.978540i	$-0.433937\pi$
$97$	4.05883	0.412112	0.206056	+	0.978540i	$0.433937\pi$
$98$	−12.7164	−1.28455
$99$	0	0
$100$	−2.20897	−0.220897
$101$	2.29830	0.228689	0.114345	−	0.993441i	$-0.463523\pi$
$101$	2.29830	0.228689	0.114345	+	0.993441i	$0.463523\pi$
$102$	0	0
$103$	−0.915633	−0.0902200	−0.0451100	−	0.998982i	$-0.514364\pi$
$103$	−0.915633	−0.0902200	−0.0451100	+	0.998982i	$0.514364\pi$
$104$	19.0278	1.86583
$105$	0	0
$106$	2.81977	0.273880
$107$	−3.41715	−0.330348	−0.165174	−	0.986264i	$-0.552819\pi$
$107$	−3.41715	−0.330348	−0.165174	+	0.986264i	$0.552819\pi$
$108$	0	0
$109$	1.10273	0.105623	0.0528113	−	0.998605i	$-0.483182\pi$
$109$	1.10273	0.105623	0.0528113	+	0.998605i	$0.483182\pi$
$110$	5.48091	0.522584
$111$	0	0
$112$	−28.5068	−2.69364
$113$	14.7956	1.39185	0.695926	−	0.718113i	$-0.254994\pi$
$113$	14.7956	1.39185	0.695926	+	0.718113i	$0.254994\pi$
$114$	0	0
$115$	0.599231	0.0558785
$116$	−3.60466	−0.334685
$117$	0	0
$118$	−28.1270	−2.58930
$119$	21.2975	1.95234
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.57687	−0.233299
$123$	0	0
$124$	16.5424	1.48555
$125$	−11.6473	−1.04177
$126$	0	0
$127$	15.9580	1.41604	0.708021	−	0.706191i	$-0.249588\pi$
$127$	15.9580	1.41604	0.708021	+	0.706191i	$0.249588\pi$
$128$	7.00045	0.618758
$129$	0	0
$130$	15.3284	1.34439
$131$	4.18518	0.365661	0.182830	−	0.983144i	$-0.441474\pi$
$131$	4.18518	0.365661	0.182830	+	0.983144i	$0.441474\pi$
$132$	0	0
$133$	8.95936	0.776875
$134$	30.2007	2.60895
$135$	0	0
$136$	41.9434	3.59661
$137$	21.6406	1.84888	0.924439	−	0.381330i	$-0.124534\pi$
$137$	21.6406	1.84888	0.924439	+	0.381330i	$0.124534\pi$
$138$	0	0
$139$	12.0339	1.02070	0.510350	−	0.859967i	$-0.329516\pi$
$139$	12.0339	1.02070	0.510350	+	0.859967i	$0.329516\pi$
$140$	−34.0966	−2.88169
$141$	0	0
$142$	−0.238869	−0.0200454
$143$	2.79669	0.233871
$144$	0	0
$145$	−1.65227	−0.137213
$146$	9.86710	0.816607
$147$	0	0
$148$	−53.4555	−4.39401
$149$	11.4815	0.940602	0.470301	−	0.882506i	$-0.344145\pi$
$149$	11.4815	0.940602	0.470301	+	0.882506i	$0.344145\pi$
$150$	0	0
$151$	−14.2375	−1.15863	−0.579314	−	0.815104i	$-0.696680\pi$
$151$	−14.2375	−1.15863	−0.579314	+	0.815104i	$0.696680\pi$
$152$	17.6446	1.43117
$153$	0	0
$154$	−8.90228	−0.717366
$155$	7.58252	0.609043
$156$	0	0
$157$	16.4211	1.31055	0.655274	−	0.755391i	$-0.272553\pi$
$157$	16.4211	1.31055	0.655274	+	0.755391i	$0.272553\pi$
$158$	22.1578	1.76278
$159$	0	0
$160$	−16.2842	−1.28738
$161$	−0.973291	−0.0767061
$162$	0	0
$163$	1.35630	0.106234	0.0531169	−	0.998588i	$-0.483084\pi$
$163$	1.35630	0.106234	0.0531169	+	0.998588i	$0.483084\pi$
$164$	46.8244	3.65637
$165$	0	0
$166$	7.53008	0.584448
$167$	7.38317	0.571327	0.285663	−	0.958330i	$-0.407786\pi$
$167$	7.38317	0.571327	0.285663	+	0.958330i	$0.407786\pi$
$168$	0	0
$169$	−5.17852	−0.398347
$170$	33.7888	2.59148
$171$	0	0
$172$	−21.5237	−1.64117
$173$	−17.4457	−1.32637	−0.663186	−	0.748455i	$-0.730796\pi$
$173$	−17.4457	−1.32637	−0.663186	+	0.748455i	$0.730796\pi$
$174$	0	0
$175$	1.64458	0.124318
$176$	−8.25164	−0.621991
$177$	0	0
$178$	13.7626	1.03155
$179$	−18.5898	−1.38946	−0.694732	−	0.719269i	$-0.744477\pi$
$179$	−18.5898	−1.38946	−0.694732	+	0.719269i	$0.744477\pi$
$180$	0	0
$181$	24.8458	1.84677	0.923385	−	0.383874i	$-0.125410\pi$
$181$	24.8458	1.84677	0.923385	+	0.383874i	$0.125410\pi$
$182$	−24.8969	−1.84548
$183$	0	0
$184$	−1.91680	−0.141309
$185$	−24.5023	−1.80145
$186$	0	0
$187$	6.16482	0.450816
$188$	−2.65610	−0.193716
$189$	0	0
$190$	14.2142	1.03120
$191$	−0.953589	−0.0689993	−0.0344996	−	0.999405i	$-0.510984\pi$
$191$	−0.953589	−0.0689993	−0.0344996	+	0.999405i	$0.510984\pi$
$192$	0	0
$193$	26.9685	1.94123	0.970617	−	0.240630i	$-0.0773540\pi$
$193$	26.9685	1.94123	0.970617	+	0.240630i	$0.0773540\pi$
$194$	−10.4591	−0.750919
$195$	0	0
$196$	22.8990	1.63564
$197$	−3.15363	−0.224687	−0.112343	−	0.993669i	$-0.535836\pi$
$197$	−3.15363	−0.224687	−0.112343	+	0.993669i	$0.535836\pi$
$198$	0	0
$199$	18.0473	1.27934	0.639669	−	0.768651i	$-0.279072\pi$
$199$	18.0473	1.27934	0.639669	+	0.768651i	$0.279072\pi$
$200$	3.23884	0.229020
$201$	0	0
$202$	−5.92242	−0.416700
$203$	2.68367	0.188357
$204$	0	0
$205$	21.4628	1.49903
$206$	2.35947	0.164392
$207$	0	0
$208$	−23.0773	−1.60012
$209$	2.59340	0.179389
$210$	0	0
$211$	−20.2850	−1.39648	−0.698238	−	0.715866i	$-0.746032\pi$
$211$	−20.2850	−1.39648	−0.698238	+	0.715866i	$0.746032\pi$
$212$	−5.07767	−0.348736
$213$	0	0
$214$	8.80556	0.601935
$215$	−9.86580	−0.672842
$216$	0	0
$217$	−12.3158	−0.836050
$218$	−2.84160	−0.192458
$219$	0	0
$220$	−9.86969	−0.665414
$221$	17.2411	1.15976
$222$	0	0
$223$	−6.13053	−0.410531	−0.205265	−	0.978706i	$-0.565806\pi$
$223$	−6.13053	−0.410531	−0.205265	+	0.978706i	$0.565806\pi$
$224$	26.4494	1.76722
$225$	0	0
$226$	−38.1264	−2.53613
$227$	4.60955	0.305947	0.152973	−	0.988230i	$-0.451115\pi$
$227$	4.60955	0.305947	0.152973	+	0.988230i	$0.451115\pi$
$228$	0	0
$229$	−7.09755	−0.469019	−0.234510	−	0.972114i	$-0.575348\pi$
$229$	−7.09755	−0.469019	−0.234510	+	0.972114i	$0.575348\pi$
$230$	−1.54414	−0.101818
$231$	0	0
$232$	5.28523	0.346992
$233$	20.0906	1.31618	0.658089	−	0.752940i	$-0.271365\pi$
$233$	20.0906	1.31618	0.658089	+	0.752940i	$0.271365\pi$
$234$	0	0
$235$	−1.21747	−0.0794191
$236$	50.6493	3.29699
$237$	0	0
$238$	−54.8809	−3.55740
$239$	−17.1900	−1.11193	−0.555966	−	0.831205i	$-0.687651\pi$
$239$	−17.1900	−1.11193	−0.555966	+	0.831205i	$0.687651\pi$
$240$	0	0
$241$	−11.5805	−0.745964	−0.372982	−	0.927839i	$-0.621665\pi$
$241$	−11.5805	−0.745964	−0.372982	+	0.927839i	$0.621665\pi$
$242$	−2.57687	−0.165648
$243$	0	0
$244$	4.64028	0.297063
$245$	10.4962	0.670576
$246$	0	0
$247$	7.25293	0.461493
$248$	−24.2548	−1.54018
$249$	0	0
$250$	30.0137	1.89823
$251$	12.3206	0.777668	0.388834	−	0.921308i	$-0.372878\pi$
$251$	12.3206	0.777668	0.388834	+	0.921308i	$0.372878\pi$
$252$	0	0
$253$	−0.281731	−0.0177123
$254$	−41.1217	−2.58021
$255$	0	0
$256$	−24.4903	−1.53064
$257$	2.92187	0.182262	0.0911308	−	0.995839i	$-0.470952\pi$
$257$	2.92187	0.182262	0.0911308	+	0.995839i	$0.470952\pi$
$258$	0	0
$259$	39.7975	2.47290
$260$	−27.6025	−1.71183
$261$	0	0
$262$	−10.7847	−0.666280
$263$	3.37206	0.207930	0.103965	−	0.994581i	$-0.466847\pi$
$263$	3.37206	0.207930	0.103965	+	0.994581i	$0.466847\pi$
$264$	0	0
$265$	−2.32745	−0.142974
$266$	−23.0871	−1.41556
$267$	0	0
$268$	−54.3836	−3.32201
$269$	5.18059	0.315866	0.157933	−	0.987450i	$-0.449517\pi$
$269$	5.18059	0.315866	0.157933	+	0.987450i	$0.449517\pi$
$270$	0	0
$271$	7.60756	0.462126	0.231063	−	0.972939i	$-0.425780\pi$
$271$	7.60756	0.462126	0.231063	+	0.972939i	$0.425780\pi$
$272$	−50.8698	−3.08444
$273$	0	0
$274$	−55.7650	−3.36889
$275$	0.476043	0.0287065
$276$	0	0
$277$	7.94120	0.477140	0.238570	−	0.971125i	$-0.423321\pi$
$277$	7.94120	0.477140	0.238570	+	0.971125i	$0.423321\pi$
$278$	−31.0098	−1.85984
$279$	0	0
$280$	49.9931	2.98766
$281$	−9.28524	−0.553911	−0.276956	−	0.960883i	$-0.589325\pi$
$281$	−9.28524	−0.553911	−0.276956	+	0.960883i	$0.589325\pi$
$282$	0	0
$283$	3.61981	0.215176	0.107588	−	0.994196i	$-0.465687\pi$
$283$	3.61981	0.215176	0.107588	+	0.994196i	$0.465687\pi$
$284$	0.430140	0.0255241
$285$	0	0
$286$	−7.20672	−0.426142
$287$	−34.8607	−2.05776
$288$	0	0
$289$	21.0050	1.23559
$290$	4.25768	0.250020
$291$	0	0
$292$	−17.7681	−1.03980
$293$	−6.40672	−0.374285	−0.187142	−	0.982333i	$-0.559923\pi$
$293$	−6.40672	−0.374285	−0.187142	+	0.982333i	$0.559923\pi$
$294$	0	0
$295$	23.2161	1.35169
$296$	78.3775	4.55560
$297$	0	0
$298$	−29.5864	−1.71389
$299$	−0.787915	−0.0455663
$300$	0	0
$301$	16.0244	0.923630
$302$	36.6882	2.11117
$303$	0	0
$304$	−21.3998	−1.22736
$305$	2.12696	0.121789
$306$	0	0
$307$	−3.47463	−0.198308	−0.0991538	−	0.995072i	$-0.531614\pi$
$307$	−3.47463	−0.198308	−0.0991538	+	0.995072i	$0.531614\pi$
$308$	16.0307	0.913433
$309$	0	0
$310$	−19.5392	−1.10975
$311$	−10.4890	−0.594774	−0.297387	−	0.954757i	$-0.596115\pi$
$311$	−10.4890	−0.594774	−0.297387	+	0.954757i	$0.596115\pi$
$312$	0	0
$313$	5.56175	0.314369	0.157184	−	0.987569i	$-0.449758\pi$
$313$	5.56175	0.314369	0.157184	+	0.987569i	$0.449758\pi$
$314$	−42.3152	−2.38798
$315$	0	0
$316$	−39.9005	−2.24458
$317$	−29.4912	−1.65639	−0.828195	−	0.560440i	$-0.810632\pi$
$317$	−29.4912	−1.65639	−0.828195	+	0.560440i	$0.810632\pi$
$318$	0	0
$319$	0.776820	0.0434936
$320$	6.86053	0.383515
$321$	0	0
$322$	2.50805	0.139768
$323$	15.9878	0.889586
$324$	0	0
$325$	1.33135	0.0738497
$326$	−3.49502	−0.193571
$327$	0	0
$328$	−68.6548	−3.79083
$329$	1.97746	0.109021
$330$	0	0
$331$	−9.48240	−0.521200	−0.260600	−	0.965447i	$-0.583920\pi$
$331$	−9.48240	−0.521200	−0.260600	+	0.965447i	$0.583920\pi$
$332$	−13.5597	−0.744186
$333$	0	0
$334$	−19.0255	−1.04103
$335$	−24.9278	−1.36195
$336$	0	0
$337$	13.8690	0.755493	0.377746	−	0.925909i	$-0.376699\pi$
$337$	13.8690	0.755493	0.377746	+	0.925909i	$0.376699\pi$
$338$	13.3444	0.725839
$339$	0	0
$340$	−60.8448	−3.29977
$341$	−3.56496	−0.193053
$342$	0	0
$343$	7.13452	0.385228
$344$	31.5585	1.70152
$345$	0	0
$346$	44.9554	2.41682
$347$	−29.8971	−1.60496	−0.802481	−	0.596677i	$-0.796487\pi$
$347$	−29.8971	−1.60496	−0.802481	+	0.596677i	$0.796487\pi$
$348$	0	0
$349$	15.5961	0.834838	0.417419	−	0.908714i	$-0.362935\pi$
$349$	15.5961	0.834838	0.417419	+	0.908714i	$0.362935\pi$
$350$	−4.23787	−0.226524
$351$	0	0
$352$	7.65609	0.408071
$353$	30.7972	1.63917	0.819585	−	0.572958i	$-0.194204\pi$
$353$	30.7972	1.63917	0.819585	+	0.572958i	$0.194204\pi$
$354$	0	0
$355$	0.197163	0.0104643
$356$	−24.7829	−1.31349
$357$	0	0
$358$	47.9034	2.53178
$359$	22.2398	1.17377	0.586886	−	0.809670i	$-0.300354\pi$
$359$	22.2398	1.17377	0.586886	+	0.809670i	$0.300354\pi$
$360$	0	0
$361$	−12.2743	−0.646016
$362$	−64.0244	−3.36505
$363$	0	0
$364$	44.8329	2.34988
$365$	−8.14433	−0.426294
$366$	0	0
$367$	29.9822	1.56506	0.782528	−	0.622615i	$-0.213930\pi$
$367$	29.9822	1.56506	0.782528	+	0.622615i	$0.213930\pi$
$368$	2.32474	0.121186
$369$	0	0
$370$	63.1394	3.28246
$371$	3.78032	0.196265
$372$	0	0
$373$	−19.0592	−0.986848	−0.493424	−	0.869789i	$-0.664255\pi$
$373$	−19.0592	−0.986848	−0.493424	+	0.869789i	$0.664255\pi$
$374$	−15.8860	−0.821443
$375$	0	0
$376$	3.89442	0.200839
$377$	2.17253	0.111891
$378$	0	0
$379$	−9.53824	−0.489946	−0.244973	−	0.969530i	$-0.578779\pi$
$379$	−9.53824	−0.489946	−0.244973	+	0.969530i	$0.578779\pi$
$380$	−25.5960	−1.31305
$381$	0	0
$382$	2.45728	0.125725
$383$	−14.5329	−0.742595	−0.371297	−	0.928514i	$-0.621087\pi$
$383$	−14.5329	−0.742595	−0.371297	+	0.928514i	$0.621087\pi$
$384$	0	0
$385$	7.34797	0.374487
$386$	−69.4944	−3.53717
$387$	0	0
$388$	18.8341	0.956157
$389$	−29.9711	−1.51959	−0.759797	−	0.650160i	$-0.774702\pi$
$389$	−29.9711	−1.51959	−0.759797	+	0.650160i	$0.774702\pi$
$390$	0	0
$391$	−1.73682	−0.0878348
$392$	−33.5749	−1.69579
$393$	0	0
$394$	8.12651	0.409408
$395$	−18.2891	−0.920227
$396$	0	0
$397$	−14.0267	−0.703981	−0.351990	−	0.936004i	$-0.614495\pi$
$397$	−14.0267	−0.703981	−0.351990	+	0.936004i	$0.614495\pi$
$398$	−46.5056	−2.33111
$399$	0	0
$400$	−3.92813	−0.196407
$401$	29.4333	1.46983	0.734914	−	0.678160i	$-0.237222\pi$
$401$	29.4333	1.46983	0.734914	+	0.678160i	$0.237222\pi$
$402$	0	0
$403$	−9.97008	−0.496645
$404$	10.6647	0.530591
$405$	0	0
$406$	−6.91547	−0.343209
$407$	11.5199	0.571019
$408$	0	0
$409$	−28.2125	−1.39502	−0.697509	−	0.716576i	$-0.745708\pi$
$409$	−28.2125	−1.39502	−0.697509	+	0.716576i	$0.745708\pi$
$410$	−55.3070	−2.73142
$411$	0	0
$412$	−4.24879	−0.209323
$413$	−37.7084	−1.85551
$414$	0	0
$415$	−6.21535	−0.305100
$416$	21.4117	1.04980
$417$	0	0
$418$	−6.68285	−0.326869
$419$	−30.8579	−1.50751	−0.753754	−	0.657157i	$-0.771759\pi$
$419$	−30.8579	−1.50751	−0.753754	+	0.657157i	$0.771759\pi$
$420$	0	0
$421$	−18.3011	−0.891943	−0.445971	−	0.895047i	$-0.647142\pi$
$421$	−18.3011	−0.891943	−0.445971	+	0.895047i	$0.647142\pi$
$422$	52.2718	2.54455
$423$	0	0
$424$	7.44498	0.361560
$425$	2.93472	0.142355
$426$	0	0
$427$	−3.45468	−0.167184
$428$	−15.8565	−0.766454
$429$	0	0
$430$	25.4229	1.22600
$431$	31.7246	1.52812	0.764060	−	0.645145i	$-0.223203\pi$
$431$	31.7246	1.52812	0.764060	+	0.645145i	$0.223203\pi$
$432$	0	0
$433$	0.271675	0.0130559	0.00652794	−	0.999979i	$-0.497922\pi$
$433$	0.271675	0.0130559	0.00652794	+	0.999979i	$0.497922\pi$
$434$	31.7362	1.52339
$435$	0	0
$436$	5.11699	0.245059
$437$	−0.730640	−0.0349513
$438$	0	0
$439$	3.87321	0.184858	0.0924291	−	0.995719i	$-0.470537\pi$
$439$	3.87321	0.184858	0.0924291	+	0.995719i	$0.470537\pi$
$440$	14.4711	0.689884
$441$	0	0
$442$	−44.4281	−2.11323
$443$	7.33241	0.348373	0.174187	−	0.984713i	$-0.444270\pi$
$443$	7.33241	0.348373	0.174187	+	0.984713i	$0.444270\pi$
$444$	0	0
$445$	−11.3597	−0.538502
$446$	15.7976	0.748038
$447$	0	0
$448$	−11.1431	−0.526463
$449$	24.0552	1.13524	0.567618	−	0.823292i	$-0.307865\pi$
$449$	24.0552	1.13524	0.567618	+	0.823292i	$0.307865\pi$
$450$	0	0
$451$	−10.0908	−0.475159
$452$	68.6557	3.22929
$453$	0	0
$454$	−11.8782	−0.557473
$455$	20.5500	0.963399
$456$	0	0
$457$	20.9079	0.978031	0.489015	−	0.872275i	$-0.337356\pi$
$457$	20.9079	0.978031	0.489015	+	0.872275i	$0.337356\pi$
$458$	18.2895	0.854612
$459$	0	0
$460$	2.78060	0.129646
$461$	−19.3270	−0.900149	−0.450075	−	0.892991i	$-0.648603\pi$
$461$	−19.3270	−0.900149	−0.450075	+	0.892991i	$0.648603\pi$
$462$	0	0
$463$	24.2928	1.12898	0.564491	−	0.825440i	$-0.309073\pi$
$463$	24.2928	1.12898	0.564491	+	0.825440i	$0.309073\pi$
$464$	−6.41004	−0.297579
$465$	0	0
$466$	−51.7709	−2.39824
$467$	−2.51424	−0.116345	−0.0581725	−	0.998307i	$-0.518527\pi$
$467$	−2.51424	−0.116345	−0.0581725	+	0.998307i	$0.518527\pi$
$468$	0	0
$469$	40.4885	1.86959
$470$	3.13727	0.144711
$471$	0	0
$472$	−74.2630	−3.41823
$473$	4.63845	0.213276
$474$	0	0
$475$	1.23457	0.0566459
$476$	98.8263	4.52969
$477$	0	0
$478$	44.2966	2.02608
$479$	24.7873	1.13256	0.566281	−	0.824212i	$-0.308382\pi$
$479$	24.7873	1.13256	0.566281	+	0.824212i	$0.308382\pi$
$480$	0	0
$481$	32.2176	1.46899
$482$	29.8414	1.35924
$483$	0	0
$484$	4.64028	0.210922
$485$	8.63296	0.392003
$486$	0	0
$487$	−23.0625	−1.04506	−0.522531	−	0.852620i	$-0.675012\pi$
$487$	−23.0625	−1.04506	−0.522531	+	0.852620i	$0.675012\pi$
$488$	−6.80367	−0.307988
$489$	0	0
$490$	−27.0473	−1.22187
$491$	−22.0914	−0.996970	−0.498485	−	0.866898i	$-0.666110\pi$
$491$	−22.0914	−0.996970	−0.498485	+	0.866898i	$0.666110\pi$
$492$	0	0
$493$	4.78896	0.215684
$494$	−18.6899	−0.840898
$495$	0	0
$496$	29.4167	1.32085
$497$	−0.320239	−0.0143647
$498$	0	0
$499$	−36.5436	−1.63592	−0.817958	−	0.575277i	$-0.804894\pi$
$499$	−36.5436	−1.63592	−0.817958	+	0.575277i	$0.804894\pi$
$500$	−54.0468	−2.41705
$501$	0	0
$502$	−31.7486	−1.41701
$503$	12.6767	0.565226	0.282613	−	0.959234i	$-0.408799\pi$
$503$	12.6767	0.565226	0.282613	+	0.959234i	$0.408799\pi$
$504$	0	0
$505$	4.88838	0.217530
$506$	0.725985	0.0322740
$507$	0	0
$508$	74.0495	3.28542
$509$	13.7556	0.609708	0.304854	−	0.952399i	$-0.401392\pi$
$509$	13.7556	0.609708	0.304854	+	0.952399i	$0.401392\pi$
$510$	0	0
$511$	13.2283	0.585186
$512$	49.1075	2.17027
$513$	0	0
$514$	−7.52930	−0.332103
$515$	−1.94751	−0.0858177
$516$	0	0
$517$	0.572400	0.0251741
$518$	−102.553	−4.50593
$519$	0	0
$520$	40.4713	1.77478
$521$	19.3878	0.849396	0.424698	−	0.905335i	$-0.360380\pi$
$521$	19.3878	0.849396	0.424698	+	0.905335i	$0.360380\pi$
$522$	0	0
$523$	38.4083	1.67948	0.839740	−	0.542989i	$-0.182707\pi$
$523$	38.4083	1.67948	0.839740	+	0.542989i	$0.182707\pi$
$524$	19.4204	0.848384
$525$	0	0
$526$	−8.68937	−0.378875
$527$	−21.9773	−0.957346
$528$	0	0
$529$	−22.9206	−0.996549
$530$	5.99754	0.260516
$531$	0	0
$532$	41.5739	1.80246
$533$	−28.2210	−1.22239
$534$	0	0
$535$	−7.26813	−0.314229
$536$	79.7384	3.44417
$537$	0	0
$538$	−13.3497	−0.575547
$539$	−4.93483	−0.212558
$540$	0	0
$541$	−4.12555	−0.177371	−0.0886856	−	0.996060i	$-0.528267\pi$
$541$	−4.12555	−0.177371	−0.0886856	+	0.996060i	$0.528267\pi$
$542$	−19.6037	−0.842051
$543$	0	0
$544$	47.1984	2.02361
$545$	2.34547	0.100469
$546$	0	0
$547$	12.0008	0.513117	0.256559	−	0.966529i	$-0.417411\pi$
$547$	12.0008	0.513117	0.256559	+	0.966529i	$0.417411\pi$
$548$	100.418	4.28965
$549$	0	0
$550$	−1.22670	−0.0523068
$551$	2.01460	0.0858250
$552$	0	0
$553$	29.7059	1.26322
$554$	−20.4635	−0.869409
$555$	0	0
$556$	55.8405	2.36816
$557$	14.5791	0.617736	0.308868	−	0.951105i	$-0.400050\pi$
$557$	14.5791	0.617736	0.308868	+	0.951105i	$0.400050\pi$
$558$	0	0
$559$	12.9723	0.548671
$560$	−60.6328	−2.56220
$561$	0	0
$562$	23.9269	1.00930
$563$	39.7603	1.67570	0.837849	−	0.545902i	$-0.183813\pi$
$563$	39.7603	1.67570	0.837849	+	0.545902i	$0.183813\pi$
$564$	0	0
$565$	31.4696	1.32394
$566$	−9.32780	−0.392077
$567$	0	0
$568$	−0.630680	−0.0264627
$569$	9.86575	0.413594	0.206797	−	0.978384i	$-0.433696\pi$
$569$	9.86575	0.413594	0.206797	+	0.978384i	$0.433696\pi$
$570$	0	0
$571$	−13.9244	−0.582717	−0.291358	−	0.956614i	$-0.594107\pi$
$571$	−13.9244	−0.582717	−0.291358	+	0.956614i	$0.594107\pi$
$572$	12.9774	0.542614
$573$	0	0
$574$	89.8315	3.74950
$575$	−0.134116	−0.00559303
$576$	0	0
$577$	11.6410	0.484621	0.242311	−	0.970199i	$-0.422095\pi$
$577$	11.6410	0.484621	0.242311	+	0.970199i	$0.422095\pi$
$578$	−54.1272	−2.25139
$579$	0	0
$580$	−7.66697	−0.318354
$581$	10.0952	0.418819
$582$	0	0
$583$	1.09426	0.0453196
$584$	26.0519	1.07804
$585$	0	0
$586$	16.5093	0.681993
$587$	−23.8947	−0.986241	−0.493120	−	0.869961i	$-0.664144\pi$
$587$	−23.8947	−0.986241	−0.493120	+	0.869961i	$0.664144\pi$
$588$	0	0
$589$	−9.24534	−0.380948
$590$	−59.8249	−2.46295
$591$	0	0
$592$	−95.0579	−3.90686
$593$	39.1456	1.60752	0.803758	−	0.594957i	$-0.202831\pi$
$593$	39.1456	1.60752	0.803758	+	0.594957i	$0.202831\pi$
$594$	0	0
$595$	45.2989	1.85707
$596$	53.2774	2.18233
$597$	0	0
$598$	2.03036	0.0830275
$599$	−12.5891	−0.514375	−0.257187	−	0.966362i	$-0.582796\pi$
$599$	−12.5891	−0.514375	−0.257187	+	0.966362i	$0.582796\pi$
$600$	0	0
$601$	2.40062	0.0979235	0.0489617	−	0.998801i	$-0.484409\pi$
$601$	2.40062	0.0979235	0.0489617	+	0.998801i	$0.484409\pi$
$602$	−41.2928	−1.68297
$603$	0	0
$604$	−66.0658	−2.68818
$605$	2.12696	0.0864732
$606$	0	0
$607$	−22.7959	−0.925258	−0.462629	−	0.886552i	$-0.653094\pi$
$607$	−22.7959	−0.925258	−0.462629	+	0.886552i	$0.653094\pi$
$608$	19.8553	0.805238
$609$	0	0
$610$	−5.48091	−0.221915
$611$	1.60083	0.0647625
$612$	0	0
$613$	6.59539	0.266385	0.133193	−	0.991090i	$-0.457477\pi$
$613$	6.59539	0.266385	0.133193	+	0.991090i	$0.457477\pi$
$614$	8.95368	0.361341
$615$	0	0
$616$	−23.5045	−0.947024
$617$	0.666054	0.0268143	0.0134072	−	0.999910i	$-0.495732\pi$
$617$	0.666054	0.0268143	0.0134072	+	0.999910i	$0.495732\pi$
$618$	0	0
$619$	−6.01785	−0.241878	−0.120939	−	0.992660i	$-0.538590\pi$
$619$	−6.01785	−0.241878	−0.120939	+	0.992660i	$0.538590\pi$
$620$	35.1850	1.41306
$621$	0	0
$622$	27.0287	1.08375
$623$	18.4508	0.739217
$624$	0	0
$625$	−22.3932	−0.895727
$626$	−14.3319	−0.572819
$627$	0	0
$628$	76.1986	3.04066
$629$	71.0180	2.83167
$630$	0	0
$631$	−13.1688	−0.524243	−0.262122	−	0.965035i	$-0.584422\pi$
$631$	−13.1688	−0.524243	−0.262122	+	0.965035i	$0.584422\pi$
$632$	58.5029	2.32712
$633$	0	0
$634$	75.9950	3.01815
$635$	33.9420	1.34695
$636$	0	0
$637$	−13.8012	−0.546823
$638$	−2.00177	−0.0792508
$639$	0	0
$640$	14.8897	0.588566
$641$	6.41485	0.253371	0.126686	−	0.991943i	$-0.459566\pi$
$641$	6.41485	0.253371	0.126686	+	0.991943i	$0.459566\pi$
$642$	0	0
$643$	38.5541	1.52043	0.760213	−	0.649673i	$-0.225094\pi$
$643$	38.5541	1.52043	0.760213	+	0.649673i	$0.225094\pi$
$644$	−4.51634	−0.177969
$645$	0	0
$646$	−41.1986	−1.62094
$647$	−11.1416	−0.438023	−0.219012	−	0.975722i	$-0.570283\pi$
$647$	−11.1416	−0.438023	−0.219012	+	0.975722i	$0.570283\pi$
$648$	0	0
$649$	−10.9151	−0.428457
$650$	−3.43071	−0.134563
$651$	0	0
$652$	6.29362	0.246477
$653$	15.5630	0.609027	0.304513	−	0.952508i	$-0.401506\pi$
$653$	15.5630	0.609027	0.304513	+	0.952508i	$0.401506\pi$
$654$	0	0
$655$	8.90171	0.347819
$656$	83.2660	3.25099
$657$	0	0
$658$	−5.09566	−0.198650
$659$	0.00841015	0.000327613 0	0.000163807	−	1.00000i	$-0.499948\pi$
$659$	0.00841015	0.000327613 0	0.000163807		1.00000i	$0.499948\pi$
$660$	0	0
$661$	15.1059	0.587552	0.293776	−	0.955874i	$-0.405088\pi$
$661$	15.1059	0.587552	0.293776	+	0.955874i	$0.405088\pi$
$662$	24.4349	0.949691
$663$	0	0
$664$	19.8815	0.771553
$665$	19.0562	0.738967
$666$	0	0
$667$	−0.218854	−0.00847408
$668$	34.2600	1.32556
$669$	0	0
$670$	64.2357	2.48164
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	47.7087	1.83904	0.919518	−	0.393047i	$-0.128579\pi$
$673$	47.7087	1.83904	0.919518	+	0.393047i	$0.128579\pi$
$674$	−35.7387	−1.37660
$675$	0	0
$676$	−24.0298	−0.924222
$677$	−7.70281	−0.296043	−0.148022	−	0.988984i	$-0.547290\pi$
$677$	−7.70281	−0.296043	−0.148022	+	0.988984i	$0.547290\pi$
$678$	0	0
$679$	−14.0220	−0.538113
$680$	89.2119	3.42112
$681$	0	0
$682$	9.18644	0.351767
$683$	−49.7323	−1.90295	−0.951477	−	0.307719i	$-0.900434\pi$
$683$	−49.7323	−1.90295	−0.951477	+	0.307719i	$0.900434\pi$
$684$	0	0
$685$	46.0286	1.75866
$686$	−18.3848	−0.701933
$687$	0	0
$688$	−38.2748	−1.45921
$689$	3.06031	0.116588
$690$	0	0
$691$	31.9180	1.21422	0.607108	−	0.794619i	$-0.292330\pi$
$691$	31.9180	1.21422	0.607108	+	0.794619i	$0.292330\pi$
$692$	−80.9529	−3.07737
$693$	0	0
$694$	77.0412	2.92444
$695$	25.5955	0.970894
$696$	0	0
$697$	−62.2082	−2.35630
$698$	−40.1891	−1.52118
$699$	0	0
$700$	7.63130	0.288436
$701$	−23.0589	−0.870924	−0.435462	−	0.900207i	$-0.643415\pi$
$701$	−23.0589	−0.870924	−0.435462	+	0.900207i	$0.643415\pi$
$702$	0	0
$703$	29.8756	1.12678
$704$	−3.22551	−0.121566
$705$	0	0
$706$	−79.3605	−2.98677
$707$	−7.93988	−0.298610
$708$	0	0
$709$	29.1991	1.09660	0.548298	−	0.836283i	$-0.315276\pi$
$709$	29.1991	1.09660	0.548298	+	0.836283i	$0.315276\pi$
$710$	−0.508064	−0.0190673
$711$	0	0
$712$	36.3372	1.36179
$713$	1.00436	0.0376135
$714$	0	0
$715$	5.94845	0.222459
$716$	−86.2616	−3.22375
$717$	0	0
$718$	−57.3092	−2.13876
$719$	8.50203	0.317072	0.158536	−	0.987353i	$-0.449323\pi$
$719$	8.50203	0.317072	0.158536	+	0.987353i	$0.449323\pi$
$720$	0	0
$721$	3.16322	0.117804
$722$	31.6293	1.17712
$723$	0	0
$724$	115.291	4.28477
$725$	0.369800	0.0137340
$726$	0	0
$727$	−39.5979	−1.46860	−0.734302	−	0.678823i	$-0.762490\pi$
$727$	−39.5979	−1.46860	−0.734302	+	0.678823i	$0.762490\pi$
$728$	−65.7349	−2.43630
$729$	0	0
$730$	20.9869	0.776761
$731$	28.5952	1.05763
$732$	0	0
$733$	−48.9978	−1.80978	−0.904888	−	0.425650i	$-0.860045\pi$
$733$	−48.9978	−1.80978	−0.904888	+	0.425650i	$0.860045\pi$
$734$	−77.2603	−2.85173
$735$	0	0
$736$	−2.15696	−0.0795066
$737$	11.7199	0.431708
$738$	0	0
$739$	19.1567	0.704689	0.352345	−	0.935870i	$-0.385385\pi$
$739$	19.1567	0.704689	0.352345	+	0.935870i	$0.385385\pi$
$740$	−113.698	−4.17961
$741$	0	0
$742$	−9.74141	−0.357618
$743$	26.8326	0.984394	0.492197	−	0.870484i	$-0.336194\pi$
$743$	26.8326	0.984394	0.492197	+	0.870484i	$0.336194\pi$
$744$	0	0
$745$	24.4207	0.894706
$746$	49.1132	1.79816
$747$	0	0
$748$	28.6065	1.04596
$749$	11.8052	0.431351
$750$	0	0
$751$	27.2699	0.995094	0.497547	−	0.867437i	$-0.334234\pi$
$751$	27.2699	0.995094	0.497547	+	0.867437i	$0.334234\pi$
$752$	−4.72324	−0.172239
$753$	0	0
$754$	−5.59833	−0.203879
$755$	−30.2825	−1.10209
$756$	0	0
$757$	34.1227	1.24021	0.620105	−	0.784519i	$-0.287090\pi$
$757$	34.1227	1.24021	0.620105	+	0.784519i	$0.287090\pi$
$758$	24.5788	0.892744
$759$	0	0
$760$	37.5294	1.36133
$761$	10.6951	0.387695	0.193848	−	0.981032i	$-0.437903\pi$
$761$	10.6951	0.387695	0.193848	+	0.981032i	$0.437903\pi$
$762$	0	0
$763$	−3.80959	−0.137916
$764$	−4.42492	−0.160088
$765$	0	0
$766$	37.4494	1.35310
$767$	−30.5263	−1.10224
$768$	0	0
$769$	−7.92527	−0.285793	−0.142896	−	0.989738i	$-0.545642\pi$
$769$	−7.92527	−0.285793	−0.142896	+	0.989738i	$0.545642\pi$
$770$	−18.9348	−0.682362
$771$	0	0
$772$	125.141	4.50393
$773$	12.7573	0.458847	0.229423	−	0.973327i	$-0.426316\pi$
$773$	12.7573	0.458847	0.229423	+	0.973327i	$0.426316\pi$
$774$	0	0
$775$	−1.69707	−0.0609606
$776$	−27.6149	−0.991318
$777$	0	0
$778$	77.2317	2.76889
$779$	−26.1696	−0.937622
$780$	0	0
$781$	−0.0926970	−0.00331696
$782$	4.47557	0.160046
$783$	0	0
$784$	40.7204	1.45430
$785$	34.9271	1.24660
$786$	0	0
$787$	−48.7341	−1.73718	−0.868591	−	0.495529i	$-0.834974\pi$
$787$	−48.7341	−1.73718	−0.868591	+	0.495529i	$0.834974\pi$
$788$	−14.6337	−0.521305
$789$	0	0
$790$	47.1288	1.67677
$791$	−51.1141	−1.81741
$792$	0	0
$793$	−2.79669	−0.0993134
$794$	36.1451	1.28274
$795$	0	0
$796$	83.7444	2.96824
$797$	36.4009	1.28938	0.644692	−	0.764442i	$-0.276985\pi$
$797$	36.4009	1.28938	0.644692	+	0.764442i	$0.276985\pi$
$798$	0	0
$799$	3.52874	0.124838
$800$	3.64463	0.128857
$801$	0	0
$802$	−75.8459	−2.67821
$803$	3.82910	0.135126
$804$	0	0
$805$	−2.07015	−0.0729632
$806$	25.6916	0.904950
$807$	0	0
$808$	−15.6368	−0.550102
$809$	26.4889	0.931299	0.465650	−	0.884969i	$-0.345821\pi$
$809$	26.4889	0.931299	0.465650	+	0.884969i	$0.345821\pi$
$810$	0	0
$811$	29.1977	1.02527	0.512635	−	0.858607i	$-0.328669\pi$
$811$	29.1977	1.02527	0.512635	+	0.858607i	$0.328669\pi$
$812$	12.4530	0.437013
$813$	0	0
$814$	−29.6853	−1.04047
$815$	2.88480	0.101050
$816$	0	0
$817$	12.0293	0.420854
$818$	72.7000	2.54190
$819$	0	0
$820$	99.5935	3.47796
$821$	55.4850	1.93644	0.968220	−	0.250102i	$-0.0804641\pi$
$821$	55.4850	1.93644	0.968220	+	0.250102i	$0.0804641\pi$
$822$	0	0
$823$	−35.5901	−1.24059	−0.620296	−	0.784368i	$-0.712988\pi$
$823$	−35.5901	−1.24059	−0.620296	+	0.784368i	$0.712988\pi$
$824$	6.22966	0.217021
$825$	0	0
$826$	97.1697	3.38097
$827$	−4.54069	−0.157895	−0.0789476	−	0.996879i	$-0.525156\pi$
$827$	−4.54069	−0.157895	−0.0789476	+	0.996879i	$0.525156\pi$
$828$	0	0
$829$	38.4881	1.33675	0.668373	−	0.743826i	$-0.266991\pi$
$829$	38.4881	1.33675	0.668373	+	0.743826i	$0.266991\pi$
$830$	16.0162	0.555930
$831$	0	0
$832$	−9.02076	−0.312739
$833$	−30.4223	−1.05407
$834$	0	0
$835$	15.7037	0.543449
$836$	12.0341	0.416207
$837$	0	0
$838$	79.5169	2.74687
$839$	30.4556	1.05144	0.525721	−	0.850657i	$-0.323795\pi$
$839$	30.4556	1.05144	0.525721	+	0.850657i	$0.323795\pi$
$840$	0	0
$841$	−28.3966	−0.979191
$842$	47.1597	1.62523
$843$	0	0
$844$	−94.1280	−3.24002
$845$	−11.0145	−0.378910
$846$	0	0
$847$	−3.45468	−0.118704
$848$	−9.02944	−0.310072
$849$	0	0
$850$	−7.56240	−0.259388
$851$	−3.24551	−0.111255
$852$	0	0
$853$	9.78630	0.335077	0.167538	−	0.985866i	$-0.446418\pi$
$853$	9.78630	0.335077	0.167538	+	0.985866i	$0.446418\pi$
$854$	8.90228	0.304630
$855$	0	0
$856$	23.2491	0.794639
$857$	3.08993	0.105550	0.0527750	−	0.998606i	$-0.483193\pi$
$857$	3.08993	0.105550	0.0527750	+	0.998606i	$0.483193\pi$
$858$	0	0
$859$	28.5965	0.975701	0.487850	−	0.872927i	$-0.337781\pi$
$859$	28.5965	0.975701	0.487850	+	0.872927i	$0.337781\pi$
$860$	−45.7801	−1.56109
$861$	0	0
$862$	−81.7503	−2.78443
$863$	−40.9319	−1.39334	−0.696669	−	0.717393i	$-0.745335\pi$
$863$	−40.9319	−1.39334	−0.696669	+	0.717393i	$0.745335\pi$
$864$	0	0
$865$	−37.1063	−1.26165
$866$	−0.700073	−0.0237894
$867$	0	0
$868$	−57.1487	−1.93975
$869$	8.59873	0.291692
$870$	0	0
$871$	32.7770	1.11061
$872$	−7.50263	−0.254071
$873$	0	0
$874$	1.88277	0.0636856
$875$	40.2378	1.36029
$876$	0	0
$877$	26.4940	0.894639	0.447320	−	0.894374i	$-0.352379\pi$
$877$	26.4940	0.894639	0.447320	+	0.894374i	$0.352379\pi$
$878$	−9.98077	−0.336835
$879$	0	0
$880$	−17.5509	−0.591641
$881$	6.43735	0.216880	0.108440	−	0.994103i	$-0.465414\pi$
$881$	6.43735	0.216880	0.108440	+	0.994103i	$0.465414\pi$
$882$	0	0
$883$	−26.3944	−0.888243	−0.444121	−	0.895967i	$-0.646484\pi$
$883$	−26.3944	−0.888243	−0.444121	+	0.895967i	$0.646484\pi$
$884$	80.0035	2.69081
$885$	0	0
$886$	−18.8947	−0.634780
$887$	−5.86026	−0.196768	−0.0983842	−	0.995149i	$-0.531367\pi$
$887$	−5.86026	−0.196768	−0.0983842	+	0.995149i	$0.531367\pi$
$888$	0	0
$889$	−55.1298	−1.84899
$890$	29.2725	0.981218
$891$	0	0
$892$	−28.4474	−0.952489
$893$	1.48446	0.0496755
$894$	0	0
$895$	−39.5396	−1.32166
$896$	−24.1843	−0.807942
$897$	0	0
$898$	−61.9873	−2.06854
$899$	−2.76933	−0.0923624
$900$	0	0
$901$	6.74592	0.224739
$902$	26.0028	0.865800
$903$	0	0
$904$	−100.664	−3.34804
$905$	52.8459	1.75666
$906$	0	0
$907$	−59.5292	−1.97664	−0.988318	−	0.152406i	$-0.951298\pi$
$907$	−59.5292	−1.97664	−0.988318	+	0.152406i	$0.951298\pi$
$908$	21.3896	0.709839
$909$	0	0
$910$	−52.9548	−1.75543
$911$	−25.3980	−0.841473	−0.420737	−	0.907183i	$-0.638228\pi$
$911$	−25.3980	−0.841473	−0.420737	+	0.907183i	$0.638228\pi$
$912$	0	0
$913$	2.92218	0.0967099
$914$	−53.8771	−1.78209
$915$	0	0
$916$	−32.9346	−1.08819
$917$	−14.4585	−0.477461
$918$	0	0
$919$	−12.9495	−0.427164	−0.213582	−	0.976925i	$-0.568513\pi$
$919$	−12.9495	−0.427164	−0.213582	+	0.976925i	$0.568513\pi$
$920$	−4.07697	−0.134414
$921$	0	0
$922$	49.8033	1.64018
$923$	−0.259245	−0.00853315
$924$	0	0
$925$	5.48396	0.180311
$926$	−62.5994	−2.05714
$927$	0	0
$928$	5.94741	0.195233
$929$	23.0505	0.756262	0.378131	−	0.925752i	$-0.376567\pi$
$929$	23.0505	0.756262	0.378131	+	0.925752i	$0.376567\pi$
$930$	0	0
$931$	−12.7980	−0.419436
$932$	93.2259	3.05372
$933$	0	0
$934$	6.47887	0.211995
$935$	13.1123	0.428819
$936$	0	0
$937$	−35.8408	−1.17087	−0.585434	−	0.810720i	$-0.699076\pi$
$937$	−35.8408	−1.17087	−0.585434	+	0.810720i	$0.699076\pi$
$938$	−104.334	−3.40662
$939$	0	0
$940$	−5.64941	−0.184263
$941$	28.7603	0.937558	0.468779	−	0.883315i	$-0.344694\pi$
$941$	28.7603	0.937558	0.468779	+	0.883315i	$0.344694\pi$
$942$	0	0
$943$	2.84290	0.0925777
$944$	90.0678	2.93146
$945$	0	0
$946$	−11.9527	−0.388616
$947$	−7.53264	−0.244778	−0.122389	−	0.992482i	$-0.539056\pi$
$947$	−7.53264	−0.244778	−0.122389	+	0.992482i	$0.539056\pi$
$948$	0	0
$949$	10.7088	0.347622
$950$	−3.18133	−0.103216
$951$	0	0
$952$	−144.901	−4.69627
$953$	18.7820	0.608410	0.304205	−	0.952607i	$-0.401609\pi$
$953$	18.7820	0.608410	0.304205	+	0.952607i	$0.401609\pi$
$954$	0	0
$955$	−2.02825	−0.0656325
$956$	−79.7666	−2.57984
$957$	0	0
$958$	−63.8738	−2.06367
$959$	−74.7612	−2.41417
$960$	0	0
$961$	−18.2911	−0.590035
$962$	−83.0206	−2.67669
$963$	0	0
$964$	−53.7367	−1.73074
$965$	57.3609	1.84651
$966$	0	0
$967$	−26.5111	−0.852540	−0.426270	−	0.904596i	$-0.640173\pi$
$967$	−26.5111	−0.852540	−0.426270	+	0.904596i	$0.640173\pi$
$968$	−6.80367	−0.218678
$969$	0	0
$970$	−22.2461	−0.714278
$971$	−22.0957	−0.709086	−0.354543	−	0.935040i	$-0.615364\pi$
$971$	−22.0957	−0.709086	−0.354543	+	0.935040i	$0.615364\pi$
$972$	0	0
$973$	−41.5732	−1.33277
$974$	59.4292	1.90423
$975$	0	0
$976$	8.25164	0.264128
$977$	−15.3382	−0.490714	−0.245357	−	0.969433i	$-0.578905\pi$
$977$	−15.3382	−0.490714	−0.245357	+	0.969433i	$0.578905\pi$
$978$	0	0
$979$	5.34082	0.170693
$980$	48.7052	1.55583
$981$	0	0
$982$	56.9267	1.81660
$983$	6.28411	0.200432	0.100216	−	0.994966i	$-0.468047\pi$
$983$	6.28411	0.200432	0.100216	+	0.994966i	$0.468047\pi$
$984$	0	0
$985$	−6.70765	−0.213723
$986$	−12.3405	−0.393003
$987$	0	0
$988$	33.6556	1.07073
$989$	−1.30680	−0.0415537
$990$	0	0
$991$	30.8847	0.981084	0.490542	−	0.871418i	$-0.336799\pi$
$991$	30.8847	0.981084	0.490542	+	0.871418i	$0.336799\pi$
$992$	−27.2936	−0.866574
$993$	0	0
$994$	0.825215	0.0261742
$995$	38.3858	1.21691
$996$	0	0
$997$	−12.0802	−0.382584	−0.191292	−	0.981533i	$-0.561268\pi$
$997$	−12.0802	−0.382584	−0.191292	+	0.981533i	$0.561268\pi$
$998$	94.1683	2.98085
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.25	✓	25	3.2	odd	2
6039.2.a.p.1.1	yes	25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57687 q^{2} +4.64028 q^{4} +2.12696 q^{5} -3.45468 q^{7} -6.80367 q^{8} +O(q^{10})\)	\(q-2.57687 q^{2} +4.64028 q^{4} +2.12696 q^{5} -3.45468 q^{7} -6.80367 q^{8} -5.48091 q^{10} -1.00000 q^{11} -2.79669 q^{13} +8.90228 q^{14} +8.25164 q^{16} -6.16482 q^{17} -2.59340 q^{19} +9.86969 q^{20} +2.57687 q^{22} +0.281731 q^{23} -0.476043 q^{25} +7.20672 q^{26} -16.0307 q^{28} -0.776820 q^{29} +3.56496 q^{31} -7.65609 q^{32} +15.8860 q^{34} -7.34797 q^{35} -11.5199 q^{37} +6.68285 q^{38} -14.4711 q^{40} +10.0908 q^{41} -4.63845 q^{43} -4.64028 q^{44} -0.725985 q^{46} -0.572400 q^{47} +4.93483 q^{49} +1.22670 q^{50} -12.9774 q^{52} -1.09426 q^{53} -2.12696 q^{55} +23.5045 q^{56} +2.00177 q^{58} +10.9151 q^{59} +1.00000 q^{61} -9.18644 q^{62} +3.22551 q^{64} -5.94845 q^{65} -11.7199 q^{67} -28.6065 q^{68} +18.9348 q^{70} +0.0926970 q^{71} -3.82910 q^{73} +29.6853 q^{74} -12.0341 q^{76} +3.45468 q^{77} -8.59873 q^{79} +17.5509 q^{80} -26.0028 q^{82} -2.92218 q^{83} -13.1123 q^{85} +11.9527 q^{86} +6.80367 q^{88} -5.34082 q^{89} +9.66168 q^{91} +1.30731 q^{92} +1.47500 q^{94} -5.51605 q^{95} +4.05883 q^{97} -12.7164 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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

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