LMFDB - Embedded newform 6039.2.a.m.1.2

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			1.2
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)

$f(q)$	$=$	$q-2.53522 q^{2} +4.42736 q^{4} +3.72510 q^{5} -0.951731 q^{7} -6.15390 q^{8} +O(q^{10})$	\(q-2.53522 q^{2} +4.42736 q^{4} +3.72510 q^{5} -0.951731 q^{7} -6.15390 q^{8} -9.44395 q^{10} +1.00000 q^{11} -1.53766 q^{13} +2.41285 q^{14} +6.74680 q^{16} -1.23784 q^{17} -4.40090 q^{19} +16.4923 q^{20} -2.53522 q^{22} -7.09979 q^{23} +8.87634 q^{25} +3.89831 q^{26} -4.21365 q^{28} -5.35857 q^{29} +1.23453 q^{31} -4.79684 q^{32} +3.13820 q^{34} -3.54529 q^{35} +10.2150 q^{37} +11.1573 q^{38} -22.9239 q^{40} -11.9783 q^{41} +8.81440 q^{43} +4.42736 q^{44} +17.9995 q^{46} -2.41608 q^{47} -6.09421 q^{49} -22.5035 q^{50} -6.80777 q^{52} -7.04502 q^{53} +3.72510 q^{55} +5.85686 q^{56} +13.5852 q^{58} +14.6121 q^{59} +1.00000 q^{61} -3.12980 q^{62} -1.33253 q^{64} -5.72793 q^{65} +10.3286 q^{67} -5.48036 q^{68} +8.98810 q^{70} +8.17395 q^{71} +15.6682 q^{73} -25.8973 q^{74} -19.4844 q^{76} -0.951731 q^{77} -7.88000 q^{79} +25.1325 q^{80} +30.3676 q^{82} +1.94530 q^{83} -4.61107 q^{85} -22.3465 q^{86} -6.15390 q^{88} +0.524054 q^{89} +1.46344 q^{91} -31.4333 q^{92} +6.12530 q^{94} -16.3938 q^{95} -15.3403 q^{97} +15.4502 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.53522	−1.79267	−0.896337	−	0.443373i	$-0.853782\pi$
$2$	−2.53522	−1.79267	−0.896337	+	0.443373i	$0.853782\pi$
$3$	0	0
$3$	0	0
$4$	4.42736	2.21368
$5$	3.72510	1.66591	0.832957	−	0.553338i	$-0.186646\pi$
$5$	3.72510	1.66591	0.832957	+	0.553338i	$0.186646\pi$
$6$	0	0
$7$	−0.951731	−0.359720	−0.179860	−	0.983692i	$-0.557565\pi$
$7$	−0.951731	−0.359720	−0.179860	+	0.983692i	$0.557565\pi$
$8$	−6.15390	−2.17573
$9$	0	0
$10$	−9.44395	−2.98644
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.53766	−0.426470	−0.213235	−	0.977001i	$-0.568400\pi$
$13$	−1.53766	−0.426470	−0.213235	+	0.977001i	$0.568400\pi$
$14$	2.41285	0.644861
$15$	0	0
$16$	6.74680	1.68670
$17$	−1.23784	−0.300220	−0.150110	−	0.988669i	$-0.547963\pi$
$17$	−1.23784	−0.300220	−0.150110	+	0.988669i	$0.547963\pi$
$18$	0	0
$19$	−4.40090	−1.00964	−0.504818	−	0.863226i	$-0.668440\pi$
$19$	−4.40090	−1.00964	−0.504818	+	0.863226i	$0.668440\pi$
$20$	16.4923	3.68780
$21$	0	0
$22$	−2.53522	−0.540512
$23$	−7.09979	−1.48041	−0.740204	−	0.672382i	$-0.765271\pi$
$23$	−7.09979	−1.48041	−0.740204	+	0.672382i	$0.765271\pi$
$24$	0	0
$25$	8.87634	1.77527
$26$	3.89831	0.764522
$27$	0	0
$28$	−4.21365	−0.796306
$29$	−5.35857	−0.995061	−0.497530	−	0.867447i	$-0.665760\pi$
$29$	−5.35857	−0.995061	−0.497530	+	0.867447i	$0.665760\pi$
$30$	0	0
$31$	1.23453	0.221727	0.110864	−	0.993836i	$-0.464638\pi$
$31$	1.23453	0.221727	0.110864	+	0.993836i	$0.464638\pi$
$32$	−4.79684	−0.847970
$33$	0	0
$34$	3.13820	0.538197
$35$	−3.54529	−0.599263
$36$	0	0
$37$	10.2150	1.67933	0.839666	−	0.543103i	$-0.182751\pi$
$37$	10.2150	1.67933	0.839666	+	0.543103i	$0.182751\pi$
$38$	11.1573	1.80995
$39$	0	0
$40$	−22.9239	−3.62458
$41$	−11.9783	−1.87069	−0.935345	−	0.353737i	$-0.884911\pi$
$41$	−11.9783	−1.87069	−0.935345	+	0.353737i	$0.884911\pi$
$42$	0	0
$43$	8.81440	1.34418	0.672092	−	0.740468i	$-0.265396\pi$
$43$	8.81440	1.34418	0.672092	+	0.740468i	$0.265396\pi$
$44$	4.42736	0.667450
$45$	0	0
$46$	17.9995	2.65389
$47$	−2.41608	−0.352421	−0.176211	−	0.984352i	$-0.556384\pi$
$47$	−2.41608	−0.352421	−0.176211	+	0.984352i	$0.556384\pi$
$48$	0	0
$49$	−6.09421	−0.870601
$50$	−22.5035	−3.18248
$51$	0	0
$52$	−6.80777	−0.944068
$53$	−7.04502	−0.967708	−0.483854	−	0.875149i	$-0.660763\pi$
$53$	−7.04502	−0.967708	−0.483854	+	0.875149i	$0.660763\pi$
$54$	0	0
$55$	3.72510	0.502292
$56$	5.85686	0.782655
$57$	0	0
$58$	13.5852	1.78382
$59$	14.6121	1.90233	0.951165	−	0.308682i	$-0.0998878\pi$
$59$	14.6121	1.90233	0.951165	+	0.308682i	$0.0998878\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−3.12980	−0.397485
$63$	0	0
$64$	−1.33253	−0.166566
$65$	−5.72793	−0.710462
$66$	0	0
$67$	10.3286	1.26184	0.630921	−	0.775847i	$-0.282677\pi$
$67$	10.3286	1.26184	0.630921	+	0.775847i	$0.282677\pi$
$68$	−5.48036	−0.664591
$69$	0	0
$70$	8.98810	1.07428
$71$	8.17395	0.970069	0.485035	−	0.874495i	$-0.338807\pi$
$71$	8.17395	0.970069	0.485035	+	0.874495i	$0.338807\pi$
$72$	0	0
$73$	15.6682	1.83382	0.916909	−	0.399096i	$-0.130676\pi$
$73$	15.6682	1.83382	0.916909	+	0.399096i	$0.130676\pi$
$74$	−25.8973	−3.01050
$75$	0	0
$76$	−19.4844	−2.23501
$77$	−0.951731	−0.108460
$78$	0	0
$79$	−7.88000	−0.886570	−0.443285	−	0.896381i	$-0.646187\pi$
$79$	−7.88000	−0.886570	−0.443285	+	0.896381i	$0.646187\pi$
$80$	25.1325	2.80990
$81$	0	0
$82$	30.3676	3.35354
$83$	1.94530	0.213524	0.106762	−	0.994285i	$-0.465952\pi$
$83$	1.94530	0.213524	0.106762	+	0.994285i	$0.465952\pi$
$84$	0	0
$85$	−4.61107	−0.500141
$86$	−22.3465	−2.40968
$87$	0	0
$88$	−6.15390	−0.656008
$89$	0.524054	0.0555496	0.0277748	−	0.999614i	$-0.491158\pi$
$89$	0.524054	0.0555496	0.0277748	+	0.999614i	$0.491158\pi$
$90$	0	0
$91$	1.46344	0.153410
$92$	−31.4333	−3.27715
$93$	0	0
$94$	6.12530	0.631776
$95$	−16.3938	−1.68197
$96$	0	0
$97$	−15.3403	−1.55757	−0.778783	−	0.627293i	$-0.784163\pi$
$97$	−15.3403	−1.55757	−0.778783	+	0.627293i	$0.784163\pi$
$98$	15.4502	1.56070
$99$	0	0
$100$	39.2987	3.92987
$101$	−10.6548	−1.06019	−0.530097	−	0.847937i	$-0.677845\pi$
$101$	−10.6548	−1.06019	−0.530097	+	0.847937i	$0.677845\pi$
$102$	0	0
$103$	−2.38207	−0.234712	−0.117356	−	0.993090i	$-0.537442\pi$
$103$	−2.38207	−0.234712	−0.117356	+	0.993090i	$0.537442\pi$
$104$	9.46261	0.927885
$105$	0	0
$106$	17.8607	1.73479
$107$	−3.15962	−0.305452	−0.152726	−	0.988269i	$-0.548805\pi$
$107$	−3.15962	−0.305452	−0.152726	+	0.988269i	$0.548805\pi$
$108$	0	0
$109$	8.86104	0.848733	0.424367	−	0.905490i	$-0.360497\pi$
$109$	8.86104	0.848733	0.424367	+	0.905490i	$0.360497\pi$
$110$	−9.44395	−0.900445
$111$	0	0
$112$	−6.42114	−0.606740
$113$	−19.2881	−1.81447	−0.907237	−	0.420619i	$-0.861813\pi$
$113$	−19.2881	−1.81447	−0.907237	+	0.420619i	$0.861813\pi$
$114$	0	0
$115$	−26.4474	−2.46623
$116$	−23.7243	−2.20275
$117$	0	0
$118$	−37.0449	−3.41026
$119$	1.17809	0.107995
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.53522	−0.229528
$123$	0	0
$124$	5.46569	0.490834
$125$	14.4397	1.29153
$126$	0	0
$127$	−0.0608953	−0.00540358	−0.00270179	−	0.999996i	$-0.500860\pi$
$127$	−0.0608953	−0.00540358	−0.00270179	+	0.999996i	$0.500860\pi$
$128$	12.9719	1.14657
$129$	0	0
$130$	14.5216	1.27363
$131$	−0.616348	−0.0538506	−0.0269253	−	0.999637i	$-0.508572\pi$
$131$	−0.616348	−0.0538506	−0.0269253	+	0.999637i	$0.508572\pi$
$132$	0	0
$133$	4.18847	0.363186
$134$	−26.1854	−2.26207
$135$	0	0
$136$	7.61754	0.653199
$137$	3.01013	0.257173	0.128586	−	0.991698i	$-0.458956\pi$
$137$	3.01013	0.257173	0.128586	+	0.991698i	$0.458956\pi$
$138$	0	0
$139$	−19.7278	−1.67329	−0.836645	−	0.547745i	$-0.815486\pi$
$139$	−19.7278	−1.67329	−0.836645	+	0.547745i	$0.815486\pi$
$140$	−15.6963	−1.32658
$141$	0	0
$142$	−20.7228	−1.73902
$143$	−1.53766	−0.128586
$144$	0	0
$145$	−19.9612	−1.65768
$146$	−39.7223	−3.28744
$147$	0	0
$148$	45.2254	3.71750
$149$	−8.52305	−0.698236	−0.349118	−	0.937079i	$-0.613519\pi$
$149$	−8.52305	−0.698236	−0.349118	+	0.937079i	$0.613519\pi$
$150$	0	0
$151$	5.34321	0.434824	0.217412	−	0.976080i	$-0.430238\pi$
$151$	5.34321	0.434824	0.217412	+	0.976080i	$0.430238\pi$
$152$	27.0827	2.19670
$153$	0	0
$154$	2.41285	0.194433
$155$	4.59873	0.369379
$156$	0	0
$157$	−2.25602	−0.180050	−0.0900249	−	0.995940i	$-0.528695\pi$
$157$	−2.25602	−0.180050	−0.0900249	+	0.995940i	$0.528695\pi$
$158$	19.9776	1.58933
$159$	0	0
$160$	−17.8687	−1.41264
$161$	6.75709	0.532533
$162$	0	0
$163$	−19.1474	−1.49974	−0.749868	−	0.661587i	$-0.769883\pi$
$163$	−19.1474	−1.49974	−0.749868	+	0.661587i	$0.769883\pi$
$164$	−53.0321	−4.14111
$165$	0	0
$166$	−4.93176	−0.382779
$167$	−11.8088	−0.913794	−0.456897	−	0.889520i	$-0.651039\pi$
$167$	−11.8088	−0.913794	−0.456897	+	0.889520i	$0.651039\pi$
$168$	0	0
$169$	−10.6356	−0.818123
$170$	11.6901	0.896589
$171$	0	0
$172$	39.0245	2.97559
$173$	−10.6238	−0.807711	−0.403855	−	0.914823i	$-0.632330\pi$
$173$	−10.6238	−0.807711	−0.403855	+	0.914823i	$0.632330\pi$
$174$	0	0
$175$	−8.44788	−0.638600
$176$	6.74680	0.508559
$177$	0	0
$178$	−1.32859	−0.0995823
$179$	−4.27447	−0.319489	−0.159744	−	0.987158i	$-0.551067\pi$
$179$	−4.27447	−0.319489	−0.159744	+	0.987158i	$0.551067\pi$
$180$	0	0
$181$	−19.9745	−1.48469	−0.742347	−	0.670016i	$-0.766287\pi$
$181$	−19.9745	−1.48469	−0.742347	+	0.670016i	$0.766287\pi$
$182$	−3.71014	−0.275014
$183$	0	0
$184$	43.6914	3.22097
$185$	38.0518	2.79762
$186$	0	0
$187$	−1.23784	−0.0905197
$188$	−10.6968	−0.780148
$189$	0	0
$190$	41.5619	3.01522
$191$	−23.3200	−1.68738	−0.843689	−	0.536833i	$-0.819621\pi$
$191$	−23.3200	−1.68738	−0.843689	+	0.536833i	$0.819621\pi$
$192$	0	0
$193$	9.75281	0.702023	0.351011	−	0.936371i	$-0.385838\pi$
$193$	9.75281	0.702023	0.351011	+	0.936371i	$0.385838\pi$
$194$	38.8910	2.79221
$195$	0	0
$196$	−26.9813	−1.92723
$197$	1.24715	0.0888560	0.0444280	−	0.999013i	$-0.485853\pi$
$197$	1.24715	0.0888560	0.0444280	+	0.999013i	$0.485853\pi$
$198$	0	0
$199$	−23.3799	−1.65736	−0.828680	−	0.559722i	$-0.810908\pi$
$199$	−23.3799	−1.65736	−0.828680	+	0.559722i	$0.810908\pi$
$200$	−54.6241	−3.86251
$201$	0	0
$202$	27.0124	1.90058
$203$	5.09991	0.357944
$204$	0	0
$205$	−44.6202	−3.11641
$206$	6.03907	0.420762
$207$	0	0
$208$	−10.3743	−0.719327
$209$	−4.40090	−0.304417
$210$	0	0
$211$	−23.1316	−1.59244	−0.796222	−	0.605004i	$-0.793171\pi$
$211$	−23.1316	−1.59244	−0.796222	+	0.605004i	$0.793171\pi$
$212$	−31.1908	−2.14220
$213$	0	0
$214$	8.01033	0.547575
$215$	32.8345	2.23929
$216$	0	0
$217$	−1.17494	−0.0797599
$218$	−22.4647	−1.52150
$219$	0	0
$220$	16.4923	1.11191
$221$	1.90337	0.128035
$222$	0	0
$223$	26.7382	1.79052	0.895260	−	0.445544i	$-0.146990\pi$
$223$	26.7382	1.79052	0.895260	+	0.445544i	$0.146990\pi$
$224$	4.56530	0.305032
$225$	0	0
$226$	48.8997	3.25276
$227$	−18.9121	−1.25524	−0.627621	−	0.778519i	$-0.715971\pi$
$227$	−18.9121	−1.25524	−0.627621	+	0.778519i	$0.715971\pi$
$228$	0	0
$229$	19.5554	1.29226	0.646128	−	0.763229i	$-0.276387\pi$
$229$	19.5554	1.29226	0.646128	+	0.763229i	$0.276387\pi$
$230$	67.0500	4.42115
$231$	0	0
$232$	32.9761	2.16499
$233$	16.5038	1.08120	0.540600	−	0.841280i	$-0.318197\pi$
$233$	16.5038	1.08120	0.540600	+	0.841280i	$0.318197\pi$
$234$	0	0
$235$	−9.00012	−0.587103
$236$	64.6929	4.21115
$237$	0	0
$238$	−2.98672	−0.193600
$239$	10.7789	0.697232	0.348616	−	0.937266i	$-0.386652\pi$
$239$	10.7789	0.697232	0.348616	+	0.937266i	$0.386652\pi$
$240$	0	0
$241$	−13.3180	−0.857890	−0.428945	−	0.903331i	$-0.641115\pi$
$241$	−13.3180	−0.857890	−0.428945	+	0.903331i	$0.641115\pi$
$242$	−2.53522	−0.162970
$243$	0	0
$244$	4.42736	0.283433
$245$	−22.7015	−1.45035
$246$	0	0
$247$	6.76708	0.430579
$248$	−7.59715	−0.482420
$249$	0	0
$250$	−36.6079	−2.31529
$251$	6.64024	0.419128	0.209564	−	0.977795i	$-0.432796\pi$
$251$	6.64024	0.419128	0.209564	+	0.977795i	$0.432796\pi$
$252$	0	0
$253$	−7.09979	−0.446360
$254$	0.154383	0.00968686
$255$	0	0
$256$	−30.2217	−1.88886
$257$	−18.2840	−1.14052	−0.570261	−	0.821464i	$-0.693158\pi$
$257$	−18.2840	−1.14052	−0.570261	+	0.821464i	$0.693158\pi$
$258$	0	0
$259$	−9.72191	−0.604090
$260$	−25.3596	−1.57274
$261$	0	0
$262$	1.56258	0.0965366
$263$	16.5782	1.02225	0.511126	−	0.859506i	$-0.329228\pi$
$263$	16.5782	1.02225	0.511126	+	0.859506i	$0.329228\pi$
$264$	0	0
$265$	−26.2434	−1.61212
$266$	−10.6187	−0.651075
$267$	0	0
$268$	45.7285	2.79332
$269$	−10.9254	−0.666132	−0.333066	−	0.942903i	$-0.608083\pi$
$269$	−10.9254	−0.666132	−0.333066	+	0.942903i	$0.608083\pi$
$270$	0	0
$271$	15.4907	0.940994	0.470497	−	0.882402i	$-0.344075\pi$
$271$	15.4907	0.940994	0.470497	+	0.882402i	$0.344075\pi$
$272$	−8.35145	−0.506381
$273$	0	0
$274$	−7.63135	−0.461027
$275$	8.87634	0.535263
$276$	0	0
$277$	−17.6114	−1.05816	−0.529082	−	0.848571i	$-0.677464\pi$
$277$	−17.6114	−1.05816	−0.529082	+	0.848571i	$0.677464\pi$
$278$	50.0144	2.99966
$279$	0	0
$280$	21.8174	1.30384
$281$	−23.9635	−1.42954	−0.714771	−	0.699358i	$-0.753469\pi$
$281$	−23.9635	−1.42954	−0.714771	+	0.699358i	$0.753469\pi$
$282$	0	0
$283$	26.7493	1.59008	0.795040	−	0.606558i	$-0.207450\pi$
$283$	26.7493	1.59008	0.795040	+	0.606558i	$0.207450\pi$
$284$	36.1890	2.14742
$285$	0	0
$286$	3.89831	0.230512
$287$	11.4001	0.672925
$288$	0	0
$289$	−15.4678	−0.909868
$290$	50.6060	2.97169
$291$	0	0
$292$	69.3686	4.05949
$293$	−13.0240	−0.760870	−0.380435	−	0.924808i	$-0.624226\pi$
$293$	−13.0240	−0.760870	−0.380435	+	0.924808i	$0.624226\pi$
$294$	0	0
$295$	54.4314	3.16912
$296$	−62.8620	−3.65378
$297$	0	0
$298$	21.6079	1.25171
$299$	10.9171	0.631350
$300$	0	0
$301$	−8.38894	−0.483530
$302$	−13.5462	−0.779499
$303$	0	0
$304$	−29.6920	−1.70295
$305$	3.72510	0.213298
$306$	0	0
$307$	1.78896	0.102101	0.0510506	−	0.998696i	$-0.483743\pi$
$307$	1.78896	0.102101	0.0510506	+	0.998696i	$0.483743\pi$
$308$	−4.21365	−0.240095
$309$	0	0
$310$	−11.6588	−0.662175
$311$	4.69404	0.266175	0.133087	−	0.991104i	$-0.457511\pi$
$311$	4.69404	0.266175	0.133087	+	0.991104i	$0.457511\pi$
$312$	0	0
$313$	26.2738	1.48508	0.742541	−	0.669800i	$-0.233620\pi$
$313$	26.2738	1.48508	0.742541	+	0.669800i	$0.233620\pi$
$314$	5.71951	0.322771
$315$	0	0
$316$	−34.8876	−1.96258
$317$	−12.4347	−0.698404	−0.349202	−	0.937047i	$-0.613547\pi$
$317$	−12.4347	−0.698404	−0.349202	+	0.937047i	$0.613547\pi$
$318$	0	0
$319$	−5.35857	−0.300022
$320$	−4.96381	−0.277485
$321$	0	0
$322$	−17.1307	−0.954658
$323$	5.44760	0.303113
$324$	0	0
$325$	−13.6488	−0.757098
$326$	48.5428	2.68854
$327$	0	0
$328$	73.7130	4.07012
$329$	2.29946	0.126773
$330$	0	0
$331$	6.49136	0.356798	0.178399	−	0.983958i	$-0.442908\pi$
$331$	6.49136	0.356798	0.178399	+	0.983958i	$0.442908\pi$
$332$	8.61253	0.472674
$333$	0	0
$334$	29.9380	1.63813
$335$	38.4751	2.10212
$336$	0	0
$337$	12.1826	0.663627	0.331814	−	0.943345i	$-0.392339\pi$
$337$	12.1826	0.663627	0.331814	+	0.943345i	$0.392339\pi$
$338$	26.9636	1.46663
$339$	0	0
$340$	−20.4149	−1.10715
$341$	1.23453	0.0668533
$342$	0	0
$343$	12.4622	0.672893
$344$	−54.2430	−2.92458
$345$	0	0
$346$	26.9336	1.44796
$347$	3.47254	0.186416	0.0932080	−	0.995647i	$-0.470288\pi$
$347$	3.47254	0.186416	0.0932080	+	0.995647i	$0.470288\pi$
$348$	0	0
$349$	−18.9457	−1.01414	−0.507069	−	0.861906i	$-0.669271\pi$
$349$	−18.9457	−1.01414	−0.507069	+	0.861906i	$0.669271\pi$
$350$	21.4173	1.14480
$351$	0	0
$352$	−4.79684	−0.255672
$353$	−23.8117	−1.26737	−0.633685	−	0.773591i	$-0.718459\pi$
$353$	−23.8117	−1.26737	−0.633685	+	0.773591i	$0.718459\pi$
$354$	0	0
$355$	30.4487	1.61605
$356$	2.32018	0.122969
$357$	0	0
$358$	10.8367	0.572740
$359$	22.4083	1.18267	0.591334	−	0.806427i	$-0.298602\pi$
$359$	22.4083	1.18267	0.591334	+	0.806427i	$0.298602\pi$
$360$	0	0
$361$	0.367910	0.0193637
$362$	50.6399	2.66157
$363$	0	0
$364$	6.47917	0.339601
$365$	58.3654	3.05498
$366$	0	0
$367$	14.9397	0.779845	0.389923	−	0.920848i	$-0.372502\pi$
$367$	14.9397	0.779845	0.389923	+	0.920848i	$0.372502\pi$
$368$	−47.9008	−2.49700
$369$	0	0
$370$	−96.4698	−5.01522
$371$	6.70496	0.348104
$372$	0	0
$373$	−30.2393	−1.56573	−0.782866	−	0.622190i	$-0.786243\pi$
$373$	−30.2393	−1.56573	−0.782866	+	0.622190i	$0.786243\pi$
$374$	3.13820	0.162272
$375$	0	0
$376$	14.8683	0.766774
$377$	8.23965	0.424364
$378$	0	0
$379$	9.74598	0.500618	0.250309	−	0.968166i	$-0.419468\pi$
$379$	9.74598	0.500618	0.250309	+	0.968166i	$0.419468\pi$
$380$	−72.5811	−3.72333
$381$	0	0
$382$	59.1215	3.02492
$383$	−2.20960	−0.112905	−0.0564526	−	0.998405i	$-0.517979\pi$
$383$	−2.20960	−0.112905	−0.0564526	+	0.998405i	$0.517979\pi$
$384$	0	0
$385$	−3.54529	−0.180685
$386$	−24.7256	−1.25850
$387$	0	0
$388$	−67.9168	−3.44795
$389$	11.8827	0.602478	0.301239	−	0.953549i	$-0.402600\pi$
$389$	11.8827	0.602478	0.301239	+	0.953549i	$0.402600\pi$
$390$	0	0
$391$	8.78839	0.444448
$392$	37.5032	1.89420
$393$	0	0
$394$	−3.16181	−0.159290
$395$	−29.3538	−1.47695
$396$	0	0
$397$	−2.12121	−0.106461	−0.0532303	−	0.998582i	$-0.516952\pi$
$397$	−2.12121	−0.106461	−0.0532303	+	0.998582i	$0.516952\pi$
$398$	59.2734	2.97111
$399$	0	0
$400$	59.8868	2.99434
$401$	2.29783	0.114748	0.0573741	−	0.998353i	$-0.481727\pi$
$401$	2.29783	0.114748	0.0573741	+	0.998353i	$0.481727\pi$
$402$	0	0
$403$	−1.89828	−0.0945601
$404$	−47.1727	−2.34693
$405$	0	0
$406$	−12.9294	−0.641676
$407$	10.2150	0.506338
$408$	0	0
$409$	2.53852	0.125522	0.0627608	−	0.998029i	$-0.480009\pi$
$409$	2.53852	0.125522	0.0627608	+	0.998029i	$0.480009\pi$
$410$	113.122	5.58670
$411$	0	0
$412$	−10.5463	−0.519577
$413$	−13.9068	−0.684307
$414$	0	0
$415$	7.24641	0.355712
$416$	7.37591	0.361634
$417$	0	0
$418$	11.1573	0.545720
$419$	−3.19459	−0.156066	−0.0780329	−	0.996951i	$-0.524864\pi$
$419$	−3.19459	−0.156066	−0.0780329	+	0.996951i	$0.524864\pi$
$420$	0	0
$421$	−7.11087	−0.346562	−0.173281	−	0.984872i	$-0.555437\pi$
$421$	−7.11087	−0.346562	−0.173281	+	0.984872i	$0.555437\pi$
$422$	58.6438	2.85473
$423$	0	0
$424$	43.3544	2.10547
$425$	−10.9875	−0.532971
$426$	0	0
$427$	−0.951731	−0.0460575
$428$	−13.9888	−0.676172
$429$	0	0
$430$	−83.2428	−4.01432
$431$	7.24051	0.348763	0.174382	−	0.984678i	$-0.444207\pi$
$431$	7.24051	0.348763	0.174382	+	0.984678i	$0.444207\pi$
$432$	0	0
$433$	−3.83179	−0.184144	−0.0920719	−	0.995752i	$-0.529349\pi$
$433$	−3.83179	−0.184144	−0.0920719	+	0.995752i	$0.529349\pi$
$434$	2.97873	0.142983
$435$	0	0
$436$	39.2310	1.87882
$437$	31.2454	1.49467
$438$	0	0
$439$	−24.5778	−1.17303	−0.586517	−	0.809937i	$-0.699501\pi$
$439$	−24.5778	−1.17303	−0.586517	+	0.809937i	$0.699501\pi$
$440$	−22.9239	−1.09285
$441$	0	0
$442$	−4.82548	−0.229525
$443$	6.10789	0.290194	0.145097	−	0.989417i	$-0.453651\pi$
$443$	6.10789	0.290194	0.145097	+	0.989417i	$0.453651\pi$
$444$	0	0
$445$	1.95215	0.0925408
$446$	−67.7873	−3.20982
$447$	0	0
$448$	1.26821	0.0599174
$449$	−19.8224	−0.935476	−0.467738	−	0.883867i	$-0.654931\pi$
$449$	−19.8224	−0.935476	−0.467738	+	0.883867i	$0.654931\pi$
$450$	0	0
$451$	−11.9783	−0.564034
$452$	−85.3955	−4.01667
$453$	0	0
$454$	47.9464	2.25024
$455$	5.45145	0.255568
$456$	0	0
$457$	31.8471	1.48975	0.744873	−	0.667207i	$-0.232510\pi$
$457$	31.8471	1.48975	0.744873	+	0.667207i	$0.232510\pi$
$458$	−49.5773	−2.31659
$459$	0	0
$460$	−117.092	−5.45945
$461$	−12.1889	−0.567694	−0.283847	−	0.958870i	$-0.591611\pi$
$461$	−12.1889	−0.567694	−0.283847	+	0.958870i	$0.591611\pi$
$462$	0	0
$463$	9.20136	0.427623	0.213812	−	0.976875i	$-0.431412\pi$
$463$	9.20136	0.427623	0.213812	+	0.976875i	$0.431412\pi$
$464$	−36.1532	−1.67837
$465$	0	0
$466$	−41.8408	−1.93824
$467$	−14.9650	−0.692498	−0.346249	−	0.938143i	$-0.612545\pi$
$467$	−14.9650	−0.692498	−0.346249	+	0.938143i	$0.612545\pi$
$468$	0	0
$469$	−9.83007	−0.453910
$470$	22.8173	1.05248
$471$	0	0
$472$	−89.9213	−4.13896
$473$	8.81440	0.405287
$474$	0	0
$475$	−39.0639	−1.79237
$476$	5.21583	0.239067
$477$	0	0
$478$	−27.3270	−1.24991
$479$	−32.0402	−1.46395	−0.731977	−	0.681330i	$-0.761402\pi$
$479$	−32.0402	−1.46395	−0.731977	+	0.681330i	$0.761402\pi$
$480$	0	0
$481$	−15.7072	−0.716185
$482$	33.7642	1.53792
$483$	0	0
$484$	4.42736	0.201244
$485$	−57.1439	−2.59477
$486$	0	0
$487$	−21.0911	−0.955728	−0.477864	−	0.878434i	$-0.658589\pi$
$487$	−21.0911	−0.955728	−0.477864	+	0.878434i	$0.658589\pi$
$488$	−6.15390	−0.278574
$489$	0	0
$490$	57.5534	2.60000
$491$	−15.5542	−0.701952	−0.350976	−	0.936384i	$-0.614150\pi$
$491$	−15.5542	−0.701952	−0.350976	+	0.936384i	$0.614150\pi$
$492$	0	0
$493$	6.63304	0.298737
$494$	−17.1561	−0.771888
$495$	0	0
$496$	8.32910	0.373987
$497$	−7.77940	−0.348954
$498$	0	0
$499$	−27.9129	−1.24956	−0.624778	−	0.780803i	$-0.714810\pi$
$499$	−27.9129	−1.24956	−0.624778	+	0.780803i	$0.714810\pi$
$500$	63.9298	2.85903
$501$	0	0
$502$	−16.8345	−0.751360
$503$	20.2569	0.903211	0.451606	−	0.892218i	$-0.350851\pi$
$503$	20.2569	0.903211	0.451606	+	0.892218i	$0.350851\pi$
$504$	0	0
$505$	−39.6902	−1.76619
$506$	17.9995	0.800178
$507$	0	0
$508$	−0.269605	−0.0119618
$509$	3.75634	0.166497	0.0832485	−	0.996529i	$-0.473470\pi$
$509$	3.75634	0.166497	0.0832485	+	0.996529i	$0.473470\pi$
$510$	0	0
$511$	−14.9119	−0.659662
$512$	50.6749	2.23954
$513$	0	0
$514$	46.3539	2.04458
$515$	−8.87343	−0.391010
$516$	0	0
$517$	−2.41608	−0.106259
$518$	24.6472	1.08294
$519$	0	0
$520$	35.2491	1.54578
$521$	−13.7087	−0.600588	−0.300294	−	0.953847i	$-0.597085\pi$
$521$	−13.7087	−0.600588	−0.300294	+	0.953847i	$0.597085\pi$
$522$	0	0
$523$	32.1809	1.40717	0.703586	−	0.710611i	$-0.251581\pi$
$523$	32.1809	1.40717	0.703586	+	0.710611i	$0.251581\pi$
$524$	−2.72880	−0.119208
$525$	0	0
$526$	−42.0293	−1.83257
$527$	−1.52814	−0.0665670
$528$	0	0
$529$	27.4070	1.19161
$530$	66.5328	2.89000
$531$	0	0
$532$	18.5439	0.803979
$533$	18.4185	0.797793
$534$	0	0
$535$	−11.7699	−0.508856
$536$	−63.5613	−2.74543
$537$	0	0
$538$	27.6983	1.19416
$539$	−6.09421	−0.262496
$540$	0	0
$541$	−24.1341	−1.03761	−0.518804	−	0.854894i	$-0.673622\pi$
$541$	−24.1341	−1.03761	−0.518804	+	0.854894i	$0.673622\pi$
$542$	−39.2724	−1.68690
$543$	0	0
$544$	5.93772	0.254577
$545$	33.0082	1.41392
$546$	0	0
$547$	−27.8925	−1.19260	−0.596299	−	0.802762i	$-0.703363\pi$
$547$	−27.8925	−1.19260	−0.596299	+	0.802762i	$0.703363\pi$
$548$	13.3269	0.569298
$549$	0	0
$550$	−22.5035	−0.959552
$551$	23.5825	1.00465
$552$	0	0
$553$	7.49964	0.318917
$554$	44.6487	1.89694
$555$	0	0
$556$	−87.3421	−3.70413
$557$	22.9190	0.971108	0.485554	−	0.874207i	$-0.338618\pi$
$557$	22.9190	0.971108	0.485554	+	0.874207i	$0.338618\pi$
$558$	0	0
$559$	−13.5535	−0.573254
$560$	−23.9193	−1.01078
$561$	0	0
$562$	60.7528	2.56270
$563$	−2.08076	−0.0876937	−0.0438469	−	0.999038i	$-0.513961\pi$
$563$	−2.08076	−0.0876937	−0.0438469	+	0.999038i	$0.513961\pi$
$564$	0	0
$565$	−71.8501	−3.02276
$566$	−67.8154	−2.85049
$567$	0	0
$568$	−50.3017	−2.11061
$569$	−46.1126	−1.93314	−0.966570	−	0.256402i	$-0.917463\pi$
$569$	−46.1126	−1.93314	−0.966570	+	0.256402i	$0.917463\pi$
$570$	0	0
$571$	18.4548	0.772308	0.386154	−	0.922434i	$-0.373803\pi$
$571$	18.4548	0.772308	0.386154	+	0.922434i	$0.373803\pi$
$572$	−6.80777	−0.284647
$573$	0	0
$574$	−28.9018	−1.20634
$575$	−63.0201	−2.62812
$576$	0	0
$577$	1.03683	0.0431638	0.0215819	−	0.999767i	$-0.493130\pi$
$577$	1.03683	0.0431638	0.0215819	+	0.999767i	$0.493130\pi$
$578$	39.2142	1.63110
$579$	0	0
$580$	−88.3753	−3.66958
$581$	−1.85140	−0.0768089
$582$	0	0
$583$	−7.04502	−0.291775
$584$	−96.4203	−3.98990
$585$	0	0
$586$	33.0187	1.36399
$587$	−24.5792	−1.01449	−0.507245	−	0.861802i	$-0.669336\pi$
$587$	−24.5792	−1.01449	−0.507245	+	0.861802i	$0.669336\pi$
$588$	0	0
$589$	−5.43302	−0.223864
$590$	−137.996	−5.68120
$591$	0	0
$592$	68.9184	2.83253
$593$	−45.7292	−1.87787	−0.938937	−	0.344089i	$-0.888188\pi$
$593$	−45.7292	−1.87787	−0.938937	+	0.344089i	$0.888188\pi$
$594$	0	0
$595$	4.38849	0.179911
$596$	−37.7346	−1.54567
$597$	0	0
$598$	−27.6772	−1.13180
$599$	4.29437	0.175463	0.0877316	−	0.996144i	$-0.472038\pi$
$599$	4.29437	0.175463	0.0877316	+	0.996144i	$0.472038\pi$
$600$	0	0
$601$	13.6658	0.557439	0.278719	−	0.960373i	$-0.410090\pi$
$601$	13.6658	0.557439	0.278719	+	0.960373i	$0.410090\pi$
$602$	21.2678	0.866812
$603$	0	0
$604$	23.6563	0.962562
$605$	3.72510	0.151447
$606$	0	0
$607$	−35.8576	−1.45542	−0.727708	−	0.685887i	$-0.759414\pi$
$607$	−35.8576	−1.45542	−0.727708	+	0.685887i	$0.759414\pi$
$608$	21.1104	0.856140
$609$	0	0
$610$	−9.44395	−0.382374
$611$	3.71510	0.150297
$612$	0	0
$613$	9.12777	0.368667	0.184333	−	0.982864i	$-0.440987\pi$
$613$	9.12777	0.368667	0.184333	+	0.982864i	$0.440987\pi$
$614$	−4.53541	−0.183034
$615$	0	0
$616$	5.85686	0.235980
$617$	−31.3560	−1.26234	−0.631172	−	0.775643i	$-0.717426\pi$
$617$	−31.3560	−1.26234	−0.631172	+	0.775643i	$0.717426\pi$
$618$	0	0
$619$	25.0164	1.00549	0.502746	−	0.864434i	$-0.332323\pi$
$619$	25.0164	1.00549	0.502746	+	0.864434i	$0.332323\pi$
$620$	20.3602	0.817686
$621$	0	0
$622$	−11.9004	−0.477164
$623$	−0.498758	−0.0199823
$624$	0	0
$625$	9.40766	0.376306
$626$	−66.6100	−2.66227
$627$	0	0
$628$	−9.98820	−0.398573
$629$	−12.6445	−0.504169
$630$	0	0
$631$	27.1969	1.08269	0.541345	−	0.840801i	$-0.317915\pi$
$631$	27.1969	1.08269	0.541345	+	0.840801i	$0.317915\pi$
$632$	48.4928	1.92894
$633$	0	0
$634$	31.5248	1.25201
$635$	−0.226841	−0.00900190
$636$	0	0
$637$	9.37082	0.371285
$638$	13.5852	0.537842
$639$	0	0
$640$	48.3217	1.91008
$641$	24.0469	0.949794	0.474897	−	0.880041i	$-0.342485\pi$
$641$	24.0469	0.949794	0.474897	+	0.880041i	$0.342485\pi$
$642$	0	0
$643$	12.8173	0.505465	0.252732	−	0.967536i	$-0.418671\pi$
$643$	12.8173	0.505465	0.252732	+	0.967536i	$0.418671\pi$
$644$	29.9161	1.17886
$645$	0	0
$646$	−13.8109	−0.543382
$647$	44.1154	1.73436	0.867179	−	0.497997i	$-0.165931\pi$
$647$	44.1154	1.73436	0.867179	+	0.497997i	$0.165931\pi$
$648$	0	0
$649$	14.6121	0.573574
$650$	34.6027	1.35723
$651$	0	0
$652$	−84.7722	−3.31994
$653$	1.44701	0.0566259	0.0283129	−	0.999599i	$-0.490987\pi$
$653$	1.44701	0.0566259	0.0283129	+	0.999599i	$0.490987\pi$
$654$	0	0
$655$	−2.29596	−0.0897105
$656$	−80.8149	−3.15529
$657$	0	0
$658$	−5.82963	−0.227263
$659$	−5.36239	−0.208889	−0.104444	−	0.994531i	$-0.533306\pi$
$659$	−5.36239	−0.208889	−0.104444	+	0.994531i	$0.533306\pi$
$660$	0	0
$661$	27.8255	1.08229	0.541143	−	0.840931i	$-0.317992\pi$
$661$	27.8255	1.08229	0.541143	+	0.840931i	$0.317992\pi$
$662$	−16.4571	−0.639622
$663$	0	0
$664$	−11.9712	−0.464571
$665$	15.6025	0.605037
$666$	0	0
$667$	38.0447	1.47310
$668$	−52.2819	−2.02285
$669$	0	0
$670$	−97.5430	−3.76842
$671$	1.00000	0.0386046
$672$	0	0
$673$	−44.2181	−1.70448	−0.852241	−	0.523149i	$-0.824757\pi$
$673$	−44.2181	−1.70448	−0.852241	+	0.523149i	$0.824757\pi$
$674$	−30.8856	−1.18967
$675$	0	0
$676$	−47.0876	−1.81106
$677$	33.1718	1.27490	0.637448	−	0.770494i	$-0.279990\pi$
$677$	33.1718	1.27490	0.637448	+	0.770494i	$0.279990\pi$
$678$	0	0
$679$	14.5998	0.560289
$680$	28.3761	1.08817
$681$	0	0
$682$	−3.12980	−0.119846
$683$	34.4018	1.31635	0.658175	−	0.752865i	$-0.271329\pi$
$683$	34.4018	1.31635	0.658175	+	0.752865i	$0.271329\pi$
$684$	0	0
$685$	11.2130	0.428427
$686$	−31.5944	−1.20628
$687$	0	0
$688$	59.4690	2.26723
$689$	10.8328	0.412699
$690$	0	0
$691$	−48.1463	−1.83157	−0.915787	−	0.401665i	$-0.868432\pi$
$691$	−48.1463	−1.83157	−0.915787	+	0.401665i	$0.868432\pi$
$692$	−47.0353	−1.78801
$693$	0	0
$694$	−8.80368	−0.334183
$695$	−73.4880	−2.78756
$696$	0	0
$697$	14.8272	0.561619
$698$	48.0315	1.81802
$699$	0	0
$700$	−37.4018	−1.41366
$701$	9.58358	0.361967	0.180983	−	0.983486i	$-0.442072\pi$
$701$	9.58358	0.361967	0.180983	+	0.983486i	$0.442072\pi$
$702$	0	0
$703$	−44.9551	−1.69551
$704$	−1.33253	−0.0502217
$705$	0	0
$706$	60.3681	2.27198
$707$	10.1405	0.381373
$708$	0	0
$709$	−7.95650	−0.298813	−0.149406	−	0.988776i	$-0.547736\pi$
$709$	−7.95650	−0.298813	−0.149406	+	0.988776i	$0.547736\pi$
$710$	−77.1944	−2.89705
$711$	0	0
$712$	−3.22498	−0.120861
$713$	−8.76487	−0.328247
$714$	0	0
$715$	−5.72793	−0.214212
$716$	−18.9246	−0.707246
$717$	0	0
$718$	−56.8102	−2.12014
$719$	23.0223	0.858587	0.429294	−	0.903165i	$-0.358763\pi$
$719$	23.0223	0.858587	0.429294	+	0.903165i	$0.358763\pi$
$720$	0	0
$721$	2.26709	0.0844307
$722$	−0.932733	−0.0347127
$723$	0	0
$724$	−88.4344	−3.28664
$725$	−47.5644	−1.76650
$726$	0	0
$727$	29.2774	1.08584	0.542920	−	0.839784i	$-0.317319\pi$
$727$	29.2774	1.08584	0.542920	+	0.839784i	$0.317319\pi$
$728$	−9.00585	−0.333779
$729$	0	0
$730$	−147.969	−5.47659
$731$	−10.9108	−0.403551
$732$	0	0
$733$	−40.6122	−1.50005	−0.750023	−	0.661412i	$-0.769957\pi$
$733$	−40.6122	−1.50005	−0.750023	+	0.661412i	$0.769957\pi$
$734$	−37.8755	−1.39801
$735$	0	0
$736$	34.0565	1.25534
$737$	10.3286	0.380460
$738$	0	0
$739$	20.9408	0.770321	0.385160	−	0.922850i	$-0.374146\pi$
$739$	20.9408	0.770321	0.385160	+	0.922850i	$0.374146\pi$
$740$	168.469	6.19304
$741$	0	0
$742$	−16.9986	−0.624038
$743$	−22.1711	−0.813377	−0.406689	−	0.913567i	$-0.633317\pi$
$743$	−22.1711	−0.813377	−0.406689	+	0.913567i	$0.633317\pi$
$744$	0	0
$745$	−31.7492	−1.16320
$746$	76.6634	2.80685
$747$	0	0
$748$	−5.48036	−0.200382
$749$	3.00710	0.109877
$750$	0	0
$751$	7.45803	0.272147	0.136074	−	0.990699i	$-0.456552\pi$
$751$	7.45803	0.272147	0.136074	+	0.990699i	$0.456552\pi$
$752$	−16.3008	−0.594429
$753$	0	0
$754$	−20.8894	−0.760746
$755$	19.9040	0.724380
$756$	0	0
$757$	−37.0595	−1.34695	−0.673475	−	0.739210i	$-0.735199\pi$
$757$	−37.0595	−1.34695	−0.673475	+	0.739210i	$0.735199\pi$
$758$	−24.7083	−0.897444
$759$	0	0
$760$	100.886	3.65951
$761$	31.6016	1.14556	0.572778	−	0.819711i	$-0.305866\pi$
$761$	31.6016	1.14556	0.572778	+	0.819711i	$0.305866\pi$
$762$	0	0
$763$	−8.43332	−0.305307
$764$	−103.246	−3.73531
$765$	0	0
$766$	5.60183	0.202402
$767$	−22.4684	−0.811287
$768$	0	0
$769$	−23.3899	−0.843460	−0.421730	−	0.906722i	$-0.638577\pi$
$769$	−23.3899	−0.843460	−0.421730	+	0.906722i	$0.638577\pi$
$770$	8.98810	0.323909
$771$	0	0
$772$	43.1792	1.55405
$773$	31.0588	1.11711	0.558555	−	0.829468i	$-0.311356\pi$
$773$	31.0588	1.11711	0.558555	+	0.829468i	$0.311356\pi$
$774$	0	0
$775$	10.9581	0.393625
$776$	94.4024	3.38885
$777$	0	0
$778$	−30.1254	−1.08005
$779$	52.7151	1.88872
$780$	0	0
$781$	8.17395	0.292487
$782$	−22.2805	−0.796751
$783$	0	0
$784$	−41.1164	−1.46844
$785$	−8.40388	−0.299947
$786$	0	0
$787$	18.5467	0.661117	0.330558	−	0.943786i	$-0.392763\pi$
$787$	18.5467	0.661117	0.330558	+	0.943786i	$0.392763\pi$
$788$	5.52160	0.196699
$789$	0	0
$790$	74.4184	2.64769
$791$	18.3571	0.652704
$792$	0	0
$793$	−1.53766	−0.0546039
$794$	5.37775	0.190849
$795$	0	0
$796$	−103.511	−3.66887
$797$	25.4905	0.902921	0.451461	−	0.892291i	$-0.350903\pi$
$797$	25.4905	0.902921	0.451461	+	0.892291i	$0.350903\pi$
$798$	0	0
$799$	2.99071	0.105804
$800$	−42.5784	−1.50537
$801$	0	0
$802$	−5.82552	−0.205706
$803$	15.6682	0.552917
$804$	0	0
$805$	25.1708	0.887154
$806$	4.81257	0.169515
$807$	0	0
$808$	65.5687	2.30670
$809$	−11.5648	−0.406597	−0.203299	−	0.979117i	$-0.565166\pi$
$809$	−11.5648	−0.406597	−0.203299	+	0.979117i	$0.565166\pi$
$810$	0	0
$811$	−24.1672	−0.848625	−0.424313	−	0.905516i	$-0.639484\pi$
$811$	−24.1672	−0.848625	−0.424313	+	0.905516i	$0.639484\pi$
$812$	22.5791	0.792373
$813$	0	0
$814$	−25.8973	−0.907698
$815$	−71.3257	−2.49843
$816$	0	0
$817$	−38.7913	−1.35714
$818$	−6.43571	−0.225019
$819$	0	0
$820$	−197.550	−6.89873
$821$	39.1543	1.36649	0.683247	−	0.730187i	$-0.260567\pi$
$821$	39.1543	1.36649	0.683247	+	0.730187i	$0.260567\pi$
$822$	0	0
$823$	36.9250	1.28712	0.643562	−	0.765394i	$-0.277456\pi$
$823$	36.9250	1.28712	0.643562	+	0.765394i	$0.277456\pi$
$824$	14.6590	0.510671
$825$	0	0
$826$	35.2568	1.22674
$827$	27.4711	0.955263	0.477631	−	0.878560i	$-0.341495\pi$
$827$	27.4711	0.955263	0.477631	+	0.878560i	$0.341495\pi$
$828$	0	0
$829$	−42.1370	−1.46348	−0.731738	−	0.681586i	$-0.761291\pi$
$829$	−42.1370	−1.46348	−0.731738	+	0.681586i	$0.761291\pi$
$830$	−18.3713	−0.637677
$831$	0	0
$832$	2.04898	0.0710356
$833$	7.54365	0.261372
$834$	0	0
$835$	−43.9890	−1.52230
$836$	−19.4844	−0.673881
$837$	0	0
$838$	8.09899	0.279775
$839$	40.3633	1.39350	0.696748	−	0.717316i	$-0.254630\pi$
$839$	40.3633	1.39350	0.696748	+	0.717316i	$0.254630\pi$
$840$	0	0
$841$	−0.285774	−0.00985429
$842$	18.0276	0.621273
$843$	0	0
$844$	−102.412	−3.52516
$845$	−39.6186	−1.36292
$846$	0	0
$847$	−0.951731	−0.0327019
$848$	−47.5313	−1.63223
$849$	0	0
$850$	27.8557	0.955443
$851$	−72.5242	−2.48610
$852$	0	0
$853$	10.7694	0.368738	0.184369	−	0.982857i	$-0.440976\pi$
$853$	10.7694	0.368738	0.184369	+	0.982857i	$0.440976\pi$
$854$	2.41285	0.0825660
$855$	0	0
$856$	19.4440	0.664581
$857$	27.9239	0.953862	0.476931	−	0.878941i	$-0.341749\pi$
$857$	27.9239	0.953862	0.476931	+	0.878941i	$0.341749\pi$
$858$	0	0
$859$	−4.54206	−0.154973	−0.0774866	−	0.996993i	$-0.524689\pi$
$859$	−4.54206	−0.154973	−0.0774866	+	0.996993i	$0.524689\pi$
$860$	145.370	4.95708
$861$	0	0
$862$	−18.3563	−0.625219
$863$	−24.6926	−0.840546	−0.420273	−	0.907398i	$-0.638066\pi$
$863$	−24.6926	−0.840546	−0.420273	+	0.907398i	$0.638066\pi$
$864$	0	0
$865$	−39.5746	−1.34558
$866$	9.71443	0.330110
$867$	0	0
$868$	−5.20187	−0.176563
$869$	−7.88000	−0.267311
$870$	0	0
$871$	−15.8819	−0.538138
$872$	−54.5299	−1.84662
$873$	0	0
$874$	−79.2142	−2.67946
$875$	−13.7427	−0.464589
$876$	0	0
$877$	41.3084	1.39488	0.697442	−	0.716641i	$-0.254321\pi$
$877$	41.3084	1.39488	0.697442	+	0.716641i	$0.254321\pi$
$878$	62.3102	2.10287
$879$	0	0
$880$	25.1325	0.847215
$881$	50.4986	1.70134	0.850670	−	0.525700i	$-0.176197\pi$
$881$	50.4986	1.70134	0.850670	+	0.525700i	$0.176197\pi$
$882$	0	0
$883$	17.2599	0.580842	0.290421	−	0.956899i	$-0.406205\pi$
$883$	17.2599	0.580842	0.290421	+	0.956899i	$0.406205\pi$
$884$	8.42693	0.283428
$885$	0	0
$886$	−15.4849	−0.520224
$887$	19.8671	0.667074	0.333537	−	0.942737i	$-0.391758\pi$
$887$	19.8671	0.667074	0.333537	+	0.942737i	$0.391758\pi$
$888$	0	0
$889$	0.0579559	0.00194378
$890$	−4.94914	−0.165896
$891$	0	0
$892$	118.380	3.96364
$893$	10.6329	0.355817
$894$	0	0
$895$	−15.9228	−0.532241
$896$	−12.3458	−0.412444
$897$	0	0
$898$	50.2542	1.67700
$899$	−6.61529	−0.220632
$900$	0	0
$901$	8.72060	0.290525
$902$	30.3676	1.01113
$903$	0	0
$904$	118.697	3.94781
$905$	−74.4070	−2.47337
$906$	0	0
$907$	−15.4581	−0.513277	−0.256638	−	0.966508i	$-0.582615\pi$
$907$	−15.4581	−0.513277	−0.256638	+	0.966508i	$0.582615\pi$
$908$	−83.7307	−2.77870
$909$	0	0
$910$	−13.8206	−0.458150
$911$	5.86489	0.194312	0.0971562	−	0.995269i	$-0.469025\pi$
$911$	5.86489	0.194312	0.0971562	+	0.995269i	$0.469025\pi$
$912$	0	0
$913$	1.94530	0.0643799
$914$	−80.7396	−2.67063
$915$	0	0
$916$	86.5787	2.86064
$917$	0.586598	0.0193712
$918$	0	0
$919$	−37.9978	−1.25343	−0.626716	−	0.779248i	$-0.715601\pi$
$919$	−37.9978	−1.25343	−0.626716	+	0.779248i	$0.715601\pi$
$920$	162.755	5.36586
$921$	0	0
$922$	30.9016	1.01769
$923$	−12.5688	−0.413705
$924$	0	0
$925$	90.6716	2.98126
$926$	−23.3275	−0.766589
$927$	0	0
$928$	25.7042	0.843781
$929$	−8.20155	−0.269084	−0.134542	−	0.990908i	$-0.542956\pi$
$929$	−8.20155	−0.269084	−0.134542	+	0.990908i	$0.542956\pi$
$930$	0	0
$931$	26.8200	0.878990
$932$	73.0683	2.39343
$933$	0	0
$934$	37.9397	1.24142
$935$	−4.61107	−0.150798
$936$	0	0
$937$	56.2620	1.83800	0.918999	−	0.394259i	$-0.128999\pi$
$937$	56.2620	1.83800	0.918999	+	0.394259i	$0.128999\pi$
$938$	24.9214	0.813713
$939$	0	0
$940$	−39.8468	−1.29966
$941$	46.9435	1.53031	0.765157	−	0.643844i	$-0.222661\pi$
$941$	46.9435	1.53031	0.765157	+	0.643844i	$0.222661\pi$
$942$	0	0
$943$	85.0431	2.76938
$944$	98.5848	3.20866
$945$	0	0
$946$	−22.3465	−0.726547
$947$	−52.9734	−1.72140	−0.860702	−	0.509110i	$-0.829975\pi$
$947$	−52.9734	−1.72140	−0.860702	+	0.509110i	$0.829975\pi$
$948$	0	0
$949$	−24.0923	−0.782069
$950$	99.0356	3.21314
$951$	0	0
$952$	−7.24985	−0.234969
$953$	0.446509	0.0144638	0.00723191	−	0.999974i	$-0.497698\pi$
$953$	0.446509	0.0144638	0.00723191	+	0.999974i	$0.497698\pi$
$954$	0	0
$955$	−86.8693	−2.81102
$956$	47.7223	1.54345
$957$	0	0
$958$	81.2291	2.62439
$959$	−2.86483	−0.0925102
$960$	0	0
$961$	−29.4759	−0.950837
$962$	39.8212	1.28389
$963$	0	0
$964$	−58.9637	−1.89909
$965$	36.3301	1.16951
$966$	0	0
$967$	−28.4533	−0.914997	−0.457499	−	0.889210i	$-0.651255\pi$
$967$	−28.4533	−0.914997	−0.457499	+	0.889210i	$0.651255\pi$
$968$	−6.15390	−0.197794
$969$	0	0
$970$	144.873	4.65158
$971$	15.7288	0.504760	0.252380	−	0.967628i	$-0.418787\pi$
$971$	15.7288	0.504760	0.252380	+	0.967628i	$0.418787\pi$
$972$	0	0
$973$	18.7756	0.601917
$974$	53.4706	1.71331
$975$	0	0
$976$	6.74680	0.215960
$977$	−31.3215	−1.00206	−0.501032	−	0.865429i	$-0.667046\pi$
$977$	−31.3215	−1.00206	−0.501032	+	0.865429i	$0.667046\pi$
$978$	0	0
$979$	0.524054	0.0167488
$980$	−100.508	−3.21060
$981$	0	0
$982$	39.4334	1.25837
$983$	−13.2756	−0.423427	−0.211714	−	0.977332i	$-0.567904\pi$
$983$	−13.2756	−0.423427	−0.211714	+	0.977332i	$0.567904\pi$
$984$	0	0
$985$	4.64576	0.148026
$986$	−16.8162	−0.535538
$987$	0	0
$988$	29.9603	0.953165
$989$	−62.5804	−1.98994
$990$	0	0
$991$	−9.57868	−0.304277	−0.152138	−	0.988359i	$-0.548616\pi$
$991$	−9.57868	−0.304277	−0.152138	+	0.988359i	$0.548616\pi$
$992$	−5.92182	−0.188018
$993$	0	0
$994$	19.7225	0.625560
$995$	−87.0925	−2.76102
$996$	0	0
$997$	−39.9918	−1.26655	−0.633277	−	0.773925i	$-0.718291\pi$
$997$	−39.9918	−1.26655	−0.633277	+	0.773925i	$0.718291\pi$
$998$	70.7656	2.24004
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.2	✓	25
3.2	odd	2		6039.2.a.p.1.24	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.2	✓	25	1.1	even	1	trivial
6039.2.a.p.1.24	yes	25	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.53522 q^{2} +4.42736 q^{4} +3.72510 q^{5} -0.951731 q^{7} -6.15390 q^{8} +O(q^{10})\)	\(q-2.53522 q^{2} +4.42736 q^{4} +3.72510 q^{5} -0.951731 q^{7} -6.15390 q^{8} -9.44395 q^{10} +1.00000 q^{11} -1.53766 q^{13} +2.41285 q^{14} +6.74680 q^{16} -1.23784 q^{17} -4.40090 q^{19} +16.4923 q^{20} -2.53522 q^{22} -7.09979 q^{23} +8.87634 q^{25} +3.89831 q^{26} -4.21365 q^{28} -5.35857 q^{29} +1.23453 q^{31} -4.79684 q^{32} +3.13820 q^{34} -3.54529 q^{35} +10.2150 q^{37} +11.1573 q^{38} -22.9239 q^{40} -11.9783 q^{41} +8.81440 q^{43} +4.42736 q^{44} +17.9995 q^{46} -2.41608 q^{47} -6.09421 q^{49} -22.5035 q^{50} -6.80777 q^{52} -7.04502 q^{53} +3.72510 q^{55} +5.85686 q^{56} +13.5852 q^{58} +14.6121 q^{59} +1.00000 q^{61} -3.12980 q^{62} -1.33253 q^{64} -5.72793 q^{65} +10.3286 q^{67} -5.48036 q^{68} +8.98810 q^{70} +8.17395 q^{71} +15.6682 q^{73} -25.8973 q^{74} -19.4844 q^{76} -0.951731 q^{77} -7.88000 q^{79} +25.1325 q^{80} +30.3676 q^{82} +1.94530 q^{83} -4.61107 q^{85} -22.3465 q^{86} -6.15390 q^{88} +0.524054 q^{89} +1.46344 q^{91} -31.4333 q^{92} +6.12530 q^{94} -16.3938 q^{95} -15.3403 q^{97} +15.4502 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Introduction

L-functions

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