LMFDB - Embedded newform 8037.2.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51668 q^{2} +4.33366 q^{4} -1.58121 q^{5} +3.57116 q^{7} -5.87307 q^{8} +O(q^{10})$	\(q-2.51668 q^{2} +4.33366 q^{4} -1.58121 q^{5} +3.57116 q^{7} -5.87307 q^{8} +3.97939 q^{10} -6.04500 q^{11} -5.34670 q^{13} -8.98746 q^{14} +6.11329 q^{16} -0.189577 q^{17} -1.00000 q^{19} -6.85242 q^{20} +15.2133 q^{22} -7.77786 q^{23} -2.49978 q^{25} +13.4559 q^{26} +15.4762 q^{28} +6.30707 q^{29} -7.81779 q^{31} -3.63904 q^{32} +0.477105 q^{34} -5.64675 q^{35} -7.85454 q^{37} +2.51668 q^{38} +9.28655 q^{40} -9.08711 q^{41} -0.672248 q^{43} -26.1970 q^{44} +19.5744 q^{46} -1.00000 q^{47} +5.75319 q^{49} +6.29113 q^{50} -23.1708 q^{52} +10.0039 q^{53} +9.55841 q^{55} -20.9737 q^{56} -15.8729 q^{58} -4.72100 q^{59} -8.43394 q^{61} +19.6749 q^{62} -3.06830 q^{64} +8.45426 q^{65} -8.88895 q^{67} -0.821564 q^{68} +14.2111 q^{70} -8.51929 q^{71} -8.58241 q^{73} +19.7673 q^{74} -4.33366 q^{76} -21.5877 q^{77} +15.8868 q^{79} -9.66639 q^{80} +22.8693 q^{82} -5.89406 q^{83} +0.299762 q^{85} +1.69183 q^{86} +35.5027 q^{88} -9.24280 q^{89} -19.0939 q^{91} -33.7066 q^{92} +2.51668 q^{94} +1.58121 q^{95} -9.88764 q^{97} -14.4789 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8	$ 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8} - 5 q^{10} - 12 q^{11} + 9 q^{13} - 3 q^{14} + 60 q^{16} - 16 q^{17} - 23 q^{19} - 25 q^{20} + 3 q^{22} - 12 q^{23} + 54 q^{25} - 5 q^{26} + 8 q^{28} - 27 q^{29} + 10 q^{31} + 34 q^{32} + 6 q^{35} + 15 q^{37} + 2 q^{38} + 3 q^{40} - 10 q^{41} + 24 q^{43} - 39 q^{44} + 43 q^{46} - 23 q^{47} + 78 q^{49} + 32 q^{50} + 38 q^{52} + 2 q^{53} + 5 q^{55} - 58 q^{56} - 11 q^{58} + 51 q^{59} + 48 q^{61} + 22 q^{62} + 125 q^{64} - 15 q^{65} + 26 q^{67} - 26 q^{68} + 86 q^{70} - 24 q^{71} + 53 q^{73} - 26 q^{74} - 30 q^{76} - 18 q^{77} + 29 q^{79} - 5 q^{80} + 47 q^{82} + 22 q^{83} - 5 q^{85} + 28 q^{86} + 62 q^{88} - 38 q^{89} + 15 q^{91} - 15 q^{92} + 2 q^{94} + 9 q^{95} + 33 q^{97} + 30 q^{98}+O(q^{100}) $ 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8 - 5 * q^10 - 12 * q^11 + 9 * q^13 - 3 * q^14 + 60 * q^16 - 16 * q^17 - 23 * q^19 - 25 * q^20 + 3 * q^22 - 12 * q^23 + 54 * q^25 - 5 * q^26 + 8 * q^28 - 27 * q^29 + 10 * q^31 + 34 * q^32 + 6 * q^35 + 15 * q^37 + 2 * q^38 + 3 * q^40 - 10 * q^41 + 24 * q^43 - 39 * q^44 + 43 * q^46 - 23 * q^47 + 78 * q^49 + 32 * q^50 + 38 * q^52 + 2 * q^53 + 5 * q^55 - 58 * q^56 - 11 * q^58 + 51 * q^59 + 48 * q^61 + 22 * q^62 + 125 * q^64 - 15 * q^65 + 26 * q^67 - 26 * q^68 + 86 * q^70 - 24 * q^71 + 53 * q^73 - 26 * q^74 - 30 * q^76 - 18 * q^77 + 29 * q^79 - 5 * q^80 + 47 * q^82 + 22 * q^83 - 5 * q^85 + 28 * q^86 + 62 * q^88 - 38 * q^89 + 15 * q^91 - 15 * q^92 + 2 * q^94 + 9 * q^95 + 33 * q^97 + 30 * 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 $

$2$	−2.51668	−1.77956	−0.889779	−	0.456391i	$-0.849142\pi$
$2$	−2.51668	−1.77956	−0.889779	+	0.456391i	$0.849142\pi$
$3$	0	0
$3$	0	0
$4$	4.33366	2.16683
$5$	−1.58121	−0.707138	−0.353569	−	0.935408i	$-0.615032\pi$
$5$	−1.58121	−0.707138	−0.353569	+	0.935408i	$0.615032\pi$
$6$	0	0
$7$	3.57116	1.34977	0.674886	−	0.737922i	$-0.264193\pi$
$7$	3.57116	1.34977	0.674886	+	0.737922i	$0.264193\pi$
$8$	−5.87307	−2.07644
$9$	0	0
$10$	3.97939	1.25839
$11$	−6.04500	−1.82264	−0.911318	−	0.411703i	$-0.864934\pi$
$11$	−6.04500	−1.82264	−0.911318	+	0.411703i	$0.864934\pi$
$12$	0	0
$13$	−5.34670	−1.48291	−0.741454	−	0.671003i	$-0.765864\pi$
$13$	−5.34670	−1.48291	−0.741454	+	0.671003i	$0.765864\pi$
$14$	−8.98746	−2.40200
$15$	0	0
$16$	6.11329	1.52832
$17$	−0.189577	−0.0459793	−0.0229896	−	0.999736i	$-0.507318\pi$
$17$	−0.189577	−0.0459793	−0.0229896	+	0.999736i	$0.507318\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−6.85242	−1.53225
$21$	0	0
$22$	15.2133	3.24349
$23$	−7.77786	−1.62180	−0.810898	−	0.585188i	$-0.801021\pi$
$23$	−7.77786	−1.62180	−0.810898	+	0.585188i	$0.801021\pi$
$24$	0	0
$25$	−2.49978	−0.499955
$26$	13.4559	2.63892
$27$	0	0
$28$	15.4762	2.92473
$29$	6.30707	1.17119	0.585597	−	0.810602i	$-0.300860\pi$
$29$	6.30707	1.17119	0.585597	+	0.810602i	$0.300860\pi$
$30$	0	0
$31$	−7.81779	−1.40412	−0.702059	−	0.712119i	$-0.747736\pi$
$31$	−7.81779	−1.40412	−0.702059	+	0.712119i	$0.747736\pi$
$32$	−3.63904	−0.643297
$33$	0	0
$34$	0.477105	0.0818229
$35$	−5.64675	−0.954476
$36$	0	0
$37$	−7.85454	−1.29128	−0.645639	−	0.763643i	$-0.723409\pi$
$37$	−7.85454	−1.29128	−0.645639	+	0.763643i	$0.723409\pi$
$38$	2.51668	0.408259
$39$	0	0
$40$	9.28655	1.46833
$41$	−9.08711	−1.41917	−0.709584	−	0.704621i	$-0.751117\pi$
$41$	−9.08711	−1.41917	−0.709584	+	0.704621i	$0.751117\pi$
$42$	0	0
$43$	−0.672248	−0.102517	−0.0512584	−	0.998685i	$-0.516323\pi$
$43$	−0.672248	−0.102517	−0.0512584	+	0.998685i	$0.516323\pi$
$44$	−26.1970	−3.94934
$45$	0	0
$46$	19.5744	2.88608
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	5.75319	0.821885
$50$	6.29113	0.889700
$51$	0	0
$52$	−23.1708	−3.21321
$53$	10.0039	1.37414	0.687070	−	0.726591i	$-0.258897\pi$
$53$	10.0039	1.37414	0.687070	+	0.726591i	$0.258897\pi$
$54$	0	0
$55$	9.55841	1.28886
$56$	−20.9737	−2.80272
$57$	0	0
$58$	−15.8729	−2.08421
$59$	−4.72100	−0.614622	−0.307311	−	0.951609i	$-0.599429\pi$
$59$	−4.72100	−0.614622	−0.307311	+	0.951609i	$0.599429\pi$
$60$	0	0
$61$	−8.43394	−1.07986	−0.539928	−	0.841711i	$-0.681548\pi$
$61$	−8.43394	−1.07986	−0.539928	+	0.841711i	$0.681548\pi$
$62$	19.6749	2.49871
$63$	0	0
$64$	−3.06830	−0.383538
$65$	8.45426	1.04862
$66$	0	0
$67$	−8.88895	−1.08596	−0.542979	−	0.839746i	$-0.682704\pi$
$67$	−8.88895	−1.08596	−0.542979	+	0.839746i	$0.682704\pi$
$68$	−0.821564	−0.0996293
$69$	0	0
$70$	14.2111	1.69855
$71$	−8.51929	−1.01105	−0.505527	−	0.862811i	$-0.668702\pi$
$71$	−8.51929	−1.01105	−0.505527	+	0.862811i	$0.668702\pi$
$72$	0	0
$73$	−8.58241	−1.00449	−0.502247	−	0.864724i	$-0.667493\pi$
$73$	−8.58241	−1.00449	−0.502247	+	0.864724i	$0.667493\pi$
$74$	19.7673	2.29791
$75$	0	0
$76$	−4.33366	−0.497105
$77$	−21.5877	−2.46014
$78$	0	0
$79$	15.8868	1.78740	0.893701	−	0.448662i	$-0.148099\pi$
$79$	15.8868	1.78740	0.893701	+	0.448662i	$0.148099\pi$
$80$	−9.66639	−1.08074
$81$	0	0
$82$	22.8693	2.52549
$83$	−5.89406	−0.646957	−0.323478	−	0.946236i	$-0.604852\pi$
$83$	−5.89406	−0.646957	−0.323478	+	0.946236i	$0.604852\pi$
$84$	0	0
$85$	0.299762	0.0325137
$86$	1.69183	0.182435
$87$	0	0
$88$	35.5027	3.78460
$89$	−9.24280	−0.979735	−0.489868	−	0.871797i	$-0.662955\pi$
$89$	−9.24280	−0.979735	−0.489868	+	0.871797i	$0.662955\pi$
$90$	0	0
$91$	−19.0939	−2.00159
$92$	−33.7066	−3.51416
$93$	0	0
$94$	2.51668	0.259575
$95$	1.58121	0.162229
$96$	0	0
$97$	−9.88764	−1.00394	−0.501969	−	0.864886i	$-0.667391\pi$
$97$	−9.88764	−1.00394	−0.501969	+	0.864886i	$0.667391\pi$
$98$	−14.4789	−1.46259
$99$	0	0
$100$	−10.8332	−1.08332
$101$	17.8771	1.77884	0.889418	−	0.457094i	$-0.151110\pi$
$101$	17.8771	1.77884	0.889418	+	0.457094i	$0.151110\pi$
$102$	0	0
$103$	9.94725	0.980131	0.490066	−	0.871686i	$-0.336973\pi$
$103$	9.94725	0.980131	0.490066	+	0.871686i	$0.336973\pi$
$104$	31.4015	3.07918
$105$	0	0
$106$	−25.1766	−2.44536
$107$	8.92095	0.862421	0.431210	−	0.902251i	$-0.358087\pi$
$107$	8.92095	0.862421	0.431210	+	0.902251i	$0.358087\pi$
$108$	0	0
$109$	10.6875	1.02368	0.511840	−	0.859081i	$-0.328964\pi$
$109$	10.6875	1.02368	0.511840	+	0.859081i	$0.328964\pi$
$110$	−24.0554	−2.29360
$111$	0	0
$112$	21.8315	2.06289
$113$	−13.3658	−1.25735	−0.628674	−	0.777669i	$-0.716402\pi$
$113$	−13.3658	−1.25735	−0.628674	+	0.777669i	$0.716402\pi$
$114$	0	0
$115$	12.2984	1.14683
$116$	27.3327	2.53778
$117$	0	0
$118$	11.8812	1.09376
$119$	−0.677012	−0.0620616
$120$	0	0
$121$	25.5420	2.32200
$122$	21.2255	1.92167
$123$	0	0
$124$	−33.8797	−3.04248
$125$	11.8587	1.06068
$126$	0	0
$127$	12.0028	1.06507	0.532537	−	0.846407i	$-0.321239\pi$
$127$	12.0028	1.06507	0.532537	+	0.846407i	$0.321239\pi$
$128$	15.0000	1.32582
$129$	0	0
$130$	−21.2766	−1.86608
$131$	−6.52280	−0.569900	−0.284950	−	0.958542i	$-0.591977\pi$
$131$	−6.52280	−0.569900	−0.284950	+	0.958542i	$0.591977\pi$
$132$	0	0
$133$	−3.57116	−0.309659
$134$	22.3706	1.93253
$135$	0	0
$136$	1.11340	0.0954734
$137$	−1.20804	−0.103210	−0.0516048	−	0.998668i	$-0.516434\pi$
$137$	−1.20804	−0.103210	−0.0516048	+	0.998668i	$0.516434\pi$
$138$	0	0
$139$	1.87207	0.158787	0.0793934	−	0.996843i	$-0.474702\pi$
$139$	1.87207	0.158787	0.0793934	+	0.996843i	$0.474702\pi$
$140$	−24.4711	−2.06819
$141$	0	0
$142$	21.4403	1.79923
$143$	32.3208	2.70280
$144$	0	0
$145$	−9.97280	−0.828196
$146$	21.5991	1.78756
$147$	0	0
$148$	−34.0389	−2.79798
$149$	−5.47386	−0.448436	−0.224218	−	0.974539i	$-0.571983\pi$
$149$	−5.47386	−0.448436	−0.224218	+	0.974539i	$0.571983\pi$
$150$	0	0
$151$	−3.62750	−0.295202	−0.147601	−	0.989047i	$-0.547155\pi$
$151$	−3.62750	−0.295202	−0.147601	+	0.989047i	$0.547155\pi$
$152$	5.87307	0.476369
$153$	0	0
$154$	54.3292	4.37797
$155$	12.3616	0.992905
$156$	0	0
$157$	4.69917	0.375035	0.187517	−	0.982261i	$-0.439956\pi$
$157$	4.69917	0.375035	0.187517	+	0.982261i	$0.439956\pi$
$158$	−39.9819	−3.18079
$159$	0	0
$160$	5.75408	0.454900
$161$	−27.7760	−2.18905
$162$	0	0
$163$	−1.30149	−0.101941	−0.0509703	−	0.998700i	$-0.516231\pi$
$163$	−1.30149	−0.101941	−0.0509703	+	0.998700i	$0.516231\pi$
$164$	−39.3804	−3.07509
$165$	0	0
$166$	14.8334	1.15130
$167$	14.3498	1.11042	0.555212	−	0.831709i	$-0.312637\pi$
$167$	14.3498	1.11042	0.555212	+	0.831709i	$0.312637\pi$
$168$	0	0
$169$	15.5872	1.19902
$170$	−0.754403	−0.0578601
$171$	0	0
$172$	−2.91329	−0.222136
$173$	20.8730	1.58694	0.793471	−	0.608608i	$-0.208272\pi$
$173$	20.8730	1.58694	0.793471	+	0.608608i	$0.208272\pi$
$174$	0	0
$175$	−8.92711	−0.674826
$176$	−36.9548	−2.78558
$177$	0	0
$178$	23.2611	1.74350
$179$	−16.6852	−1.24711	−0.623554	−	0.781780i	$-0.714312\pi$
$179$	−16.6852	−1.24711	−0.623554	+	0.781780i	$0.714312\pi$
$180$	0	0
$181$	−6.10033	−0.453434	−0.226717	−	0.973961i	$-0.572799\pi$
$181$	−6.10033	−0.453434	−0.226717	+	0.973961i	$0.572799\pi$
$182$	48.0533	3.56195
$183$	0	0
$184$	45.6799	3.36757
$185$	12.4197	0.913113
$186$	0	0
$187$	1.14600	0.0838035
$188$	−4.33366	−0.316065
$189$	0	0
$190$	−3.97939	−0.288695
$191$	21.8585	1.58163	0.790813	−	0.612058i	$-0.209658\pi$
$191$	21.8585	1.58163	0.790813	+	0.612058i	$0.209658\pi$
$192$	0	0
$193$	3.28113	0.236181	0.118090	−	0.993003i	$-0.462323\pi$
$193$	3.28113	0.236181	0.118090	+	0.993003i	$0.462323\pi$
$194$	24.8840	1.78657
$195$	0	0
$196$	24.9324	1.78088
$197$	−5.43898	−0.387511	−0.193755	−	0.981050i	$-0.562067\pi$
$197$	−5.43898	−0.387511	−0.193755	+	0.981050i	$0.562067\pi$
$198$	0	0
$199$	0.0368248	0.00261044	0.00130522	−	0.999999i	$-0.499585\pi$
$199$	0.0368248	0.00261044	0.00130522	+	0.999999i	$0.499585\pi$
$200$	14.6814	1.03813
$201$	0	0
$202$	−44.9908	−3.16554
$203$	22.5236	1.58084
$204$	0	0
$205$	14.3686	1.00355
$206$	−25.0340	−1.74420
$207$	0	0
$208$	−32.6859	−2.26636
$209$	6.04500	0.418141
$210$	0	0
$211$	8.76854	0.603652	0.301826	−	0.953363i	$-0.402404\pi$
$211$	8.76854	0.603652	0.301826	+	0.953363i	$0.402404\pi$
$212$	43.3535	2.97753
$213$	0	0
$214$	−22.4511	−1.53473
$215$	1.06296	0.0724936
$216$	0	0
$217$	−27.9186	−1.89524
$218$	−26.8971	−1.82170
$219$	0	0
$220$	41.4229	2.79273
$221$	1.01361	0.0681831
$222$	0	0
$223$	10.6503	0.713195	0.356598	−	0.934258i	$-0.383937\pi$
$223$	10.6503	0.713195	0.356598	+	0.934258i	$0.383937\pi$
$224$	−12.9956	−0.868304
$225$	0	0
$226$	33.6374	2.23752
$227$	0.633266	0.0420313	0.0210157	−	0.999779i	$-0.493310\pi$
$227$	0.633266	0.0420313	0.0210157	+	0.999779i	$0.493310\pi$
$228$	0	0
$229$	−13.5748	−0.897047	−0.448524	−	0.893771i	$-0.648050\pi$
$229$	−13.5748	−0.897047	−0.448524	+	0.893771i	$0.648050\pi$
$230$	−30.9512	−2.04086
$231$	0	0
$232$	−37.0419	−2.43192
$233$	−0.959758	−0.0628759	−0.0314379	−	0.999506i	$-0.510009\pi$
$233$	−0.959758	−0.0628759	−0.0314379	+	0.999506i	$0.510009\pi$
$234$	0	0
$235$	1.58121	0.103147
$236$	−20.4592	−1.33178
$237$	0	0
$238$	1.70382	0.110442
$239$	−2.41389	−0.156141	−0.0780707	−	0.996948i	$-0.524876\pi$
$239$	−2.41389	−0.156141	−0.0780707	+	0.996948i	$0.524876\pi$
$240$	0	0
$241$	7.14401	0.460186	0.230093	−	0.973169i	$-0.426097\pi$
$241$	7.14401	0.460186	0.230093	+	0.973169i	$0.426097\pi$
$242$	−64.2810	−4.13214
$243$	0	0
$244$	−36.5498	−2.33986
$245$	−9.09700	−0.581186
$246$	0	0
$247$	5.34670	0.340203
$248$	45.9144	2.91557
$249$	0	0
$250$	−29.8446	−1.88754
$251$	−0.179729	−0.0113444	−0.00567219	−	0.999984i	$-0.501806\pi$
$251$	−0.179729	−0.0113444	−0.00567219	+	0.999984i	$0.501806\pi$
$252$	0	0
$253$	47.0172	2.95594
$254$	−30.2071	−1.89536
$255$	0	0
$256$	−31.6135	−1.97585
$257$	−16.8536	−1.05130	−0.525649	−	0.850701i	$-0.676178\pi$
$257$	−16.8536	−1.05130	−0.525649	+	0.850701i	$0.676178\pi$
$258$	0	0
$259$	−28.0498	−1.74293
$260$	36.6379	2.27218
$261$	0	0
$262$	16.4158	1.01417
$263$	−26.7657	−1.65044	−0.825221	−	0.564810i	$-0.808950\pi$
$263$	−26.7657	−1.65044	−0.825221	+	0.564810i	$0.808950\pi$
$264$	0	0
$265$	−15.8182	−0.971707
$266$	8.98746	0.551056
$267$	0	0
$268$	−38.5217	−2.35309
$269$	14.1732	0.864157	0.432078	−	0.901836i	$-0.357780\pi$
$269$	14.1732	0.864157	0.432078	+	0.901836i	$0.357780\pi$
$270$	0	0
$271$	−21.5426	−1.30862	−0.654309	−	0.756227i	$-0.727040\pi$
$271$	−21.5426	−1.30862	−0.654309	+	0.756227i	$0.727040\pi$
$272$	−1.15894	−0.0702712
$273$	0	0
$274$	3.04024	0.183668
$275$	15.1112	0.911237
$276$	0	0
$277$	1.51892	0.0912631	0.0456316	−	0.998958i	$-0.485470\pi$
$277$	1.51892	0.0912631	0.0456316	+	0.998958i	$0.485470\pi$
$278$	−4.71139	−0.282570
$279$	0	0
$280$	33.1638	1.98191
$281$	−14.7132	−0.877716	−0.438858	−	0.898556i	$-0.644617\pi$
$281$	−14.7132	−0.877716	−0.438858	+	0.898556i	$0.644617\pi$
$282$	0	0
$283$	23.0707	1.37141	0.685704	−	0.727880i	$-0.259494\pi$
$283$	23.0707	1.37141	0.685704	+	0.727880i	$0.259494\pi$
$284$	−36.9197	−2.19078
$285$	0	0
$286$	−81.3411	−4.80980
$287$	−32.4515	−1.91555
$288$	0	0
$289$	−16.9641	−0.997886
$290$	25.0983	1.47382
$291$	0	0
$292$	−37.1932	−2.17657
$293$	26.7116	1.56051	0.780255	−	0.625461i	$-0.215089\pi$
$293$	26.7116	1.56051	0.780255	+	0.625461i	$0.215089\pi$
$294$	0	0
$295$	7.46489	0.434623
$296$	46.1302	2.68127
$297$	0	0
$298$	13.7759	0.798018
$299$	41.5859	2.40497
$300$	0	0
$301$	−2.40070	−0.138374
$302$	9.12924	0.525329
$303$	0	0
$304$	−6.11329	−0.350621
$305$	13.3358	0.763607
$306$	0	0
$307$	−3.48212	−0.198735	−0.0993675	−	0.995051i	$-0.531682\pi$
$307$	−3.48212	−0.198735	−0.0993675	+	0.995051i	$0.531682\pi$
$308$	−93.5536	−5.33071
$309$	0	0
$310$	−31.1101	−1.76693
$311$	16.8867	0.957559	0.478780	−	0.877935i	$-0.341079\pi$
$311$	16.8867	0.957559	0.478780	+	0.877935i	$0.341079\pi$
$312$	0	0
$313$	−20.2948	−1.14713	−0.573565	−	0.819160i	$-0.694440\pi$
$313$	−20.2948	−1.14713	−0.573565	+	0.819160i	$0.694440\pi$
$314$	−11.8263	−0.667396
$315$	0	0
$316$	68.8479	3.87300
$317$	−20.0616	−1.12677	−0.563386	−	0.826194i	$-0.690502\pi$
$317$	−20.0616	−1.12677	−0.563386	+	0.826194i	$0.690502\pi$
$318$	0	0
$319$	−38.1263	−2.13466
$320$	4.85163	0.271214
$321$	0	0
$322$	69.9032	3.89555
$323$	0.189577	0.0105484
$324$	0	0
$325$	13.3656	0.741388
$326$	3.27543	0.181409
$327$	0	0
$328$	53.3692	2.94682
$329$	−3.57116	−0.196884
$330$	0	0
$331$	2.77554	0.152558	0.0762788	−	0.997087i	$-0.475696\pi$
$331$	2.77554	0.152558	0.0762788	+	0.997087i	$0.475696\pi$
$332$	−25.5428	−1.40185
$333$	0	0
$334$	−36.1139	−1.97606
$335$	14.0553	0.767923
$336$	0	0
$337$	−32.9675	−1.79586	−0.897928	−	0.440142i	$-0.854928\pi$
$337$	−32.9675	−1.79586	−0.897928	+	0.440142i	$0.854928\pi$
$338$	−39.2280	−2.13372
$339$	0	0
$340$	1.29907	0.0704517
$341$	47.2586	2.55919
$342$	0	0
$343$	−4.45255	−0.240415
$344$	3.94815	0.212870
$345$	0	0
$346$	−52.5305	−2.82406
$347$	20.6381	1.10791	0.553955	−	0.832547i	$-0.313118\pi$
$347$	20.6381	1.10791	0.553955	+	0.832547i	$0.313118\pi$
$348$	0	0
$349$	−9.11625	−0.487982	−0.243991	−	0.969778i	$-0.578457\pi$
$349$	−9.11625	−0.487982	−0.243991	+	0.969778i	$0.578457\pi$
$350$	22.4666	1.20089
$351$	0	0
$352$	21.9980	1.17250
$353$	5.85428	0.311592	0.155796	−	0.987789i	$-0.450206\pi$
$353$	5.85428	0.311592	0.155796	+	0.987789i	$0.450206\pi$
$354$	0	0
$355$	13.4708	0.714955
$356$	−40.0552	−2.12292
$357$	0	0
$358$	41.9912	2.21930
$359$	−5.10784	−0.269581	−0.134791	−	0.990874i	$-0.543036\pi$
$359$	−5.10784	−0.269581	−0.134791	+	0.990874i	$0.543036\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	15.3526	0.806912
$363$	0	0
$364$	−82.7466	−4.33710
$365$	13.5706	0.710317
$366$	0	0
$367$	1.79698	0.0938017	0.0469008	−	0.998900i	$-0.485066\pi$
$367$	1.79698	0.0938017	0.0469008	+	0.998900i	$0.485066\pi$
$368$	−47.5483	−2.47863
$369$	0	0
$370$	−31.2563	−1.62494
$371$	35.7255	1.85478
$372$	0	0
$373$	8.19827	0.424490	0.212245	−	0.977216i	$-0.431922\pi$
$373$	8.19827	0.424490	0.212245	+	0.977216i	$0.431922\pi$
$374$	−2.88410	−0.149133
$375$	0	0
$376$	5.87307	0.302880
$377$	−33.7220	−1.73677
$378$	0	0
$379$	−11.2813	−0.579481	−0.289740	−	0.957105i	$-0.593569\pi$
$379$	−11.2813	−0.579481	−0.289740	+	0.957105i	$0.593569\pi$
$380$	6.85242	0.351522
$381$	0	0
$382$	−55.0108	−2.81460
$383$	7.78756	0.397926	0.198963	−	0.980007i	$-0.436243\pi$
$383$	7.78756	0.397926	0.198963	+	0.980007i	$0.436243\pi$
$384$	0	0
$385$	34.1346	1.73966
$386$	−8.25754	−0.420298
$387$	0	0
$388$	−42.8497	−2.17536
$389$	−22.4811	−1.13984	−0.569919	−	0.821701i	$-0.693026\pi$
$389$	−22.4811	−1.13984	−0.569919	+	0.821701i	$0.693026\pi$
$390$	0	0
$391$	1.47451	0.0745690
$392$	−33.7889	−1.70660
$393$	0	0
$394$	13.6881	0.689599
$395$	−25.1203	−1.26394
$396$	0	0
$397$	−3.42770	−0.172031	−0.0860157	−	0.996294i	$-0.527414\pi$
$397$	−3.42770	−0.172031	−0.0860157	+	0.996294i	$0.527414\pi$
$398$	−0.0926760	−0.00464543
$399$	0	0
$400$	−15.2819	−0.764093
$401$	36.2353	1.80950	0.904752	−	0.425938i	$-0.140056\pi$
$401$	36.2353	1.80950	0.904752	+	0.425938i	$0.140056\pi$
$402$	0	0
$403$	41.7994	2.08218
$404$	77.4732	3.85444
$405$	0	0
$406$	−56.6845	−2.81321
$407$	47.4807	2.35353
$408$	0	0
$409$	27.7954	1.37439	0.687197	−	0.726471i	$-0.258841\pi$
$409$	27.7954	1.37439	0.687197	+	0.726471i	$0.258841\pi$
$410$	−36.1612	−1.78587
$411$	0	0
$412$	43.1080	2.12378
$413$	−16.8595	−0.829600
$414$	0	0
$415$	9.31974	0.457488
$416$	19.4568	0.953951
$417$	0	0
$418$	−15.2133	−0.744107
$419$	−38.1273	−1.86264	−0.931319	−	0.364204i	$-0.881341\pi$
$419$	−38.1273	−1.86264	−0.931319	+	0.364204i	$0.881341\pi$
$420$	0	0
$421$	28.4836	1.38821	0.694104	−	0.719875i	$-0.255801\pi$
$421$	28.4836	1.38821	0.694104	+	0.719875i	$0.255801\pi$
$422$	−22.0676	−1.07423
$423$	0	0
$424$	−58.7535	−2.85332
$425$	0.473901	0.0229876
$426$	0	0
$427$	−30.1190	−1.45756
$428$	38.6604	1.86872
$429$	0	0
$430$	−2.67514	−0.129007
$431$	22.9282	1.10441	0.552206	−	0.833708i	$-0.313786\pi$
$431$	22.9282	1.10441	0.552206	+	0.833708i	$0.313786\pi$
$432$	0	0
$433$	18.2657	0.877792	0.438896	−	0.898538i	$-0.355370\pi$
$433$	18.2657	0.877792	0.438896	+	0.898538i	$0.355370\pi$
$434$	70.2621	3.37269
$435$	0	0
$436$	46.3161	2.21814
$437$	7.77786	0.372065
$438$	0	0
$439$	−0.765497	−0.0365352	−0.0182676	−	0.999833i	$-0.505815\pi$
$439$	−0.765497	−0.0365352	−0.0182676	+	0.999833i	$0.505815\pi$
$440$	−56.1372	−2.67624
$441$	0	0
$442$	−2.55094	−0.121336
$443$	−27.0261	−1.28405	−0.642024	−	0.766685i	$-0.721905\pi$
$443$	−27.0261	−1.28405	−0.642024	+	0.766685i	$0.721905\pi$
$444$	0	0
$445$	14.6148	0.692808
$446$	−26.8033	−1.26917
$447$	0	0
$448$	−10.9574	−0.517688
$449$	−26.2076	−1.23682	−0.618408	−	0.785857i	$-0.712222\pi$
$449$	−26.2076	−1.23682	−0.618408	+	0.785857i	$0.712222\pi$
$450$	0	0
$451$	54.9316	2.58663
$452$	−57.9228	−2.72446
$453$	0	0
$454$	−1.59372	−0.0747972
$455$	30.1915	1.41540
$456$	0	0
$457$	−8.57959	−0.401336	−0.200668	−	0.979659i	$-0.564311\pi$
$457$	−8.57959	−0.401336	−0.200668	+	0.979659i	$0.564311\pi$
$458$	34.1633	1.59635
$459$	0	0
$460$	53.2972	2.48499
$461$	15.0798	0.702335	0.351168	−	0.936313i	$-0.385785\pi$
$461$	15.0798	0.702335	0.351168	+	0.936313i	$0.385785\pi$
$462$	0	0
$463$	−36.2400	−1.68422	−0.842108	−	0.539309i	$-0.818686\pi$
$463$	−36.2400	−1.68422	−0.842108	+	0.539309i	$0.818686\pi$
$464$	38.5570	1.78996
$465$	0	0
$466$	2.41540	0.111891
$467$	−0.504329	−0.0233376	−0.0116688	−	0.999932i	$-0.503714\pi$
$467$	−0.504329	−0.0233376	−0.0116688	+	0.999932i	$0.503714\pi$
$468$	0	0
$469$	−31.7439	−1.46580
$470$	−3.97939	−0.183556
$471$	0	0
$472$	27.7268	1.27623
$473$	4.06374	0.186851
$474$	0	0
$475$	2.49978	0.114698
$476$	−2.93394	−0.134477
$477$	0	0
$478$	6.07497	0.277863
$479$	−15.0132	−0.685970	−0.342985	−	0.939341i	$-0.611438\pi$
$479$	−15.0132	−0.685970	−0.342985	+	0.939341i	$0.611438\pi$
$480$	0	0
$481$	41.9959	1.91485
$482$	−17.9792	−0.818928
$483$	0	0
$484$	110.690	5.03139
$485$	15.6344	0.709923
$486$	0	0
$487$	−40.0252	−1.81372	−0.906858	−	0.421436i	$-0.861526\pi$
$487$	−40.0252	−1.81372	−0.906858	+	0.421436i	$0.861526\pi$
$488$	49.5331	2.24226
$489$	0	0
$490$	22.8942	1.03426
$491$	18.5429	0.836829	0.418415	−	0.908256i	$-0.362586\pi$
$491$	18.5429	0.836829	0.418415	+	0.908256i	$0.362586\pi$
$492$	0	0
$493$	−1.19568	−0.0538507
$494$	−13.4559	−0.605411
$495$	0	0
$496$	−47.7924	−2.14594
$497$	−30.4238	−1.36469
$498$	0	0
$499$	−14.4447	−0.646632	−0.323316	−	0.946291i	$-0.604798\pi$
$499$	−14.4447	−0.646632	−0.323316	+	0.946291i	$0.604798\pi$
$500$	51.3917	2.29830
$501$	0	0
$502$	0.452319	0.0201880
$503$	−19.1066	−0.851921	−0.425961	−	0.904742i	$-0.640064\pi$
$503$	−19.1066	−0.851921	−0.425961	+	0.904742i	$0.640064\pi$
$504$	0	0
$505$	−28.2674	−1.25788
$506$	−118.327	−5.26028
$507$	0	0
$508$	52.0160	2.30784
$509$	9.52681	0.422268	0.211134	−	0.977457i	$-0.432284\pi$
$509$	9.52681	0.422268	0.211134	+	0.977457i	$0.432284\pi$
$510$	0	0
$511$	−30.6492	−1.35584
$512$	49.5610	2.19031
$513$	0	0
$514$	42.4151	1.87085
$515$	−15.7287	−0.693089
$516$	0	0
$517$	6.04500	0.265859
$518$	70.5923	3.10165
$519$	0	0
$520$	−49.6524	−2.17740
$521$	−26.8304	−1.17546	−0.587730	−	0.809057i	$-0.699978\pi$
$521$	−26.8304	−1.17546	−0.587730	+	0.809057i	$0.699978\pi$
$522$	0	0
$523$	25.2154	1.10259	0.551296	−	0.834310i	$-0.314134\pi$
$523$	25.2154	1.10259	0.551296	+	0.834310i	$0.314134\pi$
$524$	−28.2676	−1.23488
$525$	0	0
$526$	67.3605	2.93706
$527$	1.48208	0.0645603
$528$	0	0
$529$	37.4951	1.63022
$530$	39.8094	1.72921
$531$	0	0
$532$	−15.4762	−0.670978
$533$	48.5861	2.10450
$534$	0	0
$535$	−14.1059	−0.609851
$536$	52.2054	2.25493
$537$	0	0
$538$	−35.6694	−1.53782
$539$	−34.7781	−1.49800
$540$	0	0
$541$	−13.2319	−0.568882	−0.284441	−	0.958693i	$-0.591808\pi$
$541$	−13.2319	−0.568882	−0.284441	+	0.958693i	$0.591808\pi$
$542$	54.2157	2.32876
$543$	0	0
$544$	0.689879	0.0295783
$545$	−16.8992	−0.723884
$546$	0	0
$547$	8.12482	0.347392	0.173696	−	0.984799i	$-0.444429\pi$
$547$	8.12482	0.347392	0.173696	+	0.984799i	$0.444429\pi$
$548$	−5.23522	−0.223638
$549$	0	0
$550$	−38.0299	−1.62160
$551$	−6.30707	−0.268690
$552$	0	0
$553$	56.7343	2.41259
$554$	−3.82263	−0.162408
$555$	0	0
$556$	8.11291	0.344064
$557$	−30.1615	−1.27798	−0.638992	−	0.769213i	$-0.720648\pi$
$557$	−30.1615	−1.27798	−0.638992	+	0.769213i	$0.720648\pi$
$558$	0	0
$559$	3.59431	0.152023
$560$	−34.5202	−1.45875
$561$	0	0
$562$	37.0284	1.56195
$563$	18.8361	0.793846	0.396923	−	0.917852i	$-0.370078\pi$
$563$	18.8361	0.793846	0.396923	+	0.917852i	$0.370078\pi$
$564$	0	0
$565$	21.1341	0.889119
$566$	−58.0614	−2.44050
$567$	0	0
$568$	50.0344	2.09939
$569$	35.1244	1.47249	0.736246	−	0.676714i	$-0.236596\pi$
$569$	35.1244	1.47249	0.736246	+	0.676714i	$0.236596\pi$
$570$	0	0
$571$	−44.7481	−1.87265	−0.936324	−	0.351137i	$-0.885795\pi$
$571$	−44.7481	−1.87265	−0.936324	+	0.351137i	$0.885795\pi$
$572$	140.067	5.85652
$573$	0	0
$574$	81.6700	3.40884
$575$	19.4429	0.810825
$576$	0	0
$577$	−19.2444	−0.801154	−0.400577	−	0.916263i	$-0.631190\pi$
$577$	−19.2444	−0.801154	−0.400577	+	0.916263i	$0.631190\pi$
$578$	42.6931	1.77580
$579$	0	0
$580$	−43.2187	−1.79456
$581$	−21.0486	−0.873244
$582$	0	0
$583$	−60.4735	−2.50456
$584$	50.4051	2.08578
$585$	0	0
$586$	−67.2245	−2.77702
$587$	31.1728	1.28664	0.643319	−	0.765598i	$-0.277557\pi$
$587$	31.1728	1.28664	0.643319	+	0.765598i	$0.277557\pi$
$588$	0	0
$589$	7.81779	0.322127
$590$	−18.7867	−0.773437
$591$	0	0
$592$	−48.0171	−1.97349
$593$	−38.3054	−1.57301	−0.786506	−	0.617582i	$-0.788112\pi$
$593$	−38.3054	−1.57301	−0.786506	+	0.617582i	$0.788112\pi$
$594$	0	0
$595$	1.07050	0.0438861
$596$	−23.7218	−0.971685
$597$	0	0
$598$	−104.658	−4.27979
$599$	−16.1929	−0.661625	−0.330813	−	0.943696i	$-0.607323\pi$
$599$	−16.1929	−0.661625	−0.330813	+	0.943696i	$0.607323\pi$
$600$	0	0
$601$	−35.9474	−1.46632	−0.733162	−	0.680055i	$-0.761956\pi$
$601$	−35.9474	−1.46632	−0.733162	+	0.680055i	$0.761956\pi$
$602$	6.04180	0.246245
$603$	0	0
$604$	−15.7203	−0.639652
$605$	−40.3873	−1.64198
$606$	0	0
$607$	−0.398198	−0.0161623	−0.00808117	−	0.999967i	$-0.502572\pi$
$607$	−0.398198	−0.0161623	−0.00808117	+	0.999967i	$0.502572\pi$
$608$	3.63904	0.147582
$609$	0	0
$610$	−33.5620	−1.35888
$611$	5.34670	0.216304
$612$	0	0
$613$	−0.454536	−0.0183585	−0.00917927	−	0.999958i	$-0.502922\pi$
$613$	−0.454536	−0.0183585	−0.00917927	+	0.999958i	$0.502922\pi$
$614$	8.76337	0.353661
$615$	0	0
$616$	126.786	5.10835
$617$	1.32535	0.0533568	0.0266784	−	0.999644i	$-0.491507\pi$
$617$	1.32535	0.0533568	0.0266784	+	0.999644i	$0.491507\pi$
$618$	0	0
$619$	43.7272	1.75755	0.878773	−	0.477240i	$-0.158363\pi$
$619$	43.7272	1.75755	0.878773	+	0.477240i	$0.158363\pi$
$620$	53.5708	2.15146
$621$	0	0
$622$	−42.4985	−1.70403
$623$	−33.0075	−1.32242
$624$	0	0
$625$	−6.25223	−0.250089
$626$	51.0755	2.04139
$627$	0	0
$628$	20.3646	0.812636
$629$	1.48904	0.0593721
$630$	0	0
$631$	40.8407	1.62584	0.812921	−	0.582374i	$-0.197876\pi$
$631$	40.8407	1.62584	0.812921	+	0.582374i	$0.197876\pi$
$632$	−93.3041	−3.71144
$633$	0	0
$634$	50.4886	2.00516
$635$	−18.9789	−0.753155
$636$	0	0
$637$	−30.7606	−1.21878
$638$	95.9514	3.79875
$639$	0	0
$640$	−23.7181	−0.937542
$641$	−33.4034	−1.31936	−0.659678	−	0.751548i	$-0.729307\pi$
$641$	−33.4034	−1.31936	−0.659678	+	0.751548i	$0.729307\pi$
$642$	0	0
$643$	42.0781	1.65940	0.829699	−	0.558211i	$-0.188512\pi$
$643$	42.0781	1.65940	0.829699	+	0.558211i	$0.188512\pi$
$644$	−120.372	−4.74331
$645$	0	0
$646$	−0.477105	−0.0187714
$647$	−8.85114	−0.347974	−0.173987	−	0.984748i	$-0.555665\pi$
$647$	−8.85114	−0.347974	−0.173987	+	0.984748i	$0.555665\pi$
$648$	0	0
$649$	28.5385	1.12023
$650$	−33.6368	−1.31934
$651$	0	0
$652$	−5.64022	−0.220888
$653$	−12.7699	−0.499726	−0.249863	−	0.968281i	$-0.580386\pi$
$653$	−12.7699	−0.499726	−0.249863	+	0.968281i	$0.580386\pi$
$654$	0	0
$655$	10.3139	0.402998
$656$	−55.5521	−2.16895
$657$	0	0
$658$	8.98746	0.350368
$659$	−42.7180	−1.66406	−0.832029	−	0.554733i	$-0.812821\pi$
$659$	−42.7180	−1.66406	−0.832029	+	0.554733i	$0.812821\pi$
$660$	0	0
$661$	3.34588	0.130140	0.0650699	−	0.997881i	$-0.479273\pi$
$661$	3.34588	0.130140	0.0650699	+	0.997881i	$0.479273\pi$
$662$	−6.98514	−0.271485
$663$	0	0
$664$	34.6162	1.34337
$665$	5.64675	0.218972
$666$	0	0
$667$	−49.0555	−1.89944
$668$	62.1873	2.40610
$669$	0	0
$670$	−35.3726	−1.36656
$671$	50.9832	1.96818
$672$	0	0
$673$	16.1254	0.621588	0.310794	−	0.950477i	$-0.399405\pi$
$673$	16.1254	0.621588	0.310794	+	0.950477i	$0.399405\pi$
$674$	82.9686	3.19583
$675$	0	0
$676$	67.5498	2.59807
$677$	18.4747	0.710042	0.355021	−	0.934858i	$-0.384474\pi$
$677$	18.4747	0.710042	0.355021	+	0.934858i	$0.384474\pi$
$678$	0	0
$679$	−35.3103	−1.35509
$680$	−1.76052	−0.0675129
$681$	0	0
$682$	−118.935	−4.55424
$683$	−44.3707	−1.69780	−0.848899	−	0.528555i	$-0.822734\pi$
$683$	−44.3707	−1.69780	−0.848899	+	0.528555i	$0.822734\pi$
$684$	0	0
$685$	1.91016	0.0729835
$686$	11.2056	0.427833
$687$	0	0
$688$	−4.10964	−0.156679
$689$	−53.4878	−2.03772
$690$	0	0
$691$	−15.3338	−0.583327	−0.291663	−	0.956521i	$-0.594209\pi$
$691$	−15.3338	−0.583327	−0.291663	+	0.956521i	$0.594209\pi$
$692$	90.4563	3.43863
$693$	0	0
$694$	−51.9394	−1.97159
$695$	−2.96013	−0.112284
$696$	0	0
$697$	1.72271	0.0652523
$698$	22.9427	0.868392
$699$	0	0
$700$	−38.6870	−1.46223
$701$	−20.7781	−0.784779	−0.392390	−	0.919799i	$-0.628352\pi$
$701$	−20.7781	−0.784779	−0.392390	+	0.919799i	$0.628352\pi$
$702$	0	0
$703$	7.85454	0.296240
$704$	18.5479	0.699050
$705$	0	0
$706$	−14.7333	−0.554496
$707$	63.8420	2.40102
$708$	0	0
$709$	−21.4283	−0.804756	−0.402378	−	0.915474i	$-0.631816\pi$
$709$	−21.4283	−0.804756	−0.402378	+	0.915474i	$0.631816\pi$
$710$	−33.9016	−1.27230
$711$	0	0
$712$	54.2836	2.03436
$713$	60.8057	2.27719
$714$	0	0
$715$	−51.1060	−1.91126
$716$	−72.3078	−2.70227
$717$	0	0
$718$	12.8548	0.479736
$719$	−0.697247	−0.0260029	−0.0130015	−	0.999915i	$-0.504139\pi$
$719$	−0.697247	−0.0260029	−0.0130015	+	0.999915i	$0.504139\pi$
$720$	0	0
$721$	35.5232	1.32295
$722$	−2.51668	−0.0936610
$723$	0	0
$724$	−26.4368	−0.982514
$725$	−15.7663	−0.585545
$726$	0	0
$727$	−12.7707	−0.473641	−0.236820	−	0.971553i	$-0.576105\pi$
$727$	−12.7707	−0.473641	−0.236820	+	0.971553i	$0.576105\pi$
$728$	112.140	4.15618
$729$	0	0
$730$	−34.1528	−1.26405
$731$	0.127443	0.00471365
$732$	0	0
$733$	36.6509	1.35373	0.676865	−	0.736107i	$-0.263338\pi$
$733$	36.6509	1.35373	0.676865	+	0.736107i	$0.263338\pi$
$734$	−4.52242	−0.166926
$735$	0	0
$736$	28.3039	1.04330
$737$	53.7337	1.97931
$738$	0	0
$739$	9.74169	0.358354	0.179177	−	0.983817i	$-0.442657\pi$
$739$	9.74169	0.358354	0.179177	+	0.983817i	$0.442657\pi$
$740$	53.8226	1.97856
$741$	0	0
$742$	−89.9095	−3.30068
$743$	−10.1447	−0.372171	−0.186086	−	0.982534i	$-0.559580\pi$
$743$	−10.1447	−0.372171	−0.186086	+	0.982534i	$0.559580\pi$
$744$	0	0
$745$	8.65532	0.317106
$746$	−20.6324	−0.755406
$747$	0	0
$748$	4.96636	0.181588
$749$	31.8581	1.16407
$750$	0	0
$751$	−30.8209	−1.12467	−0.562336	−	0.826909i	$-0.690097\pi$
$751$	−30.8209	−1.12467	−0.562336	+	0.826909i	$0.690097\pi$
$752$	−6.11329	−0.222929
$753$	0	0
$754$	84.8675	3.09069
$755$	5.73583	0.208748
$756$	0	0
$757$	8.31715	0.302292	0.151146	−	0.988511i	$-0.451704\pi$
$757$	8.31715	0.302292	0.151146	+	0.988511i	$0.451704\pi$
$758$	28.3913	1.03122
$759$	0	0
$760$	−9.28655	−0.336859
$761$	−53.5981	−1.94293	−0.971465	−	0.237181i	$-0.923777\pi$
$761$	−53.5981	−1.94293	−0.971465	+	0.237181i	$0.923777\pi$
$762$	0	0
$763$	38.1669	1.38174
$764$	94.7273	3.42711
$765$	0	0
$766$	−19.5988	−0.708132
$767$	25.2418	0.911428
$768$	0	0
$769$	22.6070	0.815230	0.407615	−	0.913154i	$-0.366361\pi$
$769$	22.6070	0.815230	0.407615	+	0.913154i	$0.366361\pi$
$770$	−85.9058	−3.09583
$771$	0	0
$772$	14.2193	0.511764
$773$	25.6596	0.922912	0.461456	−	0.887163i	$-0.347327\pi$
$773$	25.6596	0.922912	0.461456	+	0.887163i	$0.347327\pi$
$774$	0	0
$775$	19.5427	0.701996
$776$	58.0708	2.08462
$777$	0	0
$778$	56.5777	2.02841
$779$	9.08711	0.325579
$780$	0	0
$781$	51.4991	1.84278
$782$	−3.71086	−0.132700
$783$	0	0
$784$	35.1709	1.25610
$785$	−7.43037	−0.265201
$786$	0	0
$787$	30.8612	1.10008	0.550042	−	0.835137i	$-0.314612\pi$
$787$	30.8612	1.10008	0.550042	+	0.835137i	$0.314612\pi$
$788$	−23.5707	−0.839670
$789$	0	0
$790$	63.2197	2.24926
$791$	−47.7314	−1.69713
$792$	0	0
$793$	45.0938	1.60133
$794$	8.62641	0.306140
$795$	0	0
$796$	0.159586	0.00565638
$797$	−13.1692	−0.466476	−0.233238	−	0.972420i	$-0.574932\pi$
$797$	−13.1692	−0.466476	−0.233238	+	0.972420i	$0.574932\pi$
$798$	0	0
$799$	0.189577	0.00670677
$800$	9.09678	0.321620
$801$	0	0
$802$	−91.1925	−3.22012
$803$	51.8807	1.83083
$804$	0	0
$805$	43.9197	1.54796
$806$	−105.196	−3.70536
$807$	0	0
$808$	−104.993	−3.69365
$809$	−14.8020	−0.520412	−0.260206	−	0.965553i	$-0.583790\pi$
$809$	−14.8020	−0.520412	−0.260206	+	0.965553i	$0.583790\pi$
$810$	0	0
$811$	3.99717	0.140360	0.0701799	−	0.997534i	$-0.477643\pi$
$811$	3.99717	0.140360	0.0701799	+	0.997534i	$0.477643\pi$
$812$	97.6095	3.42542
$813$	0	0
$814$	−119.494	−4.18825
$815$	2.05793	0.0720861
$816$	0	0
$817$	0.672248	0.0235190
$818$	−69.9520	−2.44581
$819$	0	0
$820$	62.2687	2.17452
$821$	16.6773	0.582041	0.291021	−	0.956717i	$-0.406005\pi$
$821$	16.6773	0.582041	0.291021	+	0.956717i	$0.406005\pi$
$822$	0	0
$823$	3.00532	0.104759	0.0523795	−	0.998627i	$-0.483319\pi$
$823$	3.00532	0.104759	0.0523795	+	0.998627i	$0.483319\pi$
$824$	−58.4208	−2.03519
$825$	0	0
$826$	42.4298	1.47632
$827$	13.4198	0.466653	0.233327	−	0.972398i	$-0.425039\pi$
$827$	13.4198	0.466653	0.233327	+	0.972398i	$0.425039\pi$
$828$	0	0
$829$	−9.00386	−0.312717	−0.156358	−	0.987700i	$-0.549975\pi$
$829$	−9.00386	−0.312717	−0.156358	+	0.987700i	$0.549975\pi$
$830$	−23.4548	−0.814127
$831$	0	0
$832$	16.4053	0.568751
$833$	−1.09068	−0.0377897
$834$	0	0
$835$	−22.6901	−0.785223
$836$	26.1970	0.906041
$837$	0	0
$838$	95.9540	3.31467
$839$	−19.6121	−0.677085	−0.338543	−	0.940951i	$-0.609934\pi$
$839$	−19.6121	−0.677085	−0.338543	+	0.940951i	$0.609934\pi$
$840$	0	0
$841$	10.7792	0.371695
$842$	−71.6841	−2.47040
$843$	0	0
$844$	37.9999	1.30801
$845$	−24.6467	−0.847872
$846$	0	0
$847$	91.2147	3.13418
$848$	61.1567	2.10013
$849$	0	0
$850$	−1.19266	−0.0409078
$851$	61.0915	2.09419
$852$	0	0
$853$	28.6088	0.979547	0.489774	−	0.871850i	$-0.337079\pi$
$853$	28.6088	0.979547	0.489774	+	0.871850i	$0.337079\pi$
$854$	75.7997	2.59381
$855$	0	0
$856$	−52.3933	−1.79077
$857$	45.6849	1.56057	0.780284	−	0.625426i	$-0.215075\pi$
$857$	45.6849	1.56057	0.780284	+	0.625426i	$0.215075\pi$
$858$	0	0
$859$	−35.5569	−1.21318	−0.606592	−	0.795013i	$-0.707464\pi$
$859$	−35.5569	−1.21318	−0.606592	+	0.795013i	$0.707464\pi$
$860$	4.60653	0.157081
$861$	0	0
$862$	−57.7028	−1.96537
$863$	41.2100	1.40280	0.701402	−	0.712766i	$-0.252558\pi$
$863$	41.2100	1.40280	0.701402	+	0.712766i	$0.252558\pi$
$864$	0	0
$865$	−33.0045	−1.12219
$866$	−45.9688	−1.56208
$867$	0	0
$868$	−120.990	−4.10666
$869$	−96.0356	−3.25778
$870$	0	0
$871$	47.5266	1.61038
$872$	−62.7686	−2.12561
$873$	0	0
$874$	−19.5744	−0.662112
$875$	42.3494	1.43167
$876$	0	0
$877$	40.4019	1.36427	0.682137	−	0.731224i	$-0.261051\pi$
$877$	40.4019	1.36427	0.682137	+	0.731224i	$0.261051\pi$
$878$	1.92651	0.0650165
$879$	0	0
$880$	58.4333	1.96979
$881$	−13.4833	−0.454262	−0.227131	−	0.973864i	$-0.572935\pi$
$881$	−13.4833	−0.454262	−0.227131	+	0.973864i	$0.572935\pi$
$882$	0	0
$883$	15.9132	0.535522	0.267761	−	0.963485i	$-0.413716\pi$
$883$	15.9132	0.535522	0.267761	+	0.963485i	$0.413716\pi$
$884$	4.39266	0.147741
$885$	0	0
$886$	68.0159	2.28504
$887$	−24.3675	−0.818181	−0.409090	−	0.912494i	$-0.634154\pi$
$887$	−24.3675	−0.818181	−0.409090	+	0.912494i	$0.634154\pi$
$888$	0	0
$889$	42.8639	1.43761
$890$	−36.7807	−1.23289
$891$	0	0
$892$	46.1547	1.54537
$893$	1.00000	0.0334637
$894$	0	0
$895$	26.3827	0.881878
$896$	53.5674	1.78956
$897$	0	0
$898$	65.9562	2.20099
$899$	−49.3074	−1.64449
$900$	0	0
$901$	−1.89651	−0.0631820
$902$	−138.245	−4.60305
$903$	0	0
$904$	78.4982	2.61081
$905$	9.64590	0.320641
$906$	0	0
$907$	26.4951	0.879755	0.439877	−	0.898058i	$-0.355022\pi$
$907$	26.4951	0.879755	0.439877	+	0.898058i	$0.355022\pi$
$908$	2.74436	0.0910747
$909$	0	0
$910$	−75.9823	−2.51879
$911$	−7.27572	−0.241055	−0.120528	−	0.992710i	$-0.538459\pi$
$911$	−7.27572	−0.241055	−0.120528	+	0.992710i	$0.538459\pi$
$912$	0	0
$913$	35.6296	1.17917
$914$	21.5921	0.714202
$915$	0	0
$916$	−58.8285	−1.94375
$917$	−23.2940	−0.769235
$918$	0	0
$919$	47.0684	1.55264	0.776321	−	0.630337i	$-0.217083\pi$
$919$	47.0684	1.55264	0.776321	+	0.630337i	$0.217083\pi$
$920$	−72.2295	−2.38134
$921$	0	0
$922$	−37.9509	−1.24985
$923$	45.5501	1.49930
$924$	0	0
$925$	19.6346	0.645582
$926$	91.2044	2.99716
$927$	0	0
$928$	−22.9517	−0.753425
$929$	36.7399	1.20540	0.602699	−	0.797969i	$-0.294092\pi$
$929$	36.7399	1.20540	0.602699	+	0.797969i	$0.294092\pi$
$930$	0	0
$931$	−5.75319	−0.188553
$932$	−4.15927	−0.136241
$933$	0	0
$934$	1.26923	0.0415306
$935$	−1.81206	−0.0592607
$936$	0	0
$937$	−46.4331	−1.51690	−0.758452	−	0.651729i	$-0.774044\pi$
$937$	−46.4331	−1.51690	−0.758452	+	0.651729i	$0.774044\pi$
$938$	79.8890	2.60847
$939$	0	0
$940$	6.85242	0.223501
$941$	19.4030	0.632521	0.316260	−	0.948672i	$-0.397573\pi$
$941$	19.4030	0.632521	0.316260	+	0.948672i	$0.397573\pi$
$942$	0	0
$943$	70.6782	2.30160
$944$	−28.8609	−0.939341
$945$	0	0
$946$	−10.2271	−0.332512
$947$	41.3786	1.34462	0.672311	−	0.740268i	$-0.265302\pi$
$947$	41.3786	1.34462	0.672311	+	0.740268i	$0.265302\pi$
$948$	0	0
$949$	45.8876	1.48957
$950$	−6.29113	−0.204111
$951$	0	0
$952$	3.97614	0.128867
$953$	37.1396	1.20307	0.601536	−	0.798846i	$-0.294556\pi$
$953$	37.1396	1.20307	0.601536	+	0.798846i	$0.294556\pi$
$954$	0	0
$955$	−34.5629	−1.11843
$956$	−10.4610	−0.338332
$957$	0	0
$958$	37.7834	1.22072
$959$	−4.31410	−0.139309
$960$	0	0
$961$	30.1179	0.971545
$962$	−105.690	−3.40759
$963$	0	0
$964$	30.9597	0.997145
$965$	−5.18815	−0.167012
$966$	0	0
$967$	−57.7392	−1.85677	−0.928384	−	0.371623i	$-0.878801\pi$
$967$	−57.7392	−1.85677	−0.928384	+	0.371623i	$0.878801\pi$
$968$	−150.010	−4.82151
$969$	0	0
$970$	−39.3468	−1.26335
$971$	−34.3388	−1.10198	−0.550992	−	0.834510i	$-0.685751\pi$
$971$	−34.3388	−1.10198	−0.550992	+	0.834510i	$0.685751\pi$
$972$	0	0
$973$	6.68546	0.214326
$974$	100.731	3.22762
$975$	0	0
$976$	−51.5591	−1.65037
$977$	46.2181	1.47865	0.739325	−	0.673349i	$-0.235145\pi$
$977$	46.2181	1.47865	0.739325	+	0.673349i	$0.235145\pi$
$978$	0	0
$979$	55.8727	1.78570
$980$	−39.4233	−1.25933
$981$	0	0
$982$	−46.6665	−1.48919
$983$	32.5383	1.03781	0.518905	−	0.854832i	$-0.326340\pi$
$983$	32.5383	1.03781	0.518905	+	0.854832i	$0.326340\pi$
$984$	0	0
$985$	8.60016	0.274024
$986$	3.00914	0.0958304
$987$	0	0
$988$	23.1708	0.737161
$989$	5.22865	0.166261
$990$	0	0
$991$	34.3830	1.09221	0.546105	−	0.837716i	$-0.316110\pi$
$991$	34.3830	1.09221	0.546105	+	0.837716i	$0.316110\pi$
$992$	28.4492	0.903264
$993$	0	0
$994$	76.5667	2.42855
$995$	−0.0582277	−0.00184594
$996$	0	0
$997$	47.3154	1.49849	0.749247	−	0.662290i	$-0.230415\pi$
$997$	47.3154	1.49849	0.749247	+	0.662290i	$0.230415\pi$
$998$	36.3525	1.15072
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.q.1.3		23
3.2	odd	2		2679.2.a.m.1.21	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.m.1.21	✓	23	3.2	odd	2
8037.2.a.q.1.3		23	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51668 q^{2} +4.33366 q^{4} -1.58121 q^{5} +3.57116 q^{7} -5.87307 q^{8} +O(q^{10})\)	\(q-2.51668 q^{2} +4.33366 q^{4} -1.58121 q^{5} +3.57116 q^{7} -5.87307 q^{8} +3.97939 q^{10} -6.04500 q^{11} -5.34670 q^{13} -8.98746 q^{14} +6.11329 q^{16} -0.189577 q^{17} -1.00000 q^{19} -6.85242 q^{20} +15.2133 q^{22} -7.77786 q^{23} -2.49978 q^{25} +13.4559 q^{26} +15.4762 q^{28} +6.30707 q^{29} -7.81779 q^{31} -3.63904 q^{32} +0.477105 q^{34} -5.64675 q^{35} -7.85454 q^{37} +2.51668 q^{38} +9.28655 q^{40} -9.08711 q^{41} -0.672248 q^{43} -26.1970 q^{44} +19.5744 q^{46} -1.00000 q^{47} +5.75319 q^{49} +6.29113 q^{50} -23.1708 q^{52} +10.0039 q^{53} +9.55841 q^{55} -20.9737 q^{56} -15.8729 q^{58} -4.72100 q^{59} -8.43394 q^{61} +19.6749 q^{62} -3.06830 q^{64} +8.45426 q^{65} -8.88895 q^{67} -0.821564 q^{68} +14.2111 q^{70} -8.51929 q^{71} -8.58241 q^{73} +19.7673 q^{74} -4.33366 q^{76} -21.5877 q^{77} +15.8868 q^{79} -9.66639 q^{80} +22.8693 q^{82} -5.89406 q^{83} +0.299762 q^{85} +1.69183 q^{86} +35.5027 q^{88} -9.24280 q^{89} -19.0939 q^{91} -33.7066 q^{92} +2.51668 q^{94} +1.58121 q^{95} -9.88764 q^{97} -14.4789 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8	\( 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8} - 5 q^{10} - 12 q^{11} + 9 q^{13} - 3 q^{14} + 60 q^{16} - 16 q^{17} - 23 q^{19} - 25 q^{20} + 3 q^{22} - 12 q^{23} + 54 q^{25} - 5 q^{26} + 8 q^{28} - 27 q^{29} + 10 q^{31} + 34 q^{32} + 6 q^{35} + 15 q^{37} + 2 q^{38} + 3 q^{40} - 10 q^{41} + 24 q^{43} - 39 q^{44} + 43 q^{46} - 23 q^{47} + 78 q^{49} + 32 q^{50} + 38 q^{52} + 2 q^{53} + 5 q^{55} - 58 q^{56} - 11 q^{58} + 51 q^{59} + 48 q^{61} + 22 q^{62} + 125 q^{64} - 15 q^{65} + 26 q^{67} - 26 q^{68} + 86 q^{70} - 24 q^{71} + 53 q^{73} - 26 q^{74} - 30 q^{76} - 18 q^{77} + 29 q^{79} - 5 q^{80} + 47 q^{82} + 22 q^{83} - 5 q^{85} + 28 q^{86} + 62 q^{88} - 38 q^{89} + 15 q^{91} - 15 q^{92} + 2 q^{94} + 9 q^{95} + 33 q^{97} + 30 q^{98}+O(q^{100}) \) 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8 - 5 * q^10 - 12 * q^11 + 9 * q^13 - 3 * q^14 + 60 * q^16 - 16 * q^17 - 23 * q^19 - 25 * q^20 + 3 * q^22 - 12 * q^23 + 54 * q^25 - 5 * q^26 + 8 * q^28 - 27 * q^29 + 10 * q^31 + 34 * q^32 + 6 * q^35 + 15 * q^37 + 2 * q^38 + 3 * q^40 - 10 * q^41 + 24 * q^43 - 39 * q^44 + 43 * q^46 - 23 * q^47 + 78 * q^49 + 32 * q^50 + 38 * q^52 + 2 * q^53 + 5 * q^55 - 58 * q^56 - 11 * q^58 + 51 * q^59 + 48 * q^61 + 22 * q^62 + 125 * q^64 - 15 * q^65 + 26 * q^67 - 26 * q^68 + 86 * q^70 - 24 * q^71 + 53 * q^73 - 26 * q^74 - 30 * q^76 - 18 * q^77 + 29 * q^79 - 5 * q^80 + 47 * q^82 + 22 * q^83 - 5 * q^85 + 28 * q^86 + 62 * q^88 - 38 * q^89 + 15 * q^91 - 15 * q^92 + 2 * q^94 + 9 * q^95 + 33 * q^97 + 30 * q^98

Introduction

L-functions

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Database

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