LMFDB - Embedded newform 6039.2.a.p.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:	$0$
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)

$f(q)$	$=$	$q-1.23685 q^{2} -0.470198 q^{4} +0.403292 q^{5} -0.303505 q^{7} +3.05527 q^{8} +O(q^{10})$	\(q-1.23685 q^{2} -0.470198 q^{4} +0.403292 q^{5} -0.303505 q^{7} +3.05527 q^{8} -0.498813 q^{10} -1.00000 q^{11} -3.53056 q^{13} +0.375391 q^{14} -2.83852 q^{16} -7.19973 q^{17} +3.67072 q^{19} -0.189627 q^{20} +1.23685 q^{22} +3.45314 q^{23} -4.83736 q^{25} +4.36677 q^{26} +0.142707 q^{28} -2.49100 q^{29} +2.86904 q^{31} -2.59971 q^{32} +8.90500 q^{34} -0.122401 q^{35} +3.00574 q^{37} -4.54014 q^{38} +1.23217 q^{40} -4.84782 q^{41} -5.93177 q^{43} +0.470198 q^{44} -4.27103 q^{46} +8.13596 q^{47} -6.90788 q^{49} +5.98309 q^{50} +1.66006 q^{52} -1.91085 q^{53} -0.403292 q^{55} -0.927290 q^{56} +3.08100 q^{58} -1.68282 q^{59} +1.00000 q^{61} -3.54858 q^{62} +8.89249 q^{64} -1.42385 q^{65} -0.0898202 q^{67} +3.38530 q^{68} +0.151392 q^{70} +13.0923 q^{71} -14.0564 q^{73} -3.71766 q^{74} -1.72597 q^{76} +0.303505 q^{77} +3.46200 q^{79} -1.14475 q^{80} +5.99604 q^{82} +2.45660 q^{83} -2.90360 q^{85} +7.33671 q^{86} -3.05527 q^{88} +12.1773 q^{89} +1.07154 q^{91} -1.62366 q^{92} -10.0630 q^{94} +1.48037 q^{95} -1.70611 q^{97} +8.54403 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.23685	−0.874586	−0.437293	−	0.899319i	$-0.644063\pi$
$2$	−1.23685	−0.874586	−0.437293	+	0.899319i	$0.644063\pi$
$3$	0	0
$3$	0	0
$4$	−0.470198	−0.235099
$5$	0.403292	0.180358	0.0901789	−	0.995926i	$-0.471256\pi$
$5$	0.403292	0.180358	0.0901789	+	0.995926i	$0.471256\pi$
$6$	0	0
$7$	−0.303505	−0.114714	−0.0573571	−	0.998354i	$-0.518267\pi$
$7$	−0.303505	−0.114714	−0.0573571	+	0.998354i	$0.518267\pi$
$8$	3.05527	1.08020
$9$	0	0
$10$	−0.498813	−0.157738
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.53056	−0.979200	−0.489600	−	0.871947i	$-0.662857\pi$
$13$	−3.53056	−0.979200	−0.489600	+	0.871947i	$0.662857\pi$
$14$	0.375391	0.100327
$15$	0	0
$16$	−2.83852	−0.709630
$17$	−7.19973	−1.74619	−0.873096	−	0.487549i	$-0.837891\pi$
$17$	−7.19973	−1.74619	−0.873096	+	0.487549i	$0.837891\pi$
$18$	0	0
$19$	3.67072	0.842122	0.421061	−	0.907032i	$-0.361658\pi$
$19$	3.67072	0.842122	0.421061	+	0.907032i	$0.361658\pi$
$20$	−0.189627	−0.0424019
$21$	0	0
$22$	1.23685	0.263698
$23$	3.45314	0.720030	0.360015	−	0.932946i	$-0.382772\pi$
$23$	3.45314	0.720030	0.360015	+	0.932946i	$0.382772\pi$
$24$	0	0
$25$	−4.83736	−0.967471
$26$	4.36677	0.856395
$27$	0	0
$28$	0.142707	0.0269692
$29$	−2.49100	−0.462567	−0.231283	−	0.972886i	$-0.574292\pi$
$29$	−2.49100	−0.462567	−0.231283	+	0.972886i	$0.574292\pi$
$30$	0	0
$31$	2.86904	0.515295	0.257648	−	0.966239i	$-0.417053\pi$
$31$	2.86904	0.515295	0.257648	+	0.966239i	$0.417053\pi$
$32$	−2.59971	−0.459568
$33$	0	0
$34$	8.90500	1.52720
$35$	−0.122401	−0.0206896
$36$	0	0
$37$	3.00574	0.494141	0.247071	−	0.968997i	$-0.420532\pi$
$37$	3.00574	0.494141	0.247071	+	0.968997i	$0.420532\pi$
$38$	−4.54014	−0.736508
$39$	0	0
$40$	1.23217	0.194822
$41$	−4.84782	−0.757103	−0.378551	−	0.925580i	$-0.623578\pi$
$41$	−4.84782	−0.757103	−0.378551	+	0.925580i	$0.623578\pi$
$42$	0	0
$43$	−5.93177	−0.904586	−0.452293	−	0.891869i	$-0.649394\pi$
$43$	−5.93177	−0.904586	−0.452293	+	0.891869i	$0.649394\pi$
$44$	0.470198	0.0708850
$45$	0	0
$46$	−4.27103	−0.629729
$47$	8.13596	1.18675	0.593376	−	0.804925i	$-0.297795\pi$
$47$	8.13596	1.18675	0.593376	+	0.804925i	$0.297795\pi$
$48$	0	0
$49$	−6.90788	−0.986841
$50$	5.98309	0.846137
$51$	0	0
$52$	1.66006	0.230209
$53$	−1.91085	−0.262475	−0.131238	−	0.991351i	$-0.541895\pi$
$53$	−1.91085	−0.262475	−0.131238	+	0.991351i	$0.541895\pi$
$54$	0	0
$55$	−0.403292	−0.0543799
$56$	−0.927290	−0.123914
$57$	0	0
$58$	3.08100	0.404555
$59$	−1.68282	−0.219084	−0.109542	−	0.993982i	$-0.534938\pi$
$59$	−1.68282	−0.219084	−0.109542	+	0.993982i	$0.534938\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−3.54858	−0.450670
$63$	0	0
$64$	8.89249	1.11156
$65$	−1.42385	−0.176606
$66$	0	0
$67$	−0.0898202	−0.0109733	−0.00548664	−	0.999985i	$-0.501746\pi$
$67$	−0.0898202	−0.0109733	−0.00548664	+	0.999985i	$0.501746\pi$
$68$	3.38530	0.410528
$69$	0	0
$70$	0.151392	0.0180948
$71$	13.0923	1.55377	0.776886	−	0.629642i	$-0.216798\pi$
$71$	13.0923	1.55377	0.776886	+	0.629642i	$0.216798\pi$
$72$	0	0
$73$	−14.0564	−1.64518	−0.822589	−	0.568637i	$-0.807471\pi$
$73$	−14.0564	−1.64518	−0.822589	+	0.568637i	$0.807471\pi$
$74$	−3.71766	−0.432169
$75$	0	0
$76$	−1.72597	−0.197982
$77$	0.303505	0.0345876
$78$	0	0
$79$	3.46200	0.389505	0.194753	−	0.980852i	$-0.437610\pi$
$79$	3.46200	0.389505	0.194753	+	0.980852i	$0.437610\pi$
$80$	−1.14475	−0.127987
$81$	0	0
$82$	5.99604	0.662152
$83$	2.45660	0.269646	0.134823	−	0.990870i	$-0.456953\pi$
$83$	2.45660	0.269646	0.134823	+	0.990870i	$0.456953\pi$
$84$	0	0
$85$	−2.90360	−0.314939
$86$	7.33671	0.791138
$87$	0	0
$88$	−3.05527	−0.325693
$89$	12.1773	1.29079	0.645397	−	0.763848i	$-0.276692\pi$
$89$	12.1773	1.29079	0.645397	+	0.763848i	$0.276692\pi$
$90$	0	0
$91$	1.07154	0.112328
$92$	−1.62366	−0.169278
$93$	0	0
$94$	−10.0630	−1.03792
$95$	1.48037	0.151883
$96$	0	0
$97$	−1.70611	−0.173229	−0.0866145	−	0.996242i	$-0.527605\pi$
$97$	−1.70611	−0.173229	−0.0866145	+	0.996242i	$0.527605\pi$
$98$	8.54403	0.863077
$99$	0	0
$100$	2.27451	0.227451
$101$	3.09935	0.308397	0.154198	−	0.988040i	$-0.450721\pi$
$101$	3.09935	0.308397	0.154198	+	0.988040i	$0.450721\pi$
$102$	0	0
$103$	−1.70468	−0.167967	−0.0839834	−	0.996467i	$-0.526764\pi$
$103$	−1.70468	−0.167967	−0.0839834	+	0.996467i	$0.526764\pi$
$104$	−10.7868	−1.05773
$105$	0	0
$106$	2.36344	0.229557
$107$	11.9236	1.15270	0.576350	−	0.817203i	$-0.304477\pi$
$107$	11.9236	1.15270	0.576350	+	0.817203i	$0.304477\pi$
$108$	0	0
$109$	−3.88221	−0.371848	−0.185924	−	0.982564i	$-0.559528\pi$
$109$	−3.88221	−0.371848	−0.185924	+	0.982564i	$0.559528\pi$
$110$	0.498813	0.0475599
$111$	0	0
$112$	0.861505	0.0814046
$113$	13.8764	1.30538	0.652692	−	0.757623i	$-0.273639\pi$
$113$	13.8764	1.30538	0.652692	+	0.757623i	$0.273639\pi$
$114$	0	0
$115$	1.39263	0.129863
$116$	1.17126	0.108749
$117$	0	0
$118$	2.08140	0.191608
$119$	2.18516	0.200313
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.23685	−0.111979
$123$	0	0
$124$	−1.34902	−0.121145
$125$	−3.96733	−0.354849
$126$	0	0
$127$	12.9779	1.15160	0.575801	−	0.817590i	$-0.304690\pi$
$127$	12.9779	1.15160	0.575801	+	0.817590i	$0.304690\pi$
$128$	−5.79927	−0.512588
$129$	0	0
$130$	1.76109	0.154457
$131$	−0.0196484	−0.00171669	−0.000858346	−	1.00000i	$-0.500273\pi$
$131$	−0.0196484	−0.00171669	−0.000858346		1.00000i	$0.500273\pi$
$132$	0	0
$133$	−1.11408	−0.0966033
$134$	0.111094	0.00959708
$135$	0	0
$136$	−21.9971	−1.88624
$137$	0.274897	0.0234860	0.0117430	−	0.999931i	$-0.496262\pi$
$137$	0.274897	0.0234860	0.0117430	+	0.999931i	$0.496262\pi$
$138$	0	0
$139$	−14.3695	−1.21881	−0.609404	−	0.792860i	$-0.708591\pi$
$139$	−14.3695	−1.21881	−0.609404	+	0.792860i	$0.708591\pi$
$140$	0.0575528	0.00486410
$141$	0	0
$142$	−16.1932	−1.35891
$143$	3.53056	0.295240
$144$	0	0
$145$	−1.00460	−0.0834275
$146$	17.3857	1.43885
$147$	0	0
$148$	−1.41329	−0.116172
$149$	15.6059	1.27848	0.639242	−	0.769006i	$-0.279248\pi$
$149$	15.6059	1.27848	0.639242	+	0.769006i	$0.279248\pi$
$150$	0	0
$151$	20.4498	1.66418	0.832091	−	0.554639i	$-0.187143\pi$
$151$	20.4498	1.66418	0.832091	+	0.554639i	$0.187143\pi$
$152$	11.2150	0.909660
$153$	0	0
$154$	−0.375391	−0.0302499
$155$	1.15706	0.0929375
$156$	0	0
$157$	−22.7960	−1.81932	−0.909659	−	0.415356i	$-0.863657\pi$
$157$	−22.7960	−1.81932	−0.909659	+	0.415356i	$0.863657\pi$
$158$	−4.28198	−0.340656
$159$	0	0
$160$	−1.04844	−0.0828867
$161$	−1.04805	−0.0825977
$162$	0	0
$163$	5.52845	0.433022	0.216511	−	0.976280i	$-0.430532\pi$
$163$	5.52845	0.433022	0.216511	+	0.976280i	$0.430532\pi$
$164$	2.27944	0.177994
$165$	0	0
$166$	−3.03844	−0.235829
$167$	−22.2926	−1.72505	−0.862526	−	0.506013i	$-0.831119\pi$
$167$	−22.2926	−1.72505	−0.862526	+	0.506013i	$0.831119\pi$
$168$	0	0
$169$	−0.535178	−0.0411675
$170$	3.59132	0.275441
$171$	0	0
$172$	2.78910	0.212667
$173$	−2.92197	−0.222153	−0.111077	−	0.993812i	$-0.535430\pi$
$173$	−2.92197	−0.222153	−0.111077	+	0.993812i	$0.535430\pi$
$174$	0	0
$175$	1.46816	0.110983
$176$	2.83852	0.213961
$177$	0	0
$178$	−15.0615	−1.12891
$179$	8.74757	0.653824	0.326912	−	0.945055i	$-0.393992\pi$
$179$	8.74757	0.653824	0.326912	+	0.945055i	$0.393992\pi$
$180$	0	0
$181$	−21.9185	−1.62919	−0.814596	−	0.580028i	$-0.803041\pi$
$181$	−21.9185	−1.62919	−0.814596	+	0.580028i	$0.803041\pi$
$182$	−1.32534	−0.0982406
$183$	0	0
$184$	10.5503	0.777777
$185$	1.21219	0.0891222
$186$	0	0
$187$	7.19973	0.526497
$188$	−3.82551	−0.279004
$189$	0	0
$190$	−1.83100	−0.132835
$191$	1.79904	0.130174	0.0650869	−	0.997880i	$-0.479268\pi$
$191$	1.79904	0.130174	0.0650869	+	0.997880i	$0.479268\pi$
$192$	0	0
$193$	6.99299	0.503366	0.251683	−	0.967810i	$-0.419016\pi$
$193$	6.99299	0.503366	0.251683	+	0.967810i	$0.419016\pi$
$194$	2.11020	0.151504
$195$	0	0
$196$	3.24807	0.232005
$197$	−19.7619	−1.40798	−0.703989	−	0.710211i	$-0.748600\pi$
$197$	−19.7619	−1.40798	−0.703989	+	0.710211i	$0.748600\pi$
$198$	0	0
$199$	−4.95005	−0.350900	−0.175450	−	0.984488i	$-0.556138\pi$
$199$	−4.95005	−0.350900	−0.175450	+	0.984488i	$0.556138\pi$
$200$	−14.7794	−1.04506
$201$	0	0
$202$	−3.83343	−0.269720
$203$	0.756031	0.0530630
$204$	0	0
$205$	−1.95509	−0.136549
$206$	2.10843	0.146901
$207$	0	0
$208$	10.0215	0.694869
$209$	−3.67072	−0.253909
$210$	0	0
$211$	−8.51717	−0.586346	−0.293173	−	0.956059i	$-0.594711\pi$
$211$	−8.51717	−0.586346	−0.293173	+	0.956059i	$0.594711\pi$
$212$	0.898477	0.0617076
$213$	0	0
$214$	−14.7477	−1.00813
$215$	−2.39223	−0.163149
$216$	0	0
$217$	−0.870770	−0.0591117
$218$	4.80172	0.325213
$219$	0	0
$220$	0.189627	0.0127847
$221$	25.4191	1.70987
$222$	0	0
$223$	−5.10025	−0.341538	−0.170769	−	0.985311i	$-0.554625\pi$
$223$	−5.10025	−0.341538	−0.170769	+	0.985311i	$0.554625\pi$
$224$	0.789026	0.0527190
$225$	0	0
$226$	−17.1631	−1.14167
$227$	−13.1071	−0.869951	−0.434976	−	0.900442i	$-0.643243\pi$
$227$	−13.1071	−0.869951	−0.434976	+	0.900442i	$0.643243\pi$
$228$	0	0
$229$	8.15429	0.538851	0.269425	−	0.963021i	$-0.413166\pi$
$229$	8.15429	0.538851	0.269425	+	0.963021i	$0.413166\pi$
$230$	−1.72247	−0.113576
$231$	0	0
$232$	−7.61067	−0.499665
$233$	10.5083	0.688419	0.344209	−	0.938893i	$-0.388147\pi$
$233$	10.5083	0.688419	0.344209	+	0.938893i	$0.388147\pi$
$234$	0	0
$235$	3.28117	0.214040
$236$	0.791258	0.0515065
$237$	0	0
$238$	−2.70271	−0.175191
$239$	21.1592	1.36868	0.684338	−	0.729165i	$-0.260091\pi$
$239$	21.1592	1.36868	0.684338	+	0.729165i	$0.260091\pi$
$240$	0	0
$241$	−29.1626	−1.87853	−0.939264	−	0.343194i	$-0.888491\pi$
$241$	−29.1626	−1.87853	−0.939264	+	0.343194i	$0.888491\pi$
$242$	−1.23685	−0.0795078
$243$	0	0
$244$	−0.470198	−0.0301013
$245$	−2.78590	−0.177984
$246$	0	0
$247$	−12.9597	−0.824606
$248$	8.76570	0.556622
$249$	0	0
$250$	4.90700	0.310346
$251$	24.0553	1.51835	0.759177	−	0.650884i	$-0.225602\pi$
$251$	24.0553	1.51835	0.759177	+	0.650884i	$0.225602\pi$
$252$	0	0
$253$	−3.45314	−0.217097
$254$	−16.0517	−1.00718
$255$	0	0
$256$	−10.6121	−0.663259
$257$	−12.6806	−0.790993	−0.395497	−	0.918467i	$-0.629427\pi$
$257$	−12.6806	−0.790993	−0.395497	+	0.918467i	$0.629427\pi$
$258$	0	0
$259$	−0.912259	−0.0566850
$260$	0.669489	0.0415199
$261$	0	0
$262$	0.0243022	0.00150140
$263$	25.0698	1.54587	0.772934	−	0.634487i	$-0.218789\pi$
$263$	25.0698	1.54587	0.772934	+	0.634487i	$0.218789\pi$
$264$	0	0
$265$	−0.770630	−0.0473394
$266$	1.37796	0.0844879
$267$	0	0
$268$	0.0422332	0.00257981
$269$	4.34238	0.264760	0.132380	−	0.991199i	$-0.457738\pi$
$269$	4.34238	0.264760	0.132380	+	0.991199i	$0.457738\pi$
$270$	0	0
$271$	17.9773	1.09204	0.546021	−	0.837771i	$-0.316142\pi$
$271$	17.9773	1.09204	0.546021	+	0.837771i	$0.316142\pi$
$272$	20.4366	1.23915
$273$	0	0
$274$	−0.340007	−0.0205406
$275$	4.83736	0.291704
$276$	0	0
$277$	15.1868	0.912485	0.456242	−	0.889856i	$-0.349195\pi$
$277$	15.1868	0.912485	0.456242	+	0.889856i	$0.349195\pi$
$278$	17.7730	1.06595
$279$	0	0
$280$	−0.373969	−0.0223489
$281$	−21.9272	−1.30807	−0.654034	−	0.756465i	$-0.726925\pi$
$281$	−21.9272	−1.30807	−0.654034	+	0.756465i	$0.726925\pi$
$282$	0	0
$283$	−21.5407	−1.28046	−0.640230	−	0.768183i	$-0.721161\pi$
$283$	−21.5407	−1.28046	−0.640230	+	0.768183i	$0.721161\pi$
$284$	−6.15598	−0.365290
$285$	0	0
$286$	−4.36677	−0.258213
$287$	1.47134	0.0868504
$288$	0	0
$289$	34.8362	2.04919
$290$	1.24254	0.0729645
$291$	0	0
$292$	6.60929	0.386779
$293$	29.3795	1.71637	0.858183	−	0.513344i	$-0.171593\pi$
$293$	29.3795	1.71637	0.858183	+	0.513344i	$0.171593\pi$
$294$	0	0
$295$	−0.678668	−0.0395136
$296$	9.18335	0.533771
$297$	0	0
$298$	−19.3022	−1.11814
$299$	−12.1915	−0.705054
$300$	0	0
$301$	1.80032	0.103769
$302$	−25.2934	−1.45547
$303$	0	0
$304$	−10.4194	−0.597595
$305$	0.403292	0.0230924
$306$	0	0
$307$	27.0803	1.54555	0.772777	−	0.634678i	$-0.218867\pi$
$307$	27.0803	1.54555	0.772777	+	0.634678i	$0.218867\pi$
$308$	−0.142707	−0.00813151
$309$	0	0
$310$	−1.43112	−0.0812819
$311$	29.9461	1.69809	0.849044	−	0.528321i	$-0.177178\pi$
$311$	29.9461	1.69809	0.849044	+	0.528321i	$0.177178\pi$
$312$	0	0
$313$	5.01342	0.283376	0.141688	−	0.989911i	$-0.454747\pi$
$313$	5.01342	0.283376	0.141688	+	0.989911i	$0.454747\pi$
$314$	28.1953	1.59115
$315$	0	0
$316$	−1.62782	−0.0915723
$317$	4.61539	0.259226	0.129613	−	0.991565i	$-0.458626\pi$
$317$	4.61539	0.259226	0.129613	+	0.991565i	$0.458626\pi$
$318$	0	0
$319$	2.49100	0.139469
$320$	3.58627	0.200479
$321$	0	0
$322$	1.29628	0.0722388
$323$	−26.4282	−1.47051
$324$	0	0
$325$	17.0786	0.947348
$326$	−6.83787	−0.378715
$327$	0	0
$328$	−14.8114	−0.817823
$329$	−2.46931	−0.136137
$330$	0	0
$331$	4.66498	0.256410	0.128205	−	0.991748i	$-0.459078\pi$
$331$	4.66498	0.256410	0.128205	+	0.991748i	$0.459078\pi$
$332$	−1.15509	−0.0633936
$333$	0	0
$334$	27.5726	1.50871
$335$	−0.0362238	−0.00197912
$336$	0	0
$337$	19.6742	1.07172	0.535861	−	0.844306i	$-0.319987\pi$
$337$	19.6742	1.07172	0.535861	+	0.844306i	$0.319987\pi$
$338$	0.661936	0.0360046
$339$	0	0
$340$	1.36526	0.0740419
$341$	−2.86904	−0.155367
$342$	0	0
$343$	4.22112	0.227919
$344$	−18.1231	−0.977134
$345$	0	0
$346$	3.61404	0.194292
$347$	3.66629	0.196817	0.0984083	−	0.995146i	$-0.468625\pi$
$347$	3.66629	0.196817	0.0984083	+	0.995146i	$0.468625\pi$
$348$	0	0
$349$	10.1270	0.542084	0.271042	−	0.962567i	$-0.412632\pi$
$349$	10.1270	0.542084	0.271042	+	0.962567i	$0.412632\pi$
$350$	−1.81590	−0.0970639
$351$	0	0
$352$	2.59971	0.138565
$353$	−16.6514	−0.886265	−0.443133	−	0.896456i	$-0.646133\pi$
$353$	−16.6514	−0.886265	−0.443133	+	0.896456i	$0.646133\pi$
$354$	0	0
$355$	5.28003	0.280235
$356$	−5.72575	−0.303464
$357$	0	0
$358$	−10.8194	−0.571825
$359$	11.9942	0.633027	0.316514	−	0.948588i	$-0.397488\pi$
$359$	11.9942	0.633027	0.316514	+	0.948588i	$0.397488\pi$
$360$	0	0
$361$	−5.52579	−0.290831
$362$	27.1100	1.42487
$363$	0	0
$364$	−0.503837	−0.0264082
$365$	−5.66884	−0.296720
$366$	0	0
$367$	9.28210	0.484522	0.242261	−	0.970211i	$-0.422111\pi$
$367$	9.28210	0.484522	0.242261	+	0.970211i	$0.422111\pi$
$368$	−9.80181	−0.510955
$369$	0	0
$370$	−1.49930	−0.0779450
$371$	0.579952	0.0301096
$372$	0	0
$373$	10.6865	0.553327	0.276663	−	0.960967i	$-0.410771\pi$
$373$	10.6865	0.553327	0.276663	+	0.960967i	$0.410771\pi$
$374$	−8.90500	−0.460467
$375$	0	0
$376$	24.8575	1.28193
$377$	8.79461	0.452945
$378$	0	0
$379$	32.1527	1.65157	0.825787	−	0.563982i	$-0.190731\pi$
$379$	32.1527	1.65157	0.825787	+	0.563982i	$0.190731\pi$
$380$	−0.696069	−0.0357076
$381$	0	0
$382$	−2.22514	−0.113848
$383$	22.4294	1.14609	0.573046	−	0.819524i	$-0.305762\pi$
$383$	22.4294	1.14609	0.573046	+	0.819524i	$0.305762\pi$
$384$	0	0
$385$	0.122401	0.00623815
$386$	−8.64929	−0.440237
$387$	0	0
$388$	0.802209	0.0407260
$389$	23.7359	1.20346	0.601729	−	0.798700i	$-0.294479\pi$
$389$	23.7359	1.20346	0.601729	+	0.798700i	$0.294479\pi$
$390$	0	0
$391$	−24.8617	−1.25731
$392$	−21.1054	−1.06599
$393$	0	0
$394$	24.4426	1.23140
$395$	1.39620	0.0702503
$396$	0	0
$397$	13.5028	0.677686	0.338843	−	0.940843i	$-0.389964\pi$
$397$	13.5028	0.677686	0.338843	+	0.940843i	$0.389964\pi$
$398$	6.12248	0.306892
$399$	0	0
$400$	13.7309	0.686546
$401$	−28.6582	−1.43112	−0.715561	−	0.698550i	$-0.753829\pi$
$401$	−28.6582	−1.43112	−0.715561	+	0.698550i	$0.753829\pi$
$402$	0	0
$403$	−10.1293	−0.504577
$404$	−1.45731	−0.0725037
$405$	0	0
$406$	−0.935098	−0.0464081
$407$	−3.00574	−0.148989
$408$	0	0
$409$	19.4603	0.962252	0.481126	−	0.876651i	$-0.340228\pi$
$409$	19.4603	0.962252	0.481126	+	0.876651i	$0.340228\pi$
$410$	2.41816	0.119424
$411$	0	0
$412$	0.801536	0.0394888
$413$	0.510744	0.0251321
$414$	0	0
$415$	0.990726	0.0486328
$416$	9.17842	0.450009
$417$	0	0
$418$	4.54014	0.222066
$419$	17.3821	0.849169	0.424585	−	0.905388i	$-0.360420\pi$
$419$	17.3821	0.849169	0.424585	+	0.905388i	$0.360420\pi$
$420$	0	0
$421$	10.5440	0.513885	0.256943	−	0.966427i	$-0.417285\pi$
$421$	10.5440	0.513885	0.256943	+	0.966427i	$0.417285\pi$
$422$	10.5345	0.512810
$423$	0	0
$424$	−5.83815	−0.283526
$425$	34.8277	1.68939
$426$	0	0
$427$	−0.303505	−0.0146876
$428$	−5.60646	−0.270998
$429$	0	0
$430$	2.95884	0.142688
$431$	−32.8548	−1.58256	−0.791281	−	0.611453i	$-0.790585\pi$
$431$	−32.8548	−1.58256	−0.791281	+	0.611453i	$0.790585\pi$
$432$	0	0
$433$	8.53171	0.410008	0.205004	−	0.978761i	$-0.434279\pi$
$433$	8.53171	0.410008	0.205004	+	0.978761i	$0.434279\pi$
$434$	1.07701	0.0516983
$435$	0	0
$436$	1.82541	0.0874211
$437$	12.6755	0.606353
$438$	0	0
$439$	14.2501	0.680122	0.340061	−	0.940403i	$-0.389552\pi$
$439$	14.2501	0.680122	0.340061	+	0.940403i	$0.389552\pi$
$440$	−1.23217	−0.0587412
$441$	0	0
$442$	−31.4396	−1.49543
$443$	−8.82606	−0.419339	−0.209669	−	0.977772i	$-0.567239\pi$
$443$	−8.82606	−0.419339	−0.209669	+	0.977772i	$0.567239\pi$
$444$	0	0
$445$	4.91102	0.232805
$446$	6.30825	0.298704
$447$	0	0
$448$	−2.69892	−0.127512
$449$	10.0380	0.473721	0.236860	−	0.971544i	$-0.423882\pi$
$449$	10.0380	0.473721	0.236860	+	0.971544i	$0.423882\pi$
$450$	0	0
$451$	4.84782	0.228275
$452$	−6.52467	−0.306895
$453$	0	0
$454$	16.2116	0.760848
$455$	0.432144	0.0202592
$456$	0	0
$457$	−20.9769	−0.981260	−0.490630	−	0.871368i	$-0.663233\pi$
$457$	−20.9769	−0.981260	−0.490630	+	0.871368i	$0.663233\pi$
$458$	−10.0857	−0.471272
$459$	0	0
$460$	−0.654810	−0.0305307
$461$	−24.8875	−1.15913	−0.579564	−	0.814927i	$-0.696777\pi$
$461$	−24.8875	−1.15913	−0.579564	+	0.814927i	$0.696777\pi$
$462$	0	0
$463$	21.9458	1.01991	0.509954	−	0.860202i	$-0.329663\pi$
$463$	21.9458	1.01991	0.509954	+	0.860202i	$0.329663\pi$
$464$	7.07074	0.328251
$465$	0	0
$466$	−12.9972	−0.602082
$467$	11.5803	0.535871	0.267936	−	0.963437i	$-0.413659\pi$
$467$	11.5803	0.535871	0.267936	+	0.963437i	$0.413659\pi$
$468$	0	0
$469$	0.0272609	0.00125879
$470$	−4.05832	−0.187196
$471$	0	0
$472$	−5.14146	−0.236655
$473$	5.93177	0.272743
$474$	0	0
$475$	−17.7566	−0.814728
$476$	−1.02746	−0.0470934
$477$	0	0
$478$	−26.1708	−1.19702
$479$	21.6662	0.989952	0.494976	−	0.868907i	$-0.335177\pi$
$479$	21.6662	0.989952	0.494976	+	0.868907i	$0.335177\pi$
$480$	0	0
$481$	−10.6119	−0.483863
$482$	36.0698	1.64294
$483$	0	0
$484$	−0.470198	−0.0213726
$485$	−0.688060	−0.0312432
$486$	0	0
$487$	11.2365	0.509173	0.254587	−	0.967050i	$-0.418061\pi$
$487$	11.2365	0.509173	0.254587	+	0.967050i	$0.418061\pi$
$488$	3.05527	0.138306
$489$	0	0
$490$	3.44574	0.155663
$491$	40.3512	1.82102	0.910512	−	0.413482i	$-0.135687\pi$
$491$	40.3512	1.82102	0.910512	+	0.413482i	$0.135687\pi$
$492$	0	0
$493$	17.9345	0.807730
$494$	16.0292	0.721189
$495$	0	0
$496$	−8.14383	−0.365669
$497$	−3.97358	−0.178240
$498$	0	0
$499$	−36.3298	−1.62635	−0.813173	−	0.582022i	$-0.802262\pi$
$499$	−36.3298	−1.62635	−0.813173	+	0.582022i	$0.802262\pi$
$500$	1.86543	0.0834245
$501$	0	0
$502$	−29.7528	−1.32793
$503$	−5.74480	−0.256148	−0.128074	−	0.991765i	$-0.540879\pi$
$503$	−5.74480	−0.256148	−0.128074	+	0.991765i	$0.540879\pi$
$504$	0	0
$505$	1.24994	0.0556217
$506$	4.27103	0.189870
$507$	0	0
$508$	−6.10218	−0.270740
$509$	40.4047	1.79091	0.895453	−	0.445156i	$-0.146852\pi$
$509$	40.4047	1.79091	0.895453	+	0.445156i	$0.146852\pi$
$510$	0	0
$511$	4.26619	0.188725
$512$	24.7242	1.09267
$513$	0	0
$514$	15.6840	0.691792
$515$	−0.687483	−0.0302941
$516$	0	0
$517$	−8.13596	−0.357819
$518$	1.12833	0.0495759
$519$	0	0
$520$	−4.35023	−0.190770
$521$	26.7769	1.17312	0.586559	−	0.809907i	$-0.300482\pi$
$521$	26.7769	1.17312	0.586559	+	0.809907i	$0.300482\pi$
$522$	0	0
$523$	−7.37927	−0.322673	−0.161336	−	0.986899i	$-0.551580\pi$
$523$	−7.37927	−0.322673	−0.161336	+	0.986899i	$0.551580\pi$
$524$	0.00923865	0.000403592 0
$525$	0	0
$526$	−31.0076	−1.35199
$527$	−20.6563	−0.899805
$528$	0	0
$529$	−11.0758	−0.481556
$530$	0.953155	0.0414024
$531$	0	0
$532$	0.523840	0.0227113
$533$	17.1155	0.741355
$534$	0	0
$535$	4.80870	0.207898
$536$	−0.274425	−0.0118533
$537$	0	0
$538$	−5.37088	−0.231555
$539$	6.90788	0.297544
$540$	0	0
$541$	26.7137	1.14851	0.574256	−	0.818676i	$-0.305291\pi$
$541$	26.7137	1.14851	0.574256	+	0.818676i	$0.305291\pi$
$542$	−22.2352	−0.955085
$543$	0	0
$544$	18.7172	0.802494
$545$	−1.56567	−0.0670657
$546$	0	0
$547$	−12.2664	−0.524472	−0.262236	−	0.965004i	$-0.584460\pi$
$547$	−12.2664	−0.524472	−0.262236	+	0.965004i	$0.584460\pi$
$548$	−0.129256	−0.00552154
$549$	0	0
$550$	−5.98309	−0.255120
$551$	−9.14377	−0.389538
$552$	0	0
$553$	−1.05073	−0.0446818
$554$	−18.7838	−0.798047
$555$	0	0
$556$	6.75653	0.286541
$557$	18.9900	0.804632	0.402316	−	0.915501i	$-0.368205\pi$
$557$	18.9900	0.804632	0.402316	+	0.915501i	$0.368205\pi$
$558$	0	0
$559$	20.9424	0.885770
$560$	0.347438	0.0146819
$561$	0	0
$562$	27.1207	1.14402
$563$	12.4311	0.523907	0.261953	−	0.965081i	$-0.415633\pi$
$563$	12.4311	0.523907	0.261953	+	0.965081i	$0.415633\pi$
$564$	0	0
$565$	5.59626	0.235436
$566$	26.6426	1.11987
$567$	0	0
$568$	40.0005	1.67838
$569$	24.3719	1.02172	0.510861	−	0.859663i	$-0.329327\pi$
$569$	24.3719	1.02172	0.510861	+	0.859663i	$0.329327\pi$
$570$	0	0
$571$	−22.3684	−0.936088	−0.468044	−	0.883705i	$-0.655041\pi$
$571$	−22.3684	−0.936088	−0.468044	+	0.883705i	$0.655041\pi$
$572$	−1.66006	−0.0694106
$573$	0	0
$574$	−1.81983	−0.0759582
$575$	−16.7041	−0.696609
$576$	0	0
$577$	−0.888594	−0.0369927	−0.0184963	−	0.999829i	$-0.505888\pi$
$577$	−0.888594	−0.0369927	−0.0184963	+	0.999829i	$0.505888\pi$
$578$	−43.0872	−1.79219
$579$	0	0
$580$	0.472361	0.0196137
$581$	−0.745590	−0.0309323
$582$	0	0
$583$	1.91085	0.0791392
$584$	−42.9461	−1.77712
$585$	0	0
$586$	−36.3380	−1.50111
$587$	38.8288	1.60264	0.801319	−	0.598238i	$-0.204132\pi$
$587$	38.8288	1.60264	0.801319	+	0.598238i	$0.204132\pi$
$588$	0	0
$589$	10.5315	0.433942
$590$	0.839411	0.0345580
$591$	0	0
$592$	−8.53186	−0.350657
$593$	−27.1156	−1.11350	−0.556751	−	0.830679i	$-0.687952\pi$
$593$	−27.1156	−1.11350	−0.556751	+	0.830679i	$0.687952\pi$
$594$	0	0
$595$	0.881256	0.0361280
$596$	−7.33785	−0.300570
$597$	0	0
$598$	15.0791	0.616630
$599$	−13.0804	−0.534451	−0.267225	−	0.963634i	$-0.586107\pi$
$599$	−13.0804	−0.534451	−0.267225	+	0.963634i	$0.586107\pi$
$600$	0	0
$601$	−9.30686	−0.379635	−0.189817	−	0.981819i	$-0.560790\pi$
$601$	−9.30686	−0.379635	−0.189817	+	0.981819i	$0.560790\pi$
$602$	−2.22673	−0.0907548
$603$	0	0
$604$	−9.61546	−0.391248
$605$	0.403292	0.0163962
$606$	0	0
$607$	32.2161	1.30761	0.653805	−	0.756663i	$-0.273172\pi$
$607$	32.2161	1.30761	0.653805	+	0.756663i	$0.273172\pi$
$608$	−9.54282	−0.387012
$609$	0	0
$610$	−0.498813	−0.0201963
$611$	−28.7245	−1.16207
$612$	0	0
$613$	7.00638	0.282985	0.141492	−	0.989939i	$-0.454810\pi$
$613$	7.00638	0.282985	0.141492	+	0.989939i	$0.454810\pi$
$614$	−33.4943	−1.35172
$615$	0	0
$616$	0.927290	0.0373616
$617$	−33.8218	−1.36161	−0.680807	−	0.732463i	$-0.738371\pi$
$617$	−33.8218	−1.36161	−0.680807	+	0.732463i	$0.738371\pi$
$618$	0	0
$619$	17.7967	0.715309	0.357655	−	0.933854i	$-0.383577\pi$
$619$	17.7967	0.715309	0.357655	+	0.933854i	$0.383577\pi$
$620$	−0.544048	−0.0218495
$621$	0	0
$622$	−37.0389	−1.48513
$623$	−3.69588	−0.148072
$624$	0	0
$625$	22.5868	0.903471
$626$	−6.20086	−0.247836
$627$	0	0
$628$	10.7186	0.427720
$629$	−21.6405	−0.862865
$630$	0	0
$631$	40.5090	1.61264	0.806319	−	0.591480i	$-0.201456\pi$
$631$	40.5090	1.61264	0.806319	+	0.591480i	$0.201456\pi$
$632$	10.5773	0.420744
$633$	0	0
$634$	−5.70856	−0.226716
$635$	5.23388	0.207700
$636$	0	0
$637$	24.3887	0.966314
$638$	−3.08100	−0.121978
$639$	0	0
$640$	−2.33880	−0.0924493
$641$	−10.7585	−0.424936	−0.212468	−	0.977168i	$-0.568150\pi$
$641$	−10.7585	−0.424936	−0.212468	+	0.977168i	$0.568150\pi$
$642$	0	0
$643$	−35.6131	−1.40445	−0.702223	−	0.711957i	$-0.747809\pi$
$643$	−35.6131	−1.40445	−0.702223	+	0.711957i	$0.747809\pi$
$644$	0.492790	0.0194186
$645$	0	0
$646$	32.6878	1.28608
$647$	−28.1304	−1.10592	−0.552961	−	0.833207i	$-0.686502\pi$
$647$	−28.1304	−1.10592	−0.552961	+	0.833207i	$0.686502\pi$
$648$	0	0
$649$	1.68282	0.0660564
$650$	−21.1236	−0.828537
$651$	0	0
$652$	−2.59946	−0.101803
$653$	13.3332	0.521769	0.260884	−	0.965370i	$-0.415986\pi$
$653$	13.3332	0.521769	0.260884	+	0.965370i	$0.415986\pi$
$654$	0	0
$655$	−0.00792406	−0.000309619 0
$656$	13.7606	0.537263
$657$	0	0
$658$	3.05417	0.119064
$659$	−19.3913	−0.755378	−0.377689	−	0.925932i	$-0.623281\pi$
$659$	−19.3913	−0.755378	−0.377689	+	0.925932i	$0.623281\pi$
$660$	0	0
$661$	−26.7209	−1.03932	−0.519661	−	0.854373i	$-0.673942\pi$
$661$	−26.7209	−1.03932	−0.519661	+	0.854373i	$0.673942\pi$
$662$	−5.76989	−0.224253
$663$	0	0
$664$	7.50556	0.291272
$665$	−0.449301	−0.0174232
$666$	0	0
$667$	−8.60178	−0.333062
$668$	10.4819	0.405558
$669$	0	0
$670$	0.0448034	0.00173091
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−41.3889	−1.59543	−0.797713	−	0.603038i	$-0.793957\pi$
$673$	−41.3889	−1.59543	−0.797713	+	0.603038i	$0.793957\pi$
$674$	−24.3341	−0.937314
$675$	0	0
$676$	0.251640	0.00967844
$677$	−26.0151	−0.999840	−0.499920	−	0.866072i	$-0.666637\pi$
$677$	−26.0151	−0.999840	−0.499920	+	0.866072i	$0.666637\pi$
$678$	0	0
$679$	0.517813	0.0198718
$680$	−8.87126	−0.340197
$681$	0	0
$682$	3.54858	0.135882
$683$	−14.8239	−0.567220	−0.283610	−	0.958940i	$-0.591532\pi$
$683$	−14.8239	−0.567220	−0.283610	+	0.958940i	$0.591532\pi$
$684$	0	0
$685$	0.110864	0.00423589
$686$	−5.22089	−0.199335
$687$	0	0
$688$	16.8374	0.641921
$689$	6.74635	0.257016
$690$	0	0
$691$	−48.7801	−1.85568	−0.927841	−	0.372975i	$-0.878338\pi$
$691$	−48.7801	−1.85568	−0.927841	+	0.372975i	$0.878338\pi$
$692$	1.37390	0.0522280
$693$	0	0
$694$	−4.53465	−0.172133
$695$	−5.79512	−0.219821
$696$	0	0
$697$	34.9030	1.32205
$698$	−12.5256	−0.474099
$699$	0	0
$700$	−0.690327	−0.0260919
$701$	51.2249	1.93474	0.967369	−	0.253371i	$-0.0815393\pi$
$701$	51.2249	1.93474	0.967369	+	0.253371i	$0.0815393\pi$
$702$	0	0
$703$	11.0333	0.416127
$704$	−8.89249	−0.335148
$705$	0	0
$706$	20.5953	0.775115
$707$	−0.940668	−0.0353775
$708$	0	0
$709$	25.1294	0.943753	0.471876	−	0.881665i	$-0.343577\pi$
$709$	25.1294	0.943753	0.471876	+	0.881665i	$0.343577\pi$
$710$	−6.53061	−0.245089
$711$	0	0
$712$	37.2050	1.39432
$713$	9.90722	0.371028
$714$	0	0
$715$	1.42385	0.0532488
$716$	−4.11309	−0.153713
$717$	0	0
$718$	−14.8350	−0.553637
$719$	3.40870	0.127123	0.0635615	−	0.997978i	$-0.479754\pi$
$719$	3.40870	0.127123	0.0635615	+	0.997978i	$0.479754\pi$
$720$	0	0
$721$	0.517378	0.0192682
$722$	6.83458	0.254357
$723$	0	0
$724$	10.3061	0.383021
$725$	12.0498	0.447520
$726$	0	0
$727$	37.1314	1.37713	0.688563	−	0.725176i	$-0.258242\pi$
$727$	37.1314	1.37713	0.688563	+	0.725176i	$0.258242\pi$
$728$	3.27385	0.121337
$729$	0	0
$730$	7.01151	0.259508
$731$	42.7071	1.57958
$732$	0	0
$733$	14.2953	0.528009	0.264004	−	0.964521i	$-0.414957\pi$
$733$	14.2953	0.528009	0.264004	+	0.964521i	$0.414957\pi$
$734$	−11.4806	−0.423756
$735$	0	0
$736$	−8.97718	−0.330903
$737$	0.0898202	0.00330857
$738$	0	0
$739$	34.9653	1.28622	0.643110	−	0.765774i	$-0.277644\pi$
$739$	34.9653	1.28622	0.643110	+	0.765774i	$0.277644\pi$
$740$	−0.569970	−0.0209525
$741$	0	0
$742$	−0.717315	−0.0263335
$743$	−23.7805	−0.872422	−0.436211	−	0.899844i	$-0.643680\pi$
$743$	−23.7805	−0.872422	−0.436211	+	0.899844i	$0.643680\pi$
$744$	0	0
$745$	6.29373	0.230584
$746$	−13.2176	−0.483932
$747$	0	0
$748$	−3.38530	−0.123779
$749$	−3.61888	−0.132231
$750$	0	0
$751$	0.735996	0.0268569	0.0134284	−	0.999910i	$-0.495725\pi$
$751$	0.735996	0.0268569	0.0134284	+	0.999910i	$0.495725\pi$
$752$	−23.0941	−0.842154
$753$	0	0
$754$	−10.8776	−0.396140
$755$	8.24725	0.300148
$756$	0	0
$757$	46.9323	1.70578	0.852892	−	0.522087i	$-0.174846\pi$
$757$	46.9323	1.70578	0.852892	+	0.522087i	$0.174846\pi$
$758$	−39.7681	−1.44444
$759$	0	0
$760$	4.52294	0.164064
$761$	−7.29456	−0.264427	−0.132214	−	0.991221i	$-0.542209\pi$
$761$	−7.29456	−0.264427	−0.132214	+	0.991221i	$0.542209\pi$
$762$	0	0
$763$	1.17827	0.0426563
$764$	−0.845904	−0.0306037
$765$	0	0
$766$	−27.7419	−1.00236
$767$	5.94129	0.214527
$768$	0	0
$769$	−34.6811	−1.25063	−0.625316	−	0.780372i	$-0.715030\pi$
$769$	−34.6811	−1.25063	−0.625316	+	0.780372i	$0.715030\pi$
$770$	−0.151392	−0.00545580
$771$	0	0
$772$	−3.28809	−0.118341
$773$	−15.5742	−0.560165	−0.280083	−	0.959976i	$-0.590362\pi$
$773$	−15.5742	−0.560165	−0.280083	+	0.959976i	$0.590362\pi$
$774$	0	0
$775$	−13.8786	−0.498533
$776$	−5.21262	−0.187122
$777$	0	0
$778$	−29.3578	−1.05253
$779$	−17.7950	−0.637573
$780$	0	0
$781$	−13.0923	−0.468480
$782$	30.7503	1.09963
$783$	0	0
$784$	19.6082	0.700291
$785$	−9.19344	−0.328128
$786$	0	0
$787$	2.43019	0.0866271	0.0433135	−	0.999062i	$-0.486209\pi$
$787$	2.43019	0.0866271	0.0433135	+	0.999062i	$0.486209\pi$
$788$	9.29201	0.331014
$789$	0	0
$790$	−1.72689	−0.0614399
$791$	−4.21157	−0.149746
$792$	0	0
$793$	−3.53056	−0.125374
$794$	−16.7010	−0.592695
$795$	0	0
$796$	2.32750	0.0824962
$797$	−16.5071	−0.584712	−0.292356	−	0.956310i	$-0.594439\pi$
$797$	−16.5071	−0.584712	−0.292356	+	0.956310i	$0.594439\pi$
$798$	0	0
$799$	−58.5768	−2.07230
$800$	12.5757	0.444619
$801$	0	0
$802$	35.4459	1.25164
$803$	14.0564	0.496040
$804$	0	0
$805$	−0.422669	−0.0148971
$806$	12.5285	0.441296
$807$	0	0
$808$	9.46934	0.333130
$809$	26.1046	0.917788	0.458894	−	0.888491i	$-0.348246\pi$
$809$	26.1046	0.917788	0.458894	+	0.888491i	$0.348246\pi$
$810$	0	0
$811$	−28.3961	−0.997122	−0.498561	−	0.866855i	$-0.666138\pi$
$811$	−28.3961	−0.997122	−0.498561	+	0.866855i	$0.666138\pi$
$812$	−0.355484	−0.0124750
$813$	0	0
$814$	3.71766	0.130304
$815$	2.22958	0.0780988
$816$	0	0
$817$	−21.7739	−0.761771
$818$	−24.0696	−0.841573
$819$	0	0
$820$	0.919279	0.0321026
$821$	−7.36854	−0.257164	−0.128582	−	0.991699i	$-0.541043\pi$
$821$	−7.36854	−0.257164	−0.128582	+	0.991699i	$0.541043\pi$
$822$	0	0
$823$	−24.5592	−0.856080	−0.428040	−	0.903760i	$-0.640796\pi$
$823$	−24.5592	−0.856080	−0.428040	+	0.903760i	$0.640796\pi$
$824$	−5.20825	−0.181438
$825$	0	0
$826$	−0.631715	−0.0219802
$827$	31.8506	1.10755	0.553776	−	0.832666i	$-0.313186\pi$
$827$	31.8506	1.10755	0.553776	+	0.832666i	$0.313186\pi$
$828$	0	0
$829$	28.0544	0.974370	0.487185	−	0.873299i	$-0.338024\pi$
$829$	28.0544	0.974370	0.487185	+	0.873299i	$0.338024\pi$
$830$	−1.22538	−0.0425336
$831$	0	0
$832$	−31.3954	−1.08844
$833$	49.7349	1.72321
$834$	0	0
$835$	−8.99042	−0.311126
$836$	1.72597	0.0596938
$837$	0	0
$838$	−21.4990	−0.742672
$839$	8.53067	0.294511	0.147256	−	0.989098i	$-0.452956\pi$
$839$	8.53067	0.294511	0.147256	+	0.989098i	$0.452956\pi$
$840$	0	0
$841$	−22.7949	−0.786032
$842$	−13.0414	−0.449437
$843$	0	0
$844$	4.00475	0.137849
$845$	−0.215833	−0.00742488
$846$	0	0
$847$	−0.303505	−0.0104286
$848$	5.42398	0.186260
$849$	0	0
$850$	−43.0767	−1.47752
$851$	10.3793	0.355797
$852$	0	0
$853$	47.7707	1.63564	0.817818	−	0.575477i	$-0.195183\pi$
$853$	47.7707	1.63564	0.817818	+	0.575477i	$0.195183\pi$
$854$	0.375391	0.0128456
$855$	0	0
$856$	36.4298	1.24515
$857$	20.2990	0.693399	0.346700	−	0.937976i	$-0.387302\pi$
$857$	20.2990	0.693399	0.346700	+	0.937976i	$0.387302\pi$
$858$	0	0
$859$	40.8031	1.39218	0.696091	−	0.717953i	$-0.254921\pi$
$859$	40.8031	1.39218	0.696091	+	0.717953i	$0.254921\pi$
$860$	1.12482	0.0383562
$861$	0	0
$862$	40.6366	1.38409
$863$	−13.1014	−0.445979	−0.222989	−	0.974821i	$-0.571581\pi$
$863$	−13.1014	−0.445979	−0.222989	+	0.974821i	$0.571581\pi$
$864$	0	0
$865$	−1.17841	−0.0400671
$866$	−10.5525	−0.358587
$867$	0	0
$868$	0.409434	0.0138971
$869$	−3.46200	−0.117440
$870$	0	0
$871$	0.317115	0.0107450
$872$	−11.8612	−0.401671
$873$	0	0
$874$	−15.6778	−0.530308
$875$	1.20410	0.0407062
$876$	0	0
$877$	−17.8525	−0.602836	−0.301418	−	0.953492i	$-0.597460\pi$
$877$	−17.8525	−0.602836	−0.301418	+	0.953492i	$0.597460\pi$
$878$	−17.6253	−0.594825
$879$	0	0
$880$	1.14475	0.0385896
$881$	23.6982	0.798412	0.399206	−	0.916861i	$-0.369286\pi$
$881$	23.6982	0.798412	0.399206	+	0.916861i	$0.369286\pi$
$882$	0	0
$883$	45.6569	1.53648	0.768239	−	0.640163i	$-0.221133\pi$
$883$	45.6569	1.53648	0.768239	+	0.640163i	$0.221133\pi$
$884$	−11.9520	−0.401989
$885$	0	0
$886$	10.9165	0.366748
$887$	23.1922	0.778716	0.389358	−	0.921086i	$-0.372697\pi$
$887$	23.1922	0.778716	0.389358	+	0.921086i	$0.372697\pi$
$888$	0	0
$889$	−3.93886	−0.132105
$890$	−6.07420	−0.203608
$891$	0	0
$892$	2.39813	0.0802952
$893$	29.8649	0.999390
$894$	0	0
$895$	3.52783	0.117922
$896$	1.76011	0.0588011
$897$	0	0
$898$	−12.4155	−0.414310
$899$	−7.14678	−0.238359
$900$	0	0
$901$	13.7576	0.458332
$902$	−5.99604	−0.199646
$903$	0	0
$904$	42.3962	1.41008
$905$	−8.83958	−0.293837
$906$	0	0
$907$	−0.587437	−0.0195055	−0.00975277	−	0.999952i	$-0.503104\pi$
$907$	−0.587437	−0.0195055	−0.00975277	+	0.999952i	$0.503104\pi$
$908$	6.16295	0.204525
$909$	0	0
$910$	−0.534499	−0.0177185
$911$	0.0732945	0.00242836	0.00121418	−	0.999999i	$-0.499614\pi$
$911$	0.0732945	0.00242836	0.00121418	+	0.999999i	$0.499614\pi$
$912$	0	0
$913$	−2.45660	−0.0813015
$914$	25.9454	0.858196
$915$	0	0
$916$	−3.83413	−0.126683
$917$	0.00596340	0.000196929 0
$918$	0	0
$919$	18.5785	0.612849	0.306424	−	0.951895i	$-0.400867\pi$
$919$	18.5785	0.612849	0.306424	+	0.951895i	$0.400867\pi$
$920$	4.25485	0.140278
$921$	0	0
$922$	30.7822	1.01376
$923$	−46.2231	−1.52145
$924$	0	0
$925$	−14.5398	−0.478067
$926$	−27.1437	−0.891997
$927$	0	0
$928$	6.47587	0.212581
$929$	8.28707	0.271890	0.135945	−	0.990716i	$-0.456593\pi$
$929$	8.28707	0.271890	0.135945	+	0.990716i	$0.456593\pi$
$930$	0	0
$931$	−25.3569	−0.831040
$932$	−4.94096	−0.161847
$933$	0	0
$934$	−14.3231	−0.468666
$935$	2.90360	0.0949577
$936$	0	0
$937$	38.3030	1.25131	0.625653	−	0.780102i	$-0.284833\pi$
$937$	38.3030	1.25131	0.625653	+	0.780102i	$0.284833\pi$
$938$	−0.0337177	−0.00110092
$939$	0	0
$940$	−1.54280	−0.0503205
$941$	−8.69924	−0.283587	−0.141793	−	0.989896i	$-0.545287\pi$
$941$	−8.69924	−0.283587	−0.141793	+	0.989896i	$0.545287\pi$
$942$	0	0
$943$	−16.7402	−0.545137
$944$	4.77671	0.155469
$945$	0	0
$946$	−7.33671	−0.238537
$947$	−16.1962	−0.526306	−0.263153	−	0.964754i	$-0.584762\pi$
$947$	−16.1962	−0.526306	−0.263153	+	0.964754i	$0.584762\pi$
$948$	0	0
$949$	49.6269	1.61096
$950$	21.9623	0.712550
$951$	0	0
$952$	6.67624	0.216378
$953$	−13.7098	−0.444104	−0.222052	−	0.975035i	$-0.571276\pi$
$953$	−13.7098	−0.444104	−0.222052	+	0.975035i	$0.571276\pi$
$954$	0	0
$955$	0.725538	0.0234779
$956$	−9.94901	−0.321774
$957$	0	0
$958$	−26.7978	−0.865799
$959$	−0.0834327	−0.00269418
$960$	0	0
$961$	−22.7686	−0.734471
$962$	13.1254	0.423180
$963$	0	0
$964$	13.7122	0.441640
$965$	2.82022	0.0907860
$966$	0	0
$967$	29.8355	0.959444	0.479722	−	0.877420i	$-0.340737\pi$
$967$	29.8355	0.959444	0.479722	+	0.877420i	$0.340737\pi$
$968$	3.05527	0.0982000
$969$	0	0
$970$	0.851028	0.0273249
$971$	−45.1242	−1.44811	−0.724053	−	0.689745i	$-0.757723\pi$
$971$	−45.1242	−1.44811	−0.724053	+	0.689745i	$0.757723\pi$
$972$	0	0
$973$	4.36123	0.139815
$974$	−13.8979	−0.445316
$975$	0	0
$976$	−2.83852	−0.0908588
$977$	60.1493	1.92435	0.962173	−	0.272439i	$-0.0878304\pi$
$977$	60.1493	1.92435	0.962173	+	0.272439i	$0.0878304\pi$
$978$	0	0
$979$	−12.1773	−0.389189
$980$	1.30992	0.0418439
$981$	0	0
$982$	−49.9085	−1.59264
$983$	−13.5937	−0.433573	−0.216786	−	0.976219i	$-0.569558\pi$
$983$	−13.5937	−0.433573	−0.216786	+	0.976219i	$0.569558\pi$
$984$	0	0
$985$	−7.96983	−0.253940
$986$	−22.1823	−0.706430
$987$	0	0
$988$	6.09362	0.193864
$989$	−20.4832	−0.651329
$990$	0	0
$991$	−20.1950	−0.641517	−0.320758	−	0.947161i	$-0.603938\pi$
$991$	−20.1950	−0.641517	−0.320758	+	0.947161i	$0.603938\pi$
$992$	−7.45868	−0.236813
$993$	0	0
$994$	4.91473	0.155886
$995$	−1.99632	−0.0632875
$996$	0	0
$997$	−39.4389	−1.24904	−0.624522	−	0.781007i	$-0.714706\pi$
$997$	−39.4389	−1.24904	−0.624522	+	0.781007i	$0.714706\pi$
$998$	44.9346	1.42238
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.19	✓	25	3.2	odd	2
6039.2.a.p.1.7	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)

\(f(q)\)	\(=\)	\(q-1.23685 q^{2} -0.470198 q^{4} +0.403292 q^{5} -0.303505 q^{7} +3.05527 q^{8} +O(q^{10})\)	\(q-1.23685 q^{2} -0.470198 q^{4} +0.403292 q^{5} -0.303505 q^{7} +3.05527 q^{8} -0.498813 q^{10} -1.00000 q^{11} -3.53056 q^{13} +0.375391 q^{14} -2.83852 q^{16} -7.19973 q^{17} +3.67072 q^{19} -0.189627 q^{20} +1.23685 q^{22} +3.45314 q^{23} -4.83736 q^{25} +4.36677 q^{26} +0.142707 q^{28} -2.49100 q^{29} +2.86904 q^{31} -2.59971 q^{32} +8.90500 q^{34} -0.122401 q^{35} +3.00574 q^{37} -4.54014 q^{38} +1.23217 q^{40} -4.84782 q^{41} -5.93177 q^{43} +0.470198 q^{44} -4.27103 q^{46} +8.13596 q^{47} -6.90788 q^{49} +5.98309 q^{50} +1.66006 q^{52} -1.91085 q^{53} -0.403292 q^{55} -0.927290 q^{56} +3.08100 q^{58} -1.68282 q^{59} +1.00000 q^{61} -3.54858 q^{62} +8.89249 q^{64} -1.42385 q^{65} -0.0898202 q^{67} +3.38530 q^{68} +0.151392 q^{70} +13.0923 q^{71} -14.0564 q^{73} -3.71766 q^{74} -1.72597 q^{76} +0.303505 q^{77} +3.46200 q^{79} -1.14475 q^{80} +5.99604 q^{82} +2.45660 q^{83} -2.90360 q^{85} +7.33671 q^{86} -3.05527 q^{88} +12.1773 q^{89} +1.07154 q^{91} -1.62366 q^{92} -10.0630 q^{94} +1.48037 q^{95} -1.70611 q^{97} +8.54403 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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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