LMFDB - Embedded newform 6039.2.a.m.1.14

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			1.14
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.228656 q^{2} -1.94772 q^{4} +0.872276 q^{5} -1.27506 q^{7} -0.902667 q^{8} +O(q^{10})$	\(q+0.228656 q^{2} -1.94772 q^{4} +0.872276 q^{5} -1.27506 q^{7} -0.902667 q^{8} +0.199451 q^{10} +1.00000 q^{11} +5.51971 q^{13} -0.291550 q^{14} +3.68903 q^{16} -7.05471 q^{17} -4.53854 q^{19} -1.69895 q^{20} +0.228656 q^{22} +4.97643 q^{23} -4.23913 q^{25} +1.26211 q^{26} +2.48346 q^{28} -4.43142 q^{29} +6.49677 q^{31} +2.64885 q^{32} -1.61310 q^{34} -1.11220 q^{35} +4.65721 q^{37} -1.03776 q^{38} -0.787375 q^{40} +2.47842 q^{41} -7.76809 q^{43} -1.94772 q^{44} +1.13789 q^{46} +0.452696 q^{47} -5.37422 q^{49} -0.969302 q^{50} -10.7508 q^{52} +6.10390 q^{53} +0.872276 q^{55} +1.15095 q^{56} -1.01327 q^{58} -0.139545 q^{59} +1.00000 q^{61} +1.48552 q^{62} -6.77239 q^{64} +4.81471 q^{65} +3.75357 q^{67} +13.7406 q^{68} -0.254312 q^{70} -9.37234 q^{71} -1.78042 q^{73} +1.06490 q^{74} +8.83980 q^{76} -1.27506 q^{77} +2.21606 q^{79} +3.21786 q^{80} +0.566706 q^{82} +8.77256 q^{83} -6.15366 q^{85} -1.77622 q^{86} -0.902667 q^{88} +1.14261 q^{89} -7.03796 q^{91} -9.69268 q^{92} +0.103511 q^{94} -3.95886 q^{95} -14.4430 q^{97} -1.22885 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.228656	0.161684	0.0808419	−	0.996727i	$-0.474239\pi$
$2$	0.228656	0.161684	0.0808419	+	0.996727i	$0.474239\pi$
$3$	0	0
$3$	0	0
$4$	−1.94772	−0.973858
$5$	0.872276	0.390094	0.195047	−	0.980794i	$-0.437514\pi$
$5$	0.872276	0.390094	0.195047	+	0.980794i	$0.437514\pi$
$6$	0	0
$7$	−1.27506	−0.481927	−0.240964	−	0.970534i	$-0.577463\pi$
$7$	−1.27506	−0.481927	−0.240964	+	0.970534i	$0.577463\pi$
$8$	−0.902667	−0.319141
$9$	0	0
$10$	0.199451	0.0630719
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.51971	1.53089	0.765445	−	0.643501i	$-0.222519\pi$
$13$	5.51971	1.53089	0.765445	+	0.643501i	$0.222519\pi$
$14$	−0.291550	−0.0779199
$15$	0	0
$16$	3.68903	0.922258
$17$	−7.05471	−1.71102	−0.855510	−	0.517787i	$-0.826756\pi$
$17$	−7.05471	−1.71102	−0.855510	+	0.517787i	$0.826756\pi$
$18$	0	0
$19$	−4.53854	−1.04121	−0.520607	−	0.853797i	$-0.674294\pi$
$19$	−4.53854	−1.04121	−0.520607	+	0.853797i	$0.674294\pi$
$20$	−1.69895	−0.379896
$21$	0	0
$22$	0.228656	0.0487495
$23$	4.97643	1.03766	0.518829	−	0.854878i	$-0.326368\pi$
$23$	4.97643	1.03766	0.518829	+	0.854878i	$0.326368\pi$
$24$	0	0
$25$	−4.23913	−0.847827
$26$	1.26211	0.247520
$27$	0	0
$28$	2.48346	0.469329
$29$	−4.43142	−0.822894	−0.411447	−	0.911434i	$-0.634976\pi$
$29$	−4.43142	−0.822894	−0.411447	+	0.911434i	$0.634976\pi$
$30$	0	0
$31$	6.49677	1.16685	0.583427	−	0.812166i	$-0.301711\pi$
$31$	6.49677	1.16685	0.583427	+	0.812166i	$0.301711\pi$
$32$	2.64885	0.468255
$33$	0	0
$34$	−1.61310	−0.276644
$35$	−1.11220	−0.187997
$36$	0	0
$37$	4.65721	0.765640	0.382820	−	0.923823i	$-0.374953\pi$
$37$	4.65721	0.765640	0.382820	+	0.923823i	$0.374953\pi$
$38$	−1.03776	−0.168347
$39$	0	0
$40$	−0.787375	−0.124495
$41$	2.47842	0.387065	0.193532	−	0.981094i	$-0.438006\pi$
$41$	2.47842	0.387065	0.193532	+	0.981094i	$0.438006\pi$
$42$	0	0
$43$	−7.76809	−1.18462	−0.592311	−	0.805709i	$-0.701784\pi$
$43$	−7.76809	−1.18462	−0.592311	+	0.805709i	$0.701784\pi$
$44$	−1.94772	−0.293629
$45$	0	0
$46$	1.13789	0.167772
$47$	0.452696	0.0660324	0.0330162	−	0.999455i	$-0.489489\pi$
$47$	0.452696	0.0660324	0.0330162	+	0.999455i	$0.489489\pi$
$48$	0	0
$49$	−5.37422	−0.767746
$50$	−0.969302	−0.137080
$51$	0	0
$52$	−10.7508	−1.49087
$53$	6.10390	0.838435	0.419218	−	0.907886i	$-0.362304\pi$
$53$	6.10390	0.838435	0.419218	+	0.907886i	$0.362304\pi$
$54$	0	0
$55$	0.872276	0.117618
$56$	1.15095	0.153803
$57$	0	0
$58$	−1.01327	−0.133049
$59$	−0.139545	−0.0181672	−0.00908361	−	0.999959i	$-0.502891\pi$
$59$	−0.139545	−0.0181672	−0.00908361	+	0.999959i	$0.502891\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	1.48552	0.188661
$63$	0	0
$64$	−6.77239	−0.846549
$65$	4.81471	0.597191
$66$	0	0
$67$	3.75357	0.458572	0.229286	−	0.973359i	$-0.426361\pi$
$67$	3.75357	0.458572	0.229286	+	0.973359i	$0.426361\pi$
$68$	13.7406	1.66629
$69$	0	0
$70$	−0.254312	−0.0303961
$71$	−9.37234	−1.11229	−0.556146	−	0.831085i	$-0.687720\pi$
$71$	−9.37234	−1.11229	−0.556146	+	0.831085i	$0.687720\pi$
$72$	0	0
$73$	−1.78042	−0.208383	−0.104191	−	0.994557i	$-0.533225\pi$
$73$	−1.78042	−0.208383	−0.104191	+	0.994557i	$0.533225\pi$
$74$	1.06490	0.123792
$75$	0	0
$76$	8.83980	1.01399
$77$	−1.27506	−0.145307
$78$	0	0
$79$	2.21606	0.249327	0.124663	−	0.992199i	$-0.460215\pi$
$79$	2.21606	0.249327	0.124663	+	0.992199i	$0.460215\pi$
$80$	3.21786	0.359767
$81$	0	0
$82$	0.566706	0.0625822
$83$	8.77256	0.962914	0.481457	−	0.876470i	$-0.340108\pi$
$83$	8.77256	0.962914	0.481457	+	0.876470i	$0.340108\pi$
$84$	0	0
$85$	−6.15366	−0.667458
$86$	−1.77622	−0.191534
$87$	0	0
$88$	−0.902667	−0.0962246
$89$	1.14261	0.121116	0.0605582	−	0.998165i	$-0.480712\pi$
$89$	1.14261	0.121116	0.0605582	+	0.998165i	$0.480712\pi$
$90$	0	0
$91$	−7.03796	−0.737778
$92$	−9.69268	−1.01053
$93$	0	0
$94$	0.103511	0.0106764
$95$	−3.95886	−0.406171
$96$	0	0
$97$	−14.4430	−1.46646	−0.733232	−	0.679979i	$-0.761989\pi$
$97$	−14.4430	−1.46646	−0.733232	+	0.679979i	$0.761989\pi$
$98$	−1.22885	−0.124132
$99$	0	0
$100$	8.25663	0.825663
$101$	−7.74593	−0.770749	−0.385375	−	0.922760i	$-0.625928\pi$
$101$	−7.74593	−0.770749	−0.385375	+	0.922760i	$0.625928\pi$
$102$	0	0
$103$	10.7975	1.06391	0.531954	−	0.846774i	$-0.321458\pi$
$103$	10.7975	1.06391	0.531954	+	0.846774i	$0.321458\pi$
$104$	−4.98246	−0.488570
$105$	0	0
$106$	1.39569	0.135561
$107$	−19.0879	−1.84529	−0.922647	−	0.385645i	$-0.873979\pi$
$107$	−19.0879	−1.84529	−0.922647	+	0.385645i	$0.873979\pi$
$108$	0	0
$109$	8.61246	0.824924	0.412462	−	0.910975i	$-0.364669\pi$
$109$	8.61246	0.824924	0.412462	+	0.910975i	$0.364669\pi$
$110$	0.199451	0.0190169
$111$	0	0
$112$	−4.70374	−0.444462
$113$	15.7297	1.47973	0.739864	−	0.672757i	$-0.234890\pi$
$113$	15.7297	1.47973	0.739864	+	0.672757i	$0.234890\pi$
$114$	0	0
$115$	4.34082	0.404784
$116$	8.63115	0.801382
$117$	0	0
$118$	−0.0319078	−0.00293735
$119$	8.99518	0.824587
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.228656	0.0207015
$123$	0	0
$124$	−12.6539	−1.13635
$125$	−8.05908	−0.720826
$126$	0	0
$127$	−19.6857	−1.74682	−0.873411	−	0.486983i	$-0.838097\pi$
$127$	−19.6857	−1.74682	−0.873411	+	0.486983i	$0.838097\pi$
$128$	−6.84625	−0.605129
$129$	0	0
$130$	1.10091	0.0965561
$131$	−0.673702	−0.0588616	−0.0294308	−	0.999567i	$-0.509369\pi$
$131$	−0.673702	−0.0588616	−0.0294308	+	0.999567i	$0.509369\pi$
$132$	0	0
$133$	5.78692	0.501789
$134$	0.858275	0.0741436
$135$	0	0
$136$	6.36806	0.546056
$137$	−1.45200	−0.124053	−0.0620265	−	0.998075i	$-0.519756\pi$
$137$	−1.45200	−0.124053	−0.0620265	+	0.998075i	$0.519756\pi$
$138$	0	0
$139$	2.97052	0.251956	0.125978	−	0.992033i	$-0.459793\pi$
$139$	2.97052	0.251956	0.125978	+	0.992033i	$0.459793\pi$
$140$	2.16626	0.183082
$141$	0	0
$142$	−2.14304	−0.179840
$143$	5.51971	0.461581
$144$	0	0
$145$	−3.86542	−0.321006
$146$	−0.407103	−0.0336921
$147$	0	0
$148$	−9.07092	−0.745625
$149$	−11.2111	−0.918445	−0.459223	−	0.888321i	$-0.651872\pi$
$149$	−11.2111	−0.918445	−0.459223	+	0.888321i	$0.651872\pi$
$150$	0	0
$151$	−8.09238	−0.658549	−0.329274	−	0.944234i	$-0.606804\pi$
$151$	−8.09238	−0.658549	−0.329274	+	0.944234i	$0.606804\pi$
$152$	4.09680	0.332294
$153$	0	0
$154$	−0.291550	−0.0234937
$155$	5.66697	0.455182
$156$	0	0
$157$	−4.24491	−0.338780	−0.169390	−	0.985549i	$-0.554180\pi$
$157$	−4.24491	−0.338780	−0.169390	+	0.985549i	$0.554180\pi$
$158$	0.506715	0.0403121
$159$	0	0
$160$	2.31053	0.182663
$161$	−6.34525	−0.500076
$162$	0	0
$163$	−1.14603	−0.0897639	−0.0448820	−	0.998992i	$-0.514291\pi$
$163$	−1.14603	−0.0897639	−0.0448820	+	0.998992i	$0.514291\pi$
$164$	−4.82727	−0.376946
$165$	0	0
$166$	2.00590	0.155688
$167$	−22.4452	−1.73686	−0.868432	−	0.495808i	$-0.834872\pi$
$167$	−22.4452	−1.73686	−0.868432	+	0.495808i	$0.834872\pi$
$168$	0	0
$169$	17.4671	1.34363
$170$	−1.40707	−0.107917
$171$	0	0
$172$	15.1300	1.15365
$173$	−21.3279	−1.62153	−0.810767	−	0.585370i	$-0.800949\pi$
$173$	−21.3279	−1.62153	−0.810767	+	0.585370i	$0.800949\pi$
$174$	0	0
$175$	5.40515	0.408591
$176$	3.68903	0.278071
$177$	0	0
$178$	0.261264	0.0195826
$179$	21.8476	1.63296	0.816482	−	0.577371i	$-0.195921\pi$
$179$	21.8476	1.63296	0.816482	+	0.577371i	$0.195921\pi$
$180$	0	0
$181$	11.4857	0.853725	0.426862	−	0.904317i	$-0.359619\pi$
$181$	11.4857	0.853725	0.426862	+	0.904317i	$0.359619\pi$
$182$	−1.60927	−0.119287
$183$	0	0
$184$	−4.49206	−0.331159
$185$	4.06237	0.298672
$186$	0	0
$187$	−7.05471	−0.515892
$188$	−0.881723	−0.0643062
$189$	0	0
$190$	−0.905216	−0.0656713
$191$	4.38460	0.317258	0.158629	−	0.987338i	$-0.449293\pi$
$191$	4.38460	0.317258	0.158629	+	0.987338i	$0.449293\pi$
$192$	0	0
$193$	21.2587	1.53023	0.765116	−	0.643893i	$-0.222682\pi$
$193$	21.2587	1.53023	0.765116	+	0.643893i	$0.222682\pi$
$194$	−3.30247	−0.237104
$195$	0	0
$196$	10.4675	0.747676
$197$	−9.17680	−0.653820	−0.326910	−	0.945055i	$-0.606007\pi$
$197$	−9.17680	−0.653820	−0.326910	+	0.945055i	$0.606007\pi$
$198$	0	0
$199$	−10.4269	−0.739146	−0.369573	−	0.929202i	$-0.620496\pi$
$199$	−10.4269	−0.739146	−0.369573	+	0.929202i	$0.620496\pi$
$200$	3.82653	0.270576
$201$	0	0
$202$	−1.77115	−0.124618
$203$	5.65032	0.396575
$204$	0	0
$205$	2.16187	0.150992
$206$	2.46890	0.172017
$207$	0	0
$208$	20.3624	1.41188
$209$	−4.53854	−0.313938
$210$	0	0
$211$	−3.89024	−0.267815	−0.133908	−	0.990994i	$-0.542753\pi$
$211$	−3.89024	−0.267815	−0.133908	+	0.990994i	$0.542753\pi$
$212$	−11.8887	−0.816517
$213$	0	0
$214$	−4.36455	−0.298354
$215$	−6.77592	−0.462114
$216$	0	0
$217$	−8.28377	−0.562339
$218$	1.96929	0.133377
$219$	0	0
$220$	−1.69895	−0.114543
$221$	−38.9399	−2.61938
$222$	0	0
$223$	−10.7363	−0.718954	−0.359477	−	0.933154i	$-0.617045\pi$
$223$	−10.7363	−0.718954	−0.359477	+	0.933154i	$0.617045\pi$
$224$	−3.37745	−0.225665
$225$	0	0
$226$	3.59669	0.239248
$227$	−15.2522	−1.01233	−0.506163	−	0.862438i	$-0.668936\pi$
$227$	−15.2522	−1.01233	−0.506163	+	0.862438i	$0.668936\pi$
$228$	0	0
$229$	10.4919	0.693322	0.346661	−	0.937990i	$-0.387315\pi$
$229$	10.4919	0.693322	0.346661	+	0.937990i	$0.387315\pi$
$230$	0.992553	0.0654470
$231$	0	0
$232$	4.00010	0.262619
$233$	−19.5688	−1.28199	−0.640996	−	0.767544i	$-0.721479\pi$
$233$	−19.5688	−1.28199	−0.640996	+	0.767544i	$0.721479\pi$
$234$	0	0
$235$	0.394876	0.0257588
$236$	0.271794	0.0176923
$237$	0	0
$238$	2.05680	0.133322
$239$	−6.22136	−0.402426	−0.201213	−	0.979548i	$-0.564488\pi$
$239$	−6.22136	−0.402426	−0.201213	+	0.979548i	$0.564488\pi$
$240$	0	0
$241$	−8.10210	−0.521902	−0.260951	−	0.965352i	$-0.584036\pi$
$241$	−8.10210	−0.521902	−0.260951	+	0.965352i	$0.584036\pi$
$242$	0.228656	0.0146985
$243$	0	0
$244$	−1.94772	−0.124690
$245$	−4.68780	−0.299493
$246$	0	0
$247$	−25.0514	−1.59398
$248$	−5.86442	−0.372391
$249$	0	0
$250$	−1.84275	−0.116546
$251$	−13.5672	−0.856357	−0.428178	−	0.903694i	$-0.640844\pi$
$251$	−13.5672	−0.856357	−0.428178	+	0.903694i	$0.640844\pi$
$252$	0	0
$253$	4.97643	0.312866
$254$	−4.50124	−0.282433
$255$	0	0
$256$	11.9794	0.748709
$257$	−10.5711	−0.659409	−0.329705	−	0.944084i	$-0.606949\pi$
$257$	−10.5711	−0.659409	−0.329705	+	0.944084i	$0.606949\pi$
$258$	0	0
$259$	−5.93822	−0.368983
$260$	−9.37768	−0.581579
$261$	0	0
$262$	−0.154046	−0.00951697
$263$	−16.1723	−0.997226	−0.498613	−	0.866825i	$-0.666157\pi$
$263$	−16.1723	−0.997226	−0.498613	+	0.866825i	$0.666157\pi$
$264$	0	0
$265$	5.32429	0.327068
$266$	1.32321	0.0811312
$267$	0	0
$268$	−7.31089	−0.446584
$269$	−26.3605	−1.60723	−0.803615	−	0.595150i	$-0.797093\pi$
$269$	−26.3605	−1.60723	−0.803615	+	0.595150i	$0.797093\pi$
$270$	0	0
$271$	−12.6166	−0.766402	−0.383201	−	0.923665i	$-0.625178\pi$
$271$	−12.6166	−0.766402	−0.383201	+	0.923665i	$0.625178\pi$
$272$	−26.0251	−1.57800
$273$	0	0
$274$	−0.332009	−0.0200574
$275$	−4.23913	−0.255629
$276$	0	0
$277$	17.4948	1.05116	0.525582	−	0.850743i	$-0.323848\pi$
$277$	17.4948	1.05116	0.525582	+	0.850743i	$0.323848\pi$
$278$	0.679225	0.0407372
$279$	0	0
$280$	1.00395	0.0599975
$281$	−30.0039	−1.78988	−0.894940	−	0.446186i	$-0.852782\pi$
$281$	−30.0039	−1.78988	−0.894940	+	0.446186i	$0.852782\pi$
$282$	0	0
$283$	−8.33845	−0.495669	−0.247835	−	0.968802i	$-0.579719\pi$
$283$	−8.33845	−0.495669	−0.247835	+	0.968802i	$0.579719\pi$
$284$	18.2547	1.08321
$285$	0	0
$286$	1.26211	0.0746302
$287$	−3.16014	−0.186537
$288$	0	0
$289$	32.7690	1.92759
$290$	−0.883849	−0.0519014
$291$	0	0
$292$	3.46776	0.202935
$293$	−19.7968	−1.15654	−0.578271	−	0.815845i	$-0.696272\pi$
$293$	−19.7968	−1.15654	−0.578271	+	0.815845i	$0.696272\pi$
$294$	0	0
$295$	−0.121722	−0.00708692
$296$	−4.20391	−0.244347
$297$	0	0
$298$	−2.56347	−0.148498
$299$	27.4684	1.58854
$300$	0	0
$301$	9.90478	0.570902
$302$	−1.85037	−0.106477
$303$	0	0
$304$	−16.7428	−0.960268
$305$	0.872276	0.0499464
$306$	0	0
$307$	24.5355	1.40031	0.700157	−	0.713989i	$-0.253113\pi$
$307$	24.5355	1.40031	0.700157	+	0.713989i	$0.253113\pi$
$308$	2.48346	0.141508
$309$	0	0
$310$	1.29578	0.0735956
$311$	−12.5596	−0.712189	−0.356095	−	0.934450i	$-0.615892\pi$
$311$	−12.5596	−0.712189	−0.356095	+	0.934450i	$0.615892\pi$
$312$	0	0
$313$	−33.1430	−1.87336	−0.936678	−	0.350192i	$-0.886116\pi$
$313$	−33.1430	−1.87336	−0.936678	+	0.350192i	$0.886116\pi$
$314$	−0.970621	−0.0547753
$315$	0	0
$316$	−4.31627	−0.242809
$317$	−19.6616	−1.10430	−0.552152	−	0.833743i	$-0.686193\pi$
$317$	−19.6616	−1.10430	−0.552152	+	0.833743i	$0.686193\pi$
$318$	0	0
$319$	−4.43142	−0.248112
$320$	−5.90740	−0.330233
$321$	0	0
$322$	−1.45088	−0.0808542
$323$	32.0181	1.78154
$324$	0	0
$325$	−23.3988	−1.29793
$326$	−0.262046	−0.0145134
$327$	0	0
$328$	−2.23719	−0.123528
$329$	−0.577214	−0.0318228
$330$	0	0
$331$	−22.7900	−1.25265	−0.626327	−	0.779561i	$-0.715442\pi$
$331$	−22.7900	−1.25265	−0.626327	+	0.779561i	$0.715442\pi$
$332$	−17.0865	−0.937742
$333$	0	0
$334$	−5.13223	−0.280823
$335$	3.27415	0.178886
$336$	0	0
$337$	2.19401	0.119515	0.0597577	−	0.998213i	$-0.480967\pi$
$337$	2.19401	0.119515	0.0597577	+	0.998213i	$0.480967\pi$
$338$	3.99396	0.217243
$339$	0	0
$340$	11.9856	0.650009
$341$	6.49677	0.351820
$342$	0	0
$343$	15.7779	0.851925
$344$	7.01200	0.378062
$345$	0	0
$346$	−4.87675	−0.262176
$347$	17.6560	0.947824	0.473912	−	0.880572i	$-0.342841\pi$
$347$	17.6560	0.947824	0.473912	+	0.880572i	$0.342841\pi$
$348$	0	0
$349$	−5.91008	−0.316359	−0.158180	−	0.987410i	$-0.550563\pi$
$349$	−5.91008	−0.316359	−0.158180	+	0.987410i	$0.550563\pi$
$350$	1.23592	0.0660626
$351$	0	0
$352$	2.64885	0.141184
$353$	30.2566	1.61039	0.805197	−	0.593007i	$-0.202059\pi$
$353$	30.2566	1.61039	0.805197	+	0.593007i	$0.202059\pi$
$354$	0	0
$355$	−8.17527	−0.433898
$356$	−2.22548	−0.117950
$357$	0	0
$358$	4.99557	0.264024
$359$	−1.17480	−0.0620035	−0.0310018	−	0.999519i	$-0.509870\pi$
$359$	−1.17480	−0.0620035	−0.0310018	+	0.999519i	$0.509870\pi$
$360$	0	0
$361$	1.59839	0.0841256
$362$	2.62627	0.138034
$363$	0	0
$364$	13.7079	0.718491
$365$	−1.55302	−0.0812888
$366$	0	0
$367$	24.0252	1.25411	0.627053	−	0.778976i	$-0.284261\pi$
$367$	24.0252	1.25411	0.627053	+	0.778976i	$0.284261\pi$
$368$	18.3582	0.956988
$369$	0	0
$370$	0.928884	0.0482904
$371$	−7.78284	−0.404065
$372$	0	0
$373$	−28.0830	−1.45408	−0.727041	−	0.686594i	$-0.759105\pi$
$373$	−28.0830	−1.45408	−0.727041	+	0.686594i	$0.759105\pi$
$374$	−1.61310	−0.0834114
$375$	0	0
$376$	−0.408634	−0.0210737
$377$	−24.4601	−1.25976
$378$	0	0
$379$	−37.6191	−1.93236	−0.966181	−	0.257863i	$-0.916982\pi$
$379$	−37.6191	−1.93236	−0.966181	+	0.257863i	$0.916982\pi$
$380$	7.71074	0.395553
$381$	0	0
$382$	1.00256	0.0512956
$383$	−35.7407	−1.82627	−0.913133	−	0.407662i	$-0.866344\pi$
$383$	−35.7407	−1.82627	−0.913133	+	0.407662i	$0.866344\pi$
$384$	0	0
$385$	−1.11220	−0.0566832
$386$	4.86091	0.247414
$387$	0	0
$388$	28.1309	1.42813
$389$	−2.59704	−0.131675	−0.0658375	−	0.997830i	$-0.520972\pi$
$389$	−2.59704	−0.131675	−0.0658375	+	0.997830i	$0.520972\pi$
$390$	0	0
$391$	−35.1073	−1.77545
$392$	4.85113	0.245019
$393$	0	0
$394$	−2.09833	−0.105712
$395$	1.93302	0.0972608
$396$	0	0
$397$	−0.903397	−0.0453402	−0.0226701	−	0.999743i	$-0.507217\pi$
$397$	−0.903397	−0.0453402	−0.0226701	+	0.999743i	$0.507217\pi$
$398$	−2.38418	−0.119508
$399$	0	0
$400$	−15.6383	−0.781915
$401$	9.59179	0.478991	0.239496	−	0.970897i	$-0.423018\pi$
$401$	9.59179	0.478991	0.239496	+	0.970897i	$0.423018\pi$
$402$	0	0
$403$	35.8602	1.78633
$404$	15.0869	0.750600
$405$	0	0
$406$	1.29198	0.0641198
$407$	4.65721	0.230849
$408$	0	0
$409$	30.5997	1.51306	0.756530	−	0.653959i	$-0.226893\pi$
$409$	30.5997	1.51306	0.756530	+	0.653959i	$0.226893\pi$
$410$	0.494324	0.0244129
$411$	0	0
$412$	−21.0304	−1.03609
$413$	0.177928	0.00875528
$414$	0	0
$415$	7.65210	0.375627
$416$	14.6209	0.716848
$417$	0	0
$418$	−1.03776	−0.0507587
$419$	−24.7763	−1.21040	−0.605200	−	0.796073i	$-0.706907\pi$
$419$	−24.7763	−1.21040	−0.605200	+	0.796073i	$0.706907\pi$
$420$	0	0
$421$	−19.8172	−0.965829	−0.482915	−	0.875667i	$-0.660422\pi$
$421$	−19.8172	−0.965829	−0.482915	+	0.875667i	$0.660422\pi$
$422$	−0.889525	−0.0433014
$423$	0	0
$424$	−5.50979	−0.267579
$425$	29.9059	1.45065
$426$	0	0
$427$	−1.27506	−0.0617045
$428$	37.1778	1.79706
$429$	0	0
$430$	−1.54935	−0.0747163
$431$	27.7911	1.33865	0.669326	−	0.742969i	$-0.266583\pi$
$431$	27.7911	1.33865	0.669326	+	0.742969i	$0.266583\pi$
$432$	0	0
$433$	38.0090	1.82660	0.913298	−	0.407292i	$-0.133527\pi$
$433$	38.0090	1.82660	0.913298	+	0.407292i	$0.133527\pi$
$434$	−1.89413	−0.0909211
$435$	0	0
$436$	−16.7746	−0.803359
$437$	−22.5858	−1.08042
$438$	0	0
$439$	36.9234	1.76226	0.881129	−	0.472875i	$-0.156784\pi$
$439$	36.9234	1.76226	0.881129	+	0.472875i	$0.156784\pi$
$440$	−0.787375	−0.0375366
$441$	0	0
$442$	−8.90383	−0.423512
$443$	1.62999	0.0774432	0.0387216	−	0.999250i	$-0.487671\pi$
$443$	1.62999	0.0774432	0.0387216	+	0.999250i	$0.487671\pi$
$444$	0	0
$445$	0.996671	0.0472468
$446$	−2.45491	−0.116243
$447$	0	0
$448$	8.63521	0.407975
$449$	−23.0716	−1.08882	−0.544409	−	0.838820i	$-0.683246\pi$
$449$	−23.0716	−1.08882	−0.544409	+	0.838820i	$0.683246\pi$
$450$	0	0
$451$	2.47842	0.116704
$452$	−30.6370	−1.44104
$453$	0	0
$454$	−3.48750	−0.163677
$455$	−6.13904	−0.287803
$456$	0	0
$457$	1.09124	0.0510458	0.0255229	−	0.999674i	$-0.491875\pi$
$457$	1.09124	0.0510458	0.0255229	+	0.999674i	$0.491875\pi$
$458$	2.39902	0.112099
$459$	0	0
$460$	−8.45469	−0.394202
$461$	41.4760	1.93173	0.965865	−	0.259045i	$-0.0834078\pi$
$461$	41.4760	1.93173	0.965865	+	0.259045i	$0.0834078\pi$
$462$	0	0
$463$	−35.5447	−1.65190	−0.825950	−	0.563743i	$-0.809361\pi$
$463$	−35.5447	−1.65190	−0.825950	+	0.563743i	$0.809361\pi$
$464$	−16.3476	−0.758920
$465$	0	0
$466$	−4.47451	−0.207277
$467$	−11.2608	−0.521089	−0.260545	−	0.965462i	$-0.583902\pi$
$467$	−11.2608	−0.521089	−0.260545	+	0.965462i	$0.583902\pi$
$468$	0	0
$469$	−4.78603	−0.220998
$470$	0.0902905	0.00416479
$471$	0	0
$472$	0.125963	0.00579791
$473$	−7.76809	−0.357177
$474$	0	0
$475$	19.2395	0.882769
$476$	−17.5201	−0.803031
$477$	0	0
$478$	−1.42255	−0.0650658
$479$	6.51581	0.297715	0.148858	−	0.988859i	$-0.452440\pi$
$479$	6.51581	0.297715	0.148858	+	0.988859i	$0.452440\pi$
$480$	0	0
$481$	25.7064	1.17211
$482$	−1.85259	−0.0843832
$483$	0	0
$484$	−1.94772	−0.0885326
$485$	−12.5983	−0.572058
$486$	0	0
$487$	5.85588	0.265355	0.132678	−	0.991159i	$-0.457643\pi$
$487$	5.85588	0.265355	0.132678	+	0.991159i	$0.457643\pi$
$488$	−0.902667	−0.0408618
$489$	0	0
$490$	−1.07189	−0.0484232
$491$	26.7877	1.20891	0.604457	−	0.796638i	$-0.293390\pi$
$491$	26.7877	1.20891	0.604457	+	0.796638i	$0.293390\pi$
$492$	0	0
$493$	31.2624	1.40799
$494$	−5.72815	−0.257722
$495$	0	0
$496$	23.9668	1.07614
$497$	11.9503	0.536044
$498$	0	0
$499$	16.6065	0.743410	0.371705	−	0.928351i	$-0.378773\pi$
$499$	16.6065	0.743410	0.371705	+	0.928351i	$0.378773\pi$
$500$	15.6968	0.701982
$501$	0	0
$502$	−3.10222	−0.138459
$503$	29.9324	1.33462	0.667311	−	0.744780i	$-0.267445\pi$
$503$	29.9324	1.33462	0.667311	+	0.744780i	$0.267445\pi$
$504$	0	0
$505$	−6.75659	−0.300664
$506$	1.13789	0.0505853
$507$	0	0
$508$	38.3421	1.70116
$509$	14.6222	0.648115	0.324058	−	0.946037i	$-0.394953\pi$
$509$	14.6222	0.648115	0.324058	+	0.946037i	$0.394953\pi$
$510$	0	0
$511$	2.27015	0.100425
$512$	16.4316	0.726183
$513$	0	0
$514$	−2.41715	−0.106616
$515$	9.41838	0.415023
$516$	0	0
$517$	0.452696	0.0199095
$518$	−1.35781	−0.0596586
$519$	0	0
$520$	−4.34608	−0.190588
$521$	−14.8066	−0.648691	−0.324345	−	0.945939i	$-0.605144\pi$
$521$	−14.8066	−0.648691	−0.324345	+	0.945939i	$0.605144\pi$
$522$	0	0
$523$	17.9722	0.785868	0.392934	−	0.919567i	$-0.371460\pi$
$523$	17.9722	0.785868	0.392934	+	0.919567i	$0.371460\pi$
$524$	1.31218	0.0573229
$525$	0	0
$526$	−3.69788	−0.161235
$527$	−45.8328	−1.99651
$528$	0	0
$529$	1.76487	0.0767333
$530$	1.21743	0.0528817
$531$	0	0
$532$	−11.2713	−0.488672
$533$	13.6802	0.592554
$534$	0	0
$535$	−16.6499	−0.719838
$536$	−3.38822	−0.146349
$537$	0	0
$538$	−6.02748	−0.259863
$539$	−5.37422	−0.231484
$540$	0	0
$541$	−39.6601	−1.70512	−0.852560	−	0.522630i	$-0.824951\pi$
$541$	−39.6601	−1.70512	−0.852560	+	0.522630i	$0.824951\pi$
$542$	−2.88485	−0.123915
$543$	0	0
$544$	−18.6869	−0.801194
$545$	7.51244	0.321798
$546$	0	0
$547$	8.78206	0.375494	0.187747	−	0.982217i	$-0.439882\pi$
$547$	8.78206	0.375494	0.187747	+	0.982217i	$0.439882\pi$
$548$	2.82809	0.120810
$549$	0	0
$550$	−0.969302	−0.0413312
$551$	20.1122	0.856808
$552$	0	0
$553$	−2.82562	−0.120157
$554$	4.00029	0.169956
$555$	0	0
$556$	−5.78573	−0.245369
$557$	−39.5326	−1.67505	−0.837525	−	0.546399i	$-0.815998\pi$
$557$	−39.5326	−1.67505	−0.837525	+	0.546399i	$0.815998\pi$
$558$	0	0
$559$	−42.8776	−1.81353
$560$	−4.10296	−0.173382
$561$	0	0
$562$	−6.86055	−0.289395
$563$	−27.6996	−1.16740	−0.583699	−	0.811970i	$-0.698395\pi$
$563$	−27.6996	−1.16740	−0.583699	+	0.811970i	$0.698395\pi$
$564$	0	0
$565$	13.7207	0.577232
$566$	−1.90663	−0.0801417
$567$	0	0
$568$	8.46010	0.354978
$569$	−39.6326	−1.66149	−0.830743	−	0.556657i	$-0.812084\pi$
$569$	−39.6326	−1.66149	−0.830743	+	0.556657i	$0.812084\pi$
$570$	0	0
$571$	21.5752	0.902895	0.451448	−	0.892298i	$-0.350908\pi$
$571$	21.5752	0.902895	0.451448	+	0.892298i	$0.350908\pi$
$572$	−10.7508	−0.449514
$573$	0	0
$574$	−0.722584	−0.0301601
$575$	−21.0958	−0.879754
$576$	0	0
$577$	17.4143	0.724965	0.362483	−	0.931990i	$-0.381929\pi$
$577$	17.4143	0.724965	0.362483	+	0.931990i	$0.381929\pi$
$578$	7.49281	0.311660
$579$	0	0
$580$	7.52874	0.312614
$581$	−11.1855	−0.464055
$582$	0	0
$583$	6.10390	0.252798
$584$	1.60713	0.0665035
$585$	0	0
$586$	−4.52665	−0.186994
$587$	46.6587	1.92581	0.962905	−	0.269841i	$-0.0869713\pi$
$587$	46.6587	1.92581	0.962905	+	0.269841i	$0.0869713\pi$
$588$	0	0
$589$	−29.4859	−1.21494
$590$	−0.0278324	−0.00114584
$591$	0	0
$592$	17.1806	0.706118
$593$	−33.5666	−1.37841	−0.689207	−	0.724565i	$-0.742041\pi$
$593$	−33.5666	−1.37841	−0.689207	+	0.724565i	$0.742041\pi$
$594$	0	0
$595$	7.84628	0.321666
$596$	21.8360	0.894436
$597$	0	0
$598$	6.28081	0.256841
$599$	−19.0292	−0.777511	−0.388755	−	0.921341i	$-0.627095\pi$
$599$	−19.0292	−0.777511	−0.388755	+	0.921341i	$0.627095\pi$
$600$	0	0
$601$	−3.94500	−0.160920	−0.0804599	−	0.996758i	$-0.525639\pi$
$601$	−3.94500	−0.160920	−0.0804599	+	0.996758i	$0.525639\pi$
$602$	2.26478	0.0923057
$603$	0	0
$604$	15.7617	0.641333
$605$	0.872276	0.0354631
$606$	0	0
$607$	39.6943	1.61114	0.805572	−	0.592498i	$-0.201858\pi$
$607$	39.6943	1.61114	0.805572	+	0.592498i	$0.201858\pi$
$608$	−12.0219	−0.487554
$609$	0	0
$610$	0.199451	0.00807552
$611$	2.49875	0.101088
$612$	0	0
$613$	18.7680	0.758032	0.379016	−	0.925390i	$-0.376263\pi$
$613$	18.7680	0.758032	0.379016	+	0.925390i	$0.376263\pi$
$614$	5.61018	0.226408
$615$	0	0
$616$	1.15095	0.0463733
$617$	29.2101	1.17596	0.587978	−	0.808877i	$-0.299924\pi$
$617$	29.2101	1.17596	0.587978	+	0.808877i	$0.299924\pi$
$618$	0	0
$619$	14.1366	0.568196	0.284098	−	0.958795i	$-0.408306\pi$
$619$	14.1366	0.568196	0.284098	+	0.958795i	$0.408306\pi$
$620$	−11.0377	−0.443283
$621$	0	0
$622$	−2.87182	−0.115150
$623$	−1.45690	−0.0583693
$624$	0	0
$625$	14.1659	0.566637
$626$	−7.57834	−0.302891
$627$	0	0
$628$	8.26788	0.329924
$629$	−32.8553	−1.31003
$630$	0	0
$631$	40.9034	1.62834	0.814168	−	0.580629i	$-0.197193\pi$
$631$	40.9034	1.62834	0.814168	+	0.580629i	$0.197193\pi$
$632$	−2.00037	−0.0795704
$633$	0	0
$634$	−4.49573	−0.178548
$635$	−17.1714	−0.681425
$636$	0	0
$637$	−29.6641	−1.17534
$638$	−1.01327	−0.0401157
$639$	0	0
$640$	−5.97182	−0.236057
$641$	6.36622	0.251451	0.125725	−	0.992065i	$-0.459874\pi$
$641$	6.36622	0.251451	0.125725	+	0.992065i	$0.459874\pi$
$642$	0	0
$643$	−5.54743	−0.218769	−0.109385	−	0.993999i	$-0.534888\pi$
$643$	−5.54743	−0.218769	−0.109385	+	0.993999i	$0.534888\pi$
$644$	12.3587	0.487003
$645$	0	0
$646$	7.32112	0.288046
$647$	−28.2144	−1.10922	−0.554611	−	0.832110i	$-0.687133\pi$
$647$	−28.2144	−1.10922	−0.554611	+	0.832110i	$0.687133\pi$
$648$	0	0
$649$	−0.139545	−0.00547762
$650$	−5.35026	−0.209854
$651$	0	0
$652$	2.23214	0.0874173
$653$	−18.9375	−0.741081	−0.370541	−	0.928816i	$-0.620828\pi$
$653$	−18.9375	−0.741081	−0.370541	+	0.928816i	$0.620828\pi$
$654$	0	0
$655$	−0.587654	−0.0229615
$656$	9.14299	0.356974
$657$	0	0
$658$	−0.131983	−0.00514524
$659$	47.8590	1.86432	0.932162	−	0.362042i	$-0.117920\pi$
$659$	47.8590	1.86432	0.932162	+	0.362042i	$0.117920\pi$
$660$	0	0
$661$	0.723538	0.0281424	0.0140712	−	0.999901i	$-0.495521\pi$
$661$	0.723538	0.0281424	0.0140712	+	0.999901i	$0.495521\pi$
$662$	−5.21107	−0.202534
$663$	0	0
$664$	−7.91871	−0.307305
$665$	5.04779	0.195745
$666$	0	0
$667$	−22.0526	−0.853882
$668$	43.7169	1.69146
$669$	0	0
$670$	0.748652	0.0289230
$671$	1.00000	0.0386046
$672$	0	0
$673$	−30.8807	−1.19037	−0.595183	−	0.803591i	$-0.702920\pi$
$673$	−30.8807	−1.19037	−0.595183	+	0.803591i	$0.702920\pi$
$674$	0.501673	0.0193237
$675$	0	0
$676$	−34.0210	−1.30850
$677$	−25.9471	−0.997229	−0.498615	−	0.866824i	$-0.666158\pi$
$677$	−25.9471	−0.997229	−0.498615	+	0.866824i	$0.666158\pi$
$678$	0	0
$679$	18.4157	0.706729
$680$	5.55470	0.213013
$681$	0	0
$682$	1.48552	0.0568836
$683$	12.5338	0.479594	0.239797	−	0.970823i	$-0.422919\pi$
$683$	12.5338	0.479594	0.239797	+	0.970823i	$0.422919\pi$
$684$	0	0
$685$	−1.26655	−0.0483923
$686$	3.60770	0.137743
$687$	0	0
$688$	−28.6567	−1.09253
$689$	33.6917	1.28355
$690$	0	0
$691$	−8.77262	−0.333726	−0.166863	−	0.985980i	$-0.553364\pi$
$691$	−8.77262	−0.333726	−0.166863	+	0.985980i	$0.553364\pi$
$692$	41.5408	1.57914
$693$	0	0
$694$	4.03714	0.153248
$695$	2.59111	0.0982865
$696$	0	0
$697$	−17.4846	−0.662276
$698$	−1.35137	−0.0511502
$699$	0	0
$700$	−10.5277	−0.397910
$701$	−18.7353	−0.707621	−0.353810	−	0.935317i	$-0.615114\pi$
$701$	−18.7353	−0.707621	−0.353810	+	0.935317i	$0.615114\pi$
$702$	0	0
$703$	−21.1370	−0.797195
$704$	−6.77239	−0.255244
$705$	0	0
$706$	6.91833	0.260375
$707$	9.87653	0.371445
$708$	0	0
$709$	−7.04862	−0.264716	−0.132358	−	0.991202i	$-0.542255\pi$
$709$	−7.04862	−0.264716	−0.132358	+	0.991202i	$0.542255\pi$
$710$	−1.86932	−0.0701543
$711$	0	0
$712$	−1.03140	−0.0386532
$713$	32.3307	1.21079
$714$	0	0
$715$	4.81471	0.180060
$716$	−42.5529	−1.59028
$717$	0	0
$718$	−0.268624	−0.0100250
$719$	−17.4437	−0.650540	−0.325270	−	0.945621i	$-0.605455\pi$
$719$	−17.4437	−0.650540	−0.325270	+	0.945621i	$0.605455\pi$
$720$	0	0
$721$	−13.7674	−0.512726
$722$	0.365480	0.0136018
$723$	0	0
$724$	−22.3709	−0.831407
$725$	18.7854	0.697671
$726$	0	0
$727$	37.9461	1.40734	0.703671	−	0.710526i	$-0.251543\pi$
$727$	37.9461	1.40734	0.703671	+	0.710526i	$0.251543\pi$
$728$	6.35293	0.235455
$729$	0	0
$730$	−0.355107	−0.0131431
$731$	54.8016	2.02691
$732$	0	0
$733$	−50.3118	−1.85831	−0.929154	−	0.369693i	$-0.879463\pi$
$733$	−50.3118	−1.85831	−0.929154	+	0.369693i	$0.879463\pi$
$734$	5.49350	0.202769
$735$	0	0
$736$	13.1818	0.485889
$737$	3.75357	0.138265
$738$	0	0
$739$	−35.3697	−1.30110	−0.650549	−	0.759465i	$-0.725461\pi$
$739$	−35.3697	−1.30110	−0.650549	+	0.759465i	$0.725461\pi$
$740$	−7.91235	−0.290864
$741$	0	0
$742$	−1.77959	−0.0653308
$743$	−2.96335	−0.108715	−0.0543574	−	0.998522i	$-0.517311\pi$
$743$	−2.96335	−0.108715	−0.0543574	+	0.998522i	$0.517311\pi$
$744$	0	0
$745$	−9.77914	−0.358280
$746$	−6.42133	−0.235102
$747$	0	0
$748$	13.7406	0.502405
$749$	24.3382	0.889298
$750$	0	0
$751$	−13.2766	−0.484468	−0.242234	−	0.970218i	$-0.577880\pi$
$751$	−13.2766	−0.484468	−0.242234	+	0.970218i	$0.577880\pi$
$752$	1.67001	0.0608990
$753$	0	0
$754$	−5.59294	−0.203683
$755$	−7.05879	−0.256896
$756$	0	0
$757$	11.2374	0.408432	0.204216	−	0.978926i	$-0.434536\pi$
$757$	11.2374	0.408432	0.204216	+	0.978926i	$0.434536\pi$
$758$	−8.60181	−0.312432
$759$	0	0
$760$	3.57354	0.129626
$761$	0.978109	0.0354564	0.0177282	−	0.999843i	$-0.494357\pi$
$761$	0.978109	0.0354564	0.0177282	+	0.999843i	$0.494357\pi$
$762$	0	0
$763$	−10.9814	−0.397553
$764$	−8.53996	−0.308965
$765$	0	0
$766$	−8.17232	−0.295278
$767$	−0.770248	−0.0278120
$768$	0	0
$769$	−2.30774	−0.0832191	−0.0416096	−	0.999134i	$-0.513249\pi$
$769$	−2.30774	−0.0832191	−0.0416096	+	0.999134i	$0.513249\pi$
$770$	−0.254312	−0.00916476
$771$	0	0
$772$	−41.4058	−1.49023
$773$	−33.6378	−1.20987	−0.604935	−	0.796275i	$-0.706801\pi$
$773$	−33.6378	−1.20987	−0.604935	+	0.796275i	$0.706801\pi$
$774$	0	0
$775$	−27.5407	−0.989290
$776$	13.0372	0.468009
$777$	0	0
$778$	−0.593827	−0.0212897
$779$	−11.2484	−0.403017
$780$	0	0
$781$	−9.37234	−0.335369
$782$	−8.02748	−0.287062
$783$	0	0
$784$	−19.8257	−0.708060
$785$	−3.70273	−0.132156
$786$	0	0
$787$	21.3015	0.759318	0.379659	−	0.925126i	$-0.376041\pi$
$787$	21.3015	0.759318	0.379659	+	0.925126i	$0.376041\pi$
$788$	17.8738	0.636728
$789$	0	0
$790$	0.441996	0.0157255
$791$	−20.0563	−0.713121
$792$	0	0
$793$	5.51971	0.196010
$794$	−0.206567	−0.00733077
$795$	0	0
$796$	20.3087	0.719824
$797$	−26.9259	−0.953765	−0.476882	−	0.878967i	$-0.658233\pi$
$797$	−26.9259	−0.953765	−0.476882	+	0.878967i	$0.658233\pi$
$798$	0	0
$799$	−3.19364	−0.112983
$800$	−11.2288	−0.396999
$801$	0	0
$802$	2.19322	0.0774451
$803$	−1.78042	−0.0628297
$804$	0	0
$805$	−5.53481	−0.195076
$806$	8.19964	0.288820
$807$	0	0
$808$	6.99200	0.245978
$809$	−18.6486	−0.655648	−0.327824	−	0.944739i	$-0.606315\pi$
$809$	−18.6486	−0.655648	−0.327824	+	0.944739i	$0.606315\pi$
$810$	0	0
$811$	51.6205	1.81264	0.906321	−	0.422590i	$-0.138879\pi$
$811$	51.6205	1.81264	0.906321	+	0.422590i	$0.138879\pi$
$812$	−11.0052	−0.386208
$813$	0	0
$814$	1.06490	0.0373246
$815$	−0.999654	−0.0350163
$816$	0	0
$817$	35.2558	1.23344
$818$	6.99680	0.244637
$819$	0	0
$820$	−4.21071	−0.147044
$821$	43.8024	1.52872	0.764358	−	0.644792i	$-0.223056\pi$
$821$	43.8024	1.52872	0.764358	+	0.644792i	$0.223056\pi$
$822$	0	0
$823$	29.0240	1.01171	0.505856	−	0.862618i	$-0.331177\pi$
$823$	29.0240	1.01171	0.505856	+	0.862618i	$0.331177\pi$
$824$	−9.74653	−0.339536
$825$	0	0
$826$	0.0406843	0.00141559
$827$	−0.0272059	−0.000946042 0	−0.000473021	−	1.00000i	$-0.500151\pi$
$827$	−0.0272059	−0.000946042 0	−0.000473021		1.00000i	$0.500151\pi$
$828$	0	0
$829$	−20.2095	−0.701905	−0.350952	−	0.936393i	$-0.614142\pi$
$829$	−20.2095	−0.701905	−0.350952	+	0.936393i	$0.614142\pi$
$830$	1.74969	0.0607328
$831$	0	0
$832$	−37.3816	−1.29597
$833$	37.9136	1.31363
$834$	0	0
$835$	−19.5784	−0.677540
$836$	8.83980	0.305731
$837$	0	0
$838$	−5.66523	−0.195702
$839$	2.45841	0.0848738	0.0424369	−	0.999099i	$-0.486488\pi$
$839$	2.45841	0.0848738	0.0424369	+	0.999099i	$0.486488\pi$
$840$	0	0
$841$	−9.36254	−0.322846
$842$	−4.53130	−0.156159
$843$	0	0
$844$	7.57708	0.260814
$845$	15.2362	0.524140
$846$	0	0
$847$	−1.27506	−0.0438116
$848$	22.5175	0.773254
$849$	0	0
$850$	6.83814	0.234546
$851$	23.1763	0.794473
$852$	0	0
$853$	14.1701	0.485175	0.242588	−	0.970129i	$-0.422004\pi$
$853$	14.1701	0.485175	0.242588	+	0.970129i	$0.422004\pi$
$854$	−0.291550	−0.00997662
$855$	0	0
$856$	17.2300	0.588909
$857$	−2.30725	−0.0788141	−0.0394070	−	0.999223i	$-0.512547\pi$
$857$	−2.30725	−0.0788141	−0.0394070	+	0.999223i	$0.512547\pi$
$858$	0	0
$859$	−0.0148953	−0.000508220 0	−0.000254110	−	1.00000i	$-0.500081\pi$
$859$	−0.0148953	−0.000508220 0	−0.000254110		1.00000i	$0.500081\pi$
$860$	13.1976	0.450033
$861$	0	0
$862$	6.35459	0.216438
$863$	37.5589	1.27852	0.639260	−	0.768991i	$-0.279241\pi$
$863$	37.5589	1.27852	0.639260	+	0.768991i	$0.279241\pi$
$864$	0	0
$865$	−18.6039	−0.632550
$866$	8.69097	0.295331
$867$	0	0
$868$	16.1344	0.547638
$869$	2.21606	0.0751748
$870$	0	0
$871$	20.7186	0.702023
$872$	−7.77418	−0.263267
$873$	0	0
$874$	−5.16436	−0.174687
$875$	10.2758	0.347386
$876$	0	0
$877$	23.1102	0.780378	0.390189	−	0.920735i	$-0.372410\pi$
$877$	23.1102	0.780378	0.390189	+	0.920735i	$0.372410\pi$
$878$	8.44274	0.284929
$879$	0	0
$880$	3.21786	0.108474
$881$	−21.9178	−0.738430	−0.369215	−	0.929344i	$-0.620373\pi$
$881$	−21.9178	−0.738430	−0.369215	+	0.929344i	$0.620373\pi$
$882$	0	0
$883$	−44.0472	−1.48231	−0.741153	−	0.671336i	$-0.765721\pi$
$883$	−44.0472	−1.48231	−0.741153	+	0.671336i	$0.765721\pi$
$884$	75.8440	2.55091
$885$	0	0
$886$	0.372706	0.0125213
$887$	44.2865	1.48700	0.743498	−	0.668738i	$-0.233165\pi$
$887$	44.2865	1.48700	0.743498	+	0.668738i	$0.233165\pi$
$888$	0	0
$889$	25.1004	0.841842
$890$	0.227894	0.00763904
$891$	0	0
$892$	20.9112	0.700159
$893$	−2.05458	−0.0687539
$894$	0	0
$895$	19.0571	0.637009
$896$	8.72938	0.291628
$897$	0	0
$898$	−5.27546	−0.176044
$899$	−28.7899	−0.960196
$900$	0	0
$901$	−43.0613	−1.43458
$902$	0.566706	0.0188692
$903$	0	0
$904$	−14.1987	−0.472242
$905$	10.0187	0.333033
$906$	0	0
$907$	−11.7897	−0.391471	−0.195736	−	0.980657i	$-0.562709\pi$
$907$	−11.7897	−0.391471	−0.195736	+	0.980657i	$0.562709\pi$
$908$	29.7070	0.985861
$909$	0	0
$910$	−1.40373	−0.0465330
$911$	50.2024	1.66328	0.831640	−	0.555315i	$-0.187402\pi$
$911$	50.2024	1.66328	0.831640	+	0.555315i	$0.187402\pi$
$912$	0	0
$913$	8.77256	0.290329
$914$	0.249517	0.00825329
$915$	0	0
$916$	−20.4352	−0.675198
$917$	0.859010	0.0283670
$918$	0	0
$919$	−4.09392	−0.135046	−0.0675230	−	0.997718i	$-0.521510\pi$
$919$	−4.09392	−0.135046	−0.0675230	+	0.997718i	$0.521510\pi$
$920$	−3.91832	−0.129183
$921$	0	0
$922$	9.48372	0.312330
$923$	−51.7325	−1.70280
$924$	0	0
$925$	−19.7425	−0.649131
$926$	−8.12748	−0.267086
$927$	0	0
$928$	−11.7382	−0.385324
$929$	−1.87956	−0.0616662	−0.0308331	−	0.999525i	$-0.509816\pi$
$929$	−1.87956	−0.0616662	−0.0308331	+	0.999525i	$0.509816\pi$
$930$	0	0
$931$	24.3911	0.799387
$932$	38.1144	1.24848
$933$	0	0
$934$	−2.57485	−0.0842517
$935$	−6.15366	−0.201246
$936$	0	0
$937$	−21.8551	−0.713974	−0.356987	−	0.934109i	$-0.616196\pi$
$937$	−21.8551	−0.713974	−0.356987	+	0.934109i	$0.616196\pi$
$938$	−1.09435	−0.0357319
$939$	0	0
$940$	−0.769106	−0.0250855
$941$	−16.1328	−0.525915	−0.262957	−	0.964807i	$-0.584698\pi$
$941$	−16.1328	−0.525915	−0.262957	+	0.964807i	$0.584698\pi$
$942$	0	0
$943$	12.3337	0.401641
$944$	−0.514787	−0.0167549
$945$	0	0
$946$	−1.77622	−0.0577498
$947$	3.33314	0.108312	0.0541562	−	0.998532i	$-0.482753\pi$
$947$	3.33314	0.108312	0.0541562	+	0.998532i	$0.482753\pi$
$948$	0	0
$949$	−9.82741	−0.319011
$950$	4.39922	0.142729
$951$	0	0
$952$	−8.11966	−0.263160
$953$	51.2844	1.66126	0.830632	−	0.556822i	$-0.187980\pi$
$953$	51.2844	1.66126	0.830632	+	0.556822i	$0.187980\pi$
$954$	0	0
$955$	3.82458	0.123761
$956$	12.1174	0.391906
$957$	0	0
$958$	1.48988	0.0481357
$959$	1.85139	0.0597846
$960$	0	0
$961$	11.2080	0.361548
$962$	5.87792	0.189512
$963$	0	0
$964$	15.7806	0.508259
$965$	18.5434	0.596934
$966$	0	0
$967$	9.99705	0.321484	0.160742	−	0.986996i	$-0.448611\pi$
$967$	9.99705	0.321484	0.160742	+	0.986996i	$0.448611\pi$
$968$	−0.902667	−0.0290128
$969$	0	0
$970$	−2.88067	−0.0924926
$971$	−3.60803	−0.115787	−0.0578936	−	0.998323i	$-0.518438\pi$
$971$	−3.60803	−0.115787	−0.0578936	+	0.998323i	$0.518438\pi$
$972$	0	0
$973$	−3.78759	−0.121425
$974$	1.33898	0.0429037
$975$	0	0
$976$	3.68903	0.118083
$977$	−51.4571	−1.64626	−0.823130	−	0.567853i	$-0.807774\pi$
$977$	−51.4571	−1.64626	−0.823130	+	0.567853i	$0.807774\pi$
$978$	0	0
$979$	1.14261	0.0365180
$980$	9.13052	0.291664
$981$	0	0
$982$	6.12516	0.195462
$983$	35.9137	1.14547	0.572734	−	0.819741i	$-0.305883\pi$
$983$	35.9137	1.14547	0.572734	+	0.819741i	$0.305883\pi$
$984$	0	0
$985$	−8.00470	−0.255051
$986$	7.14832	0.227649
$987$	0	0
$988$	48.7931	1.55231
$989$	−38.6574	−1.22923
$990$	0	0
$991$	23.0440	0.732017	0.366009	−	0.930612i	$-0.380724\pi$
$991$	23.0440	0.732017	0.366009	+	0.930612i	$0.380724\pi$
$992$	17.2090	0.546386
$993$	0	0
$994$	2.73250	0.0866697
$995$	−9.09517	−0.288336
$996$	0	0
$997$	39.8343	1.26157	0.630783	−	0.775959i	$-0.282734\pi$
$997$	39.8343	1.26157	0.630783	+	0.775959i	$0.282734\pi$
$998$	3.79717	0.120197
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.14	✓	25
3.2	odd	2		6039.2.a.p.1.12	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.14	✓	25	1.1	even	1	trivial
6039.2.a.p.1.12	yes	25	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.228656 q^{2} -1.94772 q^{4} +0.872276 q^{5} -1.27506 q^{7} -0.902667 q^{8} +O(q^{10})\)	\(q+0.228656 q^{2} -1.94772 q^{4} +0.872276 q^{5} -1.27506 q^{7} -0.902667 q^{8} +0.199451 q^{10} +1.00000 q^{11} +5.51971 q^{13} -0.291550 q^{14} +3.68903 q^{16} -7.05471 q^{17} -4.53854 q^{19} -1.69895 q^{20} +0.228656 q^{22} +4.97643 q^{23} -4.23913 q^{25} +1.26211 q^{26} +2.48346 q^{28} -4.43142 q^{29} +6.49677 q^{31} +2.64885 q^{32} -1.61310 q^{34} -1.11220 q^{35} +4.65721 q^{37} -1.03776 q^{38} -0.787375 q^{40} +2.47842 q^{41} -7.76809 q^{43} -1.94772 q^{44} +1.13789 q^{46} +0.452696 q^{47} -5.37422 q^{49} -0.969302 q^{50} -10.7508 q^{52} +6.10390 q^{53} +0.872276 q^{55} +1.15095 q^{56} -1.01327 q^{58} -0.139545 q^{59} +1.00000 q^{61} +1.48552 q^{62} -6.77239 q^{64} +4.81471 q^{65} +3.75357 q^{67} +13.7406 q^{68} -0.254312 q^{70} -9.37234 q^{71} -1.78042 q^{73} +1.06490 q^{74} +8.83980 q^{76} -1.27506 q^{77} +2.21606 q^{79} +3.21786 q^{80} +0.566706 q^{82} +8.77256 q^{83} -6.15366 q^{85} -1.77622 q^{86} -0.902667 q^{88} +1.14261 q^{89} -7.03796 q^{91} -9.69268 q^{92} +0.103511 q^{94} -3.95886 q^{95} -14.4430 q^{97} -1.22885 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 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

Modular forms

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Database

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