LMFDB - Embedded newform 8037.2.a.s.1.9

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:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.39021 q^{2} -0.0673298 q^{4} -3.64442 q^{5} -1.02491 q^{7} +2.87401 q^{8} +O(q^{10})$	\(q-1.39021 q^{2} -0.0673298 q^{4} -3.64442 q^{5} -1.02491 q^{7} +2.87401 q^{8} +5.06649 q^{10} +1.10772 q^{11} +3.36711 q^{13} +1.42484 q^{14} -3.86081 q^{16} -3.23610 q^{17} -1.00000 q^{19} +0.245378 q^{20} -1.53995 q^{22} -2.24495 q^{23} +8.28179 q^{25} -4.68097 q^{26} +0.0690072 q^{28} -5.05546 q^{29} +2.95273 q^{31} -0.380711 q^{32} +4.49885 q^{34} +3.73521 q^{35} -2.69839 q^{37} +1.39021 q^{38} -10.4741 q^{40} -1.56318 q^{41} +5.22420 q^{43} -0.0745824 q^{44} +3.12094 q^{46} +1.00000 q^{47} -5.94955 q^{49} -11.5134 q^{50} -0.226707 q^{52} -8.65966 q^{53} -4.03699 q^{55} -2.94561 q^{56} +7.02813 q^{58} +5.92854 q^{59} +10.4410 q^{61} -4.10490 q^{62} +8.25088 q^{64} -12.2711 q^{65} +13.8912 q^{67} +0.217886 q^{68} -5.19271 q^{70} +4.56398 q^{71} -14.7845 q^{73} +3.75131 q^{74} +0.0673298 q^{76} -1.13531 q^{77} -3.89576 q^{79} +14.0704 q^{80} +2.17314 q^{82} +16.2620 q^{83} +11.7937 q^{85} -7.26271 q^{86} +3.18359 q^{88} -0.170723 q^{89} -3.45099 q^{91} +0.151152 q^{92} -1.39021 q^{94} +3.64442 q^{95} -10.7624 q^{97} +8.27110 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{2} + 32 q^{4} - 10 q^{5} + 6 q^{7} - 15 q^{8}+O(q^{10}) $ 24 * q - 6 * q^2 + 32 * q^4 - 10 * q^5 + 6 * q^7 - 15 * q^8	$ 24 q - 6 q^{2} + 32 q^{4} - 10 q^{5} + 6 q^{7} - 15 q^{8} + 5 q^{10} - 15 q^{11} - 3 q^{13} - 27 q^{14} + 48 q^{16} - 22 q^{17} - 24 q^{19} - 29 q^{20} - 9 q^{22} - 15 q^{23} + 58 q^{25} - q^{26} + 18 q^{28} - 36 q^{29} - 16 q^{31} - 26 q^{32} + 10 q^{34} - 37 q^{35} + 18 q^{37} + 6 q^{38} + 7 q^{40} - 24 q^{41} + 2 q^{43} - 57 q^{44} - 23 q^{46} + 24 q^{47} + 64 q^{49} - 4 q^{50} - 40 q^{52} - 42 q^{53} + 8 q^{55} - 38 q^{56} - 5 q^{58} - 39 q^{59} + 6 q^{61} + 14 q^{62} + 53 q^{64} - 47 q^{65} - 26 q^{67} - 80 q^{68} - 46 q^{70} - 6 q^{71} + 3 q^{73} - 62 q^{74} - 32 q^{76} - 13 q^{77} + 10 q^{79} - 17 q^{80} + 15 q^{82} - 44 q^{83} + 21 q^{85} + 22 q^{86} - 88 q^{88} - 32 q^{89} - 17 q^{91} + 37 q^{92} - 6 q^{94} + 10 q^{95} + 28 q^{97} - 44 q^{98}+O(q^{100}) $ 24 * q - 6 * q^2 + 32 * q^4 - 10 * q^5 + 6 * q^7 - 15 * q^8 + 5 * q^10 - 15 * q^11 - 3 * q^13 - 27 * q^14 + 48 * q^16 - 22 * q^17 - 24 * q^19 - 29 * q^20 - 9 * q^22 - 15 * q^23 + 58 * q^25 - q^26 + 18 * q^28 - 36 * q^29 - 16 * q^31 - 26 * q^32 + 10 * q^34 - 37 * q^35 + 18 * q^37 + 6 * q^38 + 7 * q^40 - 24 * q^41 + 2 * q^43 - 57 * q^44 - 23 * q^46 + 24 * q^47 + 64 * q^49 - 4 * q^50 - 40 * q^52 - 42 * q^53 + 8 * q^55 - 38 * q^56 - 5 * q^58 - 39 * q^59 + 6 * q^61 + 14 * q^62 + 53 * q^64 - 47 * q^65 - 26 * q^67 - 80 * q^68 - 46 * q^70 - 6 * q^71 + 3 * q^73 - 62 * q^74 - 32 * q^76 - 13 * q^77 + 10 * q^79 - 17 * q^80 + 15 * q^82 - 44 * q^83 + 21 * q^85 + 22 * q^86 - 88 * q^88 - 32 * q^89 - 17 * q^91 + 37 * q^92 - 6 * q^94 + 10 * q^95 + 28 * q^97 - 44 * 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 $

$2$	−1.39021	−0.983023	−0.491512	−	0.870871i	$-0.663556\pi$
$2$	−1.39021	−0.983023	−0.491512	+	0.870871i	$0.663556\pi$
$3$	0	0
$3$	0	0
$4$	−0.0673298	−0.0336649
$5$	−3.64442	−1.62983	−0.814917	−	0.579578i	$-0.803217\pi$
$5$	−3.64442	−1.62983	−0.814917	+	0.579578i	$0.803217\pi$
$6$	0	0
$7$	−1.02491	−0.387381	−0.193690	−	0.981063i	$-0.562046\pi$
$7$	−1.02491	−0.387381	−0.193690	+	0.981063i	$0.562046\pi$
$8$	2.87401	1.01612
$9$	0	0
$10$	5.06649	1.60216
$11$	1.10772	0.333989	0.166995	−	0.985958i	$-0.446594\pi$
$11$	1.10772	0.333989	0.166995	+	0.985958i	$0.446594\pi$
$12$	0	0
$13$	3.36711	0.933867	0.466934	−	0.884292i	$-0.345359\pi$
$13$	3.36711	0.933867	0.466934	+	0.884292i	$0.345359\pi$
$14$	1.42484	0.380804
$15$	0	0
$16$	−3.86081	−0.965202
$17$	−3.23610	−0.784870	−0.392435	−	0.919780i	$-0.628367\pi$
$17$	−3.23610	−0.784870	−0.392435	+	0.919780i	$0.628367\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.245378	0.0548682
$21$	0	0
$22$	−1.53995	−0.328319
$23$	−2.24495	−0.468105	−0.234052	−	0.972224i	$-0.575199\pi$
$23$	−2.24495	−0.468105	−0.234052	+	0.972224i	$0.575199\pi$
$24$	0	0
$25$	8.28179	1.65636
$26$	−4.68097	−0.918013
$27$	0	0
$28$	0.0690072	0.0130411
$29$	−5.05546	−0.938776	−0.469388	−	0.882992i	$-0.655525\pi$
$29$	−5.05546	−0.938776	−0.469388	+	0.882992i	$0.655525\pi$
$30$	0	0
$31$	2.95273	0.530326	0.265163	−	0.964204i	$-0.414574\pi$
$31$	2.95273	0.530326	0.265163	+	0.964204i	$0.414574\pi$
$32$	−0.380711	−0.0673008
$33$	0	0
$34$	4.49885	0.771546
$35$	3.73521	0.631366
$36$	0	0
$37$	−2.69839	−0.443612	−0.221806	−	0.975091i	$-0.571195\pi$
$37$	−2.69839	−0.443612	−0.221806	+	0.975091i	$0.571195\pi$
$38$	1.39021	0.225521
$39$	0	0
$40$	−10.4741	−1.65610
$41$	−1.56318	−0.244128	−0.122064	−	0.992522i	$-0.538951\pi$
$41$	−1.56318	−0.244128	−0.122064	+	0.992522i	$0.538951\pi$
$42$	0	0
$43$	5.22420	0.796683	0.398342	−	0.917237i	$-0.369586\pi$
$43$	5.22420	0.796683	0.398342	+	0.917237i	$0.369586\pi$
$44$	−0.0745824	−0.0112437
$45$	0	0
$46$	3.12094	0.460158
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−5.94955	−0.849936
$50$	−11.5134	−1.62824
$51$	0	0
$52$	−0.226707	−0.0314385
$53$	−8.65966	−1.18950	−0.594748	−	0.803912i	$-0.702748\pi$
$53$	−8.65966	−1.18950	−0.594748	+	0.803912i	$0.702748\pi$
$54$	0	0
$55$	−4.03699	−0.544347
$56$	−2.94561	−0.393624
$57$	0	0
$58$	7.02813	0.922839
$59$	5.92854	0.771831	0.385915	−	0.922534i	$-0.373886\pi$
$59$	5.92854	0.771831	0.385915	+	0.922534i	$0.373886\pi$
$60$	0	0
$61$	10.4410	1.33683	0.668416	−	0.743788i	$-0.266973\pi$
$61$	10.4410	1.33683	0.668416	+	0.743788i	$0.266973\pi$
$62$	−4.10490	−0.521323
$63$	0	0
$64$	8.25088	1.03136
$65$	−12.2711	−1.52205
$66$	0	0
$67$	13.8912	1.69708	0.848542	−	0.529128i	$-0.177481\pi$
$67$	13.8912	1.69708	0.848542	+	0.529128i	$0.177481\pi$
$68$	0.217886	0.0264226
$69$	0	0
$70$	−5.19271	−0.620648
$71$	4.56398	0.541644	0.270822	−	0.962629i	$-0.412704\pi$
$71$	4.56398	0.541644	0.270822	+	0.962629i	$0.412704\pi$
$72$	0	0
$73$	−14.7845	−1.73040	−0.865198	−	0.501430i	$-0.832807\pi$
$73$	−14.7845	−1.73040	−0.865198	+	0.501430i	$0.832807\pi$
$74$	3.75131	0.436081
$75$	0	0
$76$	0.0673298	0.00772326
$77$	−1.13531	−0.129381
$78$	0	0
$79$	−3.89576	−0.438307	−0.219154	−	0.975690i	$-0.570330\pi$
$79$	−3.89576	−0.438307	−0.219154	+	0.975690i	$0.570330\pi$
$80$	14.0704	1.57312
$81$	0	0
$82$	2.17314	0.239983
$83$	16.2620	1.78499	0.892494	−	0.451059i	$-0.148953\pi$
$83$	16.2620	1.78499	0.892494	+	0.451059i	$0.148953\pi$
$84$	0	0
$85$	11.7937	1.27921
$86$	−7.26271	−0.783159
$87$	0	0
$88$	3.18359	0.339372
$89$	−0.170723	−0.0180966	−0.00904832	−	0.999959i	$-0.502880\pi$
$89$	−0.170723	−0.0180966	−0.00904832	+	0.999959i	$0.502880\pi$
$90$	0	0
$91$	−3.45099	−0.361762
$92$	0.151152	0.0157587
$93$	0	0
$94$	−1.39021	−0.143389
$95$	3.64442	0.373910
$96$	0	0
$97$	−10.7624	−1.09275	−0.546377	−	0.837539i	$-0.683993\pi$
$97$	−10.7624	−1.09275	−0.546377	+	0.837539i	$0.683993\pi$
$98$	8.27110	0.835507
$99$	0	0
$100$	−0.557611	−0.0557611
$101$	0.802226	0.0798244	0.0399122	−	0.999203i	$-0.487292\pi$
$101$	0.802226	0.0798244	0.0399122	+	0.999203i	$0.487292\pi$
$102$	0	0
$103$	−0.162552	−0.0160167	−0.00800836	−	0.999968i	$-0.502549\pi$
$103$	−0.162552	−0.0160167	−0.00800836	+	0.999968i	$0.502549\pi$
$104$	9.67710	0.948918
$105$	0	0
$106$	12.0387	1.16930
$107$	−2.53336	−0.244909	−0.122454	−	0.992474i	$-0.539077\pi$
$107$	−2.53336	−0.244909	−0.122454	+	0.992474i	$0.539077\pi$
$108$	0	0
$109$	5.41241	0.518415	0.259207	−	0.965822i	$-0.416539\pi$
$109$	5.41241	0.518415	0.259207	+	0.965822i	$0.416539\pi$
$110$	5.61224	0.535106
$111$	0	0
$112$	3.95699	0.373901
$113$	7.35188	0.691606	0.345803	−	0.938307i	$-0.387606\pi$
$113$	7.35188	0.691606	0.345803	+	0.938307i	$0.387606\pi$
$114$	0	0
$115$	8.18154	0.762933
$116$	0.340383	0.0316038
$117$	0	0
$118$	−8.24189	−0.758728
$119$	3.31672	0.304044
$120$	0	0
$121$	−9.77296	−0.888451
$122$	−14.5151	−1.31414
$123$	0	0
$124$	−0.198807	−0.0178534
$125$	−11.9602	−1.06975
$126$	0	0
$127$	−1.95706	−0.173661	−0.0868306	−	0.996223i	$-0.527674\pi$
$127$	−1.95706	−0.173661	−0.0868306	+	0.996223i	$0.527674\pi$
$128$	−10.7090	−0.946550
$129$	0	0
$130$	17.0594	1.49621
$131$	9.70630	0.848044	0.424022	−	0.905652i	$-0.360618\pi$
$131$	9.70630	0.848044	0.424022	+	0.905652i	$0.360618\pi$
$132$	0	0
$133$	1.02491	0.0888712
$134$	−19.3117	−1.66827
$135$	0	0
$136$	−9.30060	−0.797520
$137$	−8.80435	−0.752207	−0.376103	−	0.926578i	$-0.622736\pi$
$137$	−8.80435	−0.752207	−0.376103	+	0.926578i	$0.622736\pi$
$138$	0	0
$139$	11.1056	0.941969	0.470984	−	0.882142i	$-0.343899\pi$
$139$	11.1056	0.941969	0.470984	+	0.882142i	$0.343899\pi$
$140$	−0.251491	−0.0212549
$141$	0	0
$142$	−6.34486	−0.532449
$143$	3.72980	0.311902
$144$	0	0
$145$	18.4242	1.53005
$146$	20.5535	1.70102
$147$	0	0
$148$	0.181682	0.0149342
$149$	−9.44905	−0.774096	−0.387048	−	0.922060i	$-0.626505\pi$
$149$	−9.44905	−0.774096	−0.387048	+	0.922060i	$0.626505\pi$
$150$	0	0
$151$	−7.97049	−0.648630	−0.324315	−	0.945949i	$-0.605134\pi$
$151$	−7.97049	−0.648630	−0.324315	+	0.945949i	$0.605134\pi$
$152$	−2.87401	−0.233113
$153$	0	0
$154$	1.57832	0.127185
$155$	−10.7610	−0.864343
$156$	0	0
$157$	3.41025	0.272167	0.136084	−	0.990697i	$-0.456548\pi$
$157$	3.41025	0.272167	0.136084	+	0.990697i	$0.456548\pi$
$158$	5.41591	0.430866
$159$	0	0
$160$	1.38747	0.109689
$161$	2.30088	0.181335
$162$	0	0
$163$	1.19237	0.0933940	0.0466970	−	0.998909i	$-0.485130\pi$
$163$	1.19237	0.0933940	0.0466970	+	0.998909i	$0.485130\pi$
$164$	0.105249	0.00821854
$165$	0	0
$166$	−22.6075	−1.75469
$167$	25.0809	1.94082	0.970410	−	0.241464i	$-0.0776277\pi$
$167$	25.0809	1.94082	0.970410	+	0.241464i	$0.0776277\pi$
$168$	0	0
$169$	−1.66260	−0.127892
$170$	−16.3957	−1.25749
$171$	0	0
$172$	−0.351744	−0.0268203
$173$	21.8689	1.66266	0.831332	−	0.555776i	$-0.187579\pi$
$173$	21.8689	1.66266	0.831332	+	0.555776i	$0.187579\pi$
$174$	0	0
$175$	−8.48811	−0.641641
$176$	−4.27668	−0.322367
$177$	0	0
$178$	0.237341	0.0177894
$179$	24.2698	1.81401	0.907005	−	0.421119i	$-0.138363\pi$
$179$	24.2698	1.81401	0.907005	+	0.421119i	$0.138363\pi$
$180$	0	0
$181$	22.7153	1.68842	0.844208	−	0.536016i	$-0.180071\pi$
$181$	22.7153	1.68842	0.844208	+	0.536016i	$0.180071\pi$
$182$	4.79758	0.355621
$183$	0	0
$184$	−6.45202	−0.475649
$185$	9.83406	0.723015
$186$	0	0
$187$	−3.58469	−0.262138
$188$	−0.0673298	−0.00491053
$189$	0	0
$190$	−5.06649	−0.367562
$191$	19.4265	1.40565	0.702825	−	0.711362i	$-0.251922\pi$
$191$	19.4265	1.40565	0.702825	+	0.711362i	$0.251922\pi$
$192$	0	0
$193$	−0.934137	−0.0672407	−0.0336203	−	0.999435i	$-0.510704\pi$
$193$	−0.934137	−0.0672407	−0.0336203	+	0.999435i	$0.510704\pi$
$194$	14.9619	1.07420
$195$	0	0
$196$	0.400582	0.0286130
$197$	1.89949	0.135333	0.0676666	−	0.997708i	$-0.478445\pi$
$197$	1.89949	0.135333	0.0676666	+	0.997708i	$0.478445\pi$
$198$	0	0
$199$	−2.02347	−0.143440	−0.0717200	−	0.997425i	$-0.522849\pi$
$199$	−2.02347	−0.143440	−0.0717200	+	0.997425i	$0.522849\pi$
$200$	23.8020	1.68305
$201$	0	0
$202$	−1.11526	−0.0784693
$203$	5.18141	0.363664
$204$	0	0
$205$	5.69689	0.397888
$206$	0.225980	0.0157448
$207$	0	0
$208$	−12.9997	−0.901370
$209$	−1.10772	−0.0766224
$210$	0	0
$211$	−18.6849	−1.28632	−0.643161	−	0.765731i	$-0.722377\pi$
$211$	−18.6849	−1.28632	−0.643161	+	0.765731i	$0.722377\pi$
$212$	0.583053	0.0400442
$213$	0	0
$214$	3.52188	0.240751
$215$	−19.0392	−1.29846
$216$	0	0
$217$	−3.02629	−0.205438
$218$	−7.52436	−0.509614
$219$	0	0
$220$	0.271809	0.0183254
$221$	−10.8963	−0.732965
$222$	0	0
$223$	25.3331	1.69643	0.848216	−	0.529650i	$-0.177677\pi$
$223$	25.3331	1.69643	0.848216	+	0.529650i	$0.177677\pi$
$224$	0.390196	0.0260710
$225$	0	0
$226$	−10.2206	−0.679865
$227$	−27.4378	−1.82111	−0.910557	−	0.413384i	$-0.864347\pi$
$227$	−27.4378	−1.82111	−0.910557	+	0.413384i	$0.864347\pi$
$228$	0	0
$229$	−1.33308	−0.0880921	−0.0440461	−	0.999030i	$-0.514025\pi$
$229$	−1.33308	−0.0880921	−0.0440461	+	0.999030i	$0.514025\pi$
$230$	−11.3740	−0.749981
$231$	0	0
$232$	−14.5295	−0.953906
$233$	−8.61085	−0.564115	−0.282058	−	0.959397i	$-0.591017\pi$
$233$	−8.61085	−0.564115	−0.282058	+	0.959397i	$0.591017\pi$
$234$	0	0
$235$	−3.64442	−0.237736
$236$	−0.399168	−0.0259836
$237$	0	0
$238$	−4.61093	−0.298882
$239$	18.1772	1.17578	0.587892	−	0.808940i	$-0.299958\pi$
$239$	18.1772	1.17578	0.587892	+	0.808940i	$0.299958\pi$
$240$	0	0
$241$	7.71567	0.497010	0.248505	−	0.968631i	$-0.420061\pi$
$241$	7.71567	0.497010	0.248505	+	0.968631i	$0.420061\pi$
$242$	13.5864	0.873368
$243$	0	0
$244$	−0.702989	−0.0450043
$245$	21.6827	1.38525
$246$	0	0
$247$	−3.36711	−0.214244
$248$	8.48618	0.538873
$249$	0	0
$250$	16.6272	1.05159
$251$	4.14187	0.261433	0.130716	−	0.991420i	$-0.458272\pi$
$251$	4.14187	0.261433	0.130716	+	0.991420i	$0.458272\pi$
$252$	0	0
$253$	−2.48677	−0.156342
$254$	2.72072	0.170713
$255$	0	0
$256$	−1.61406	−0.100879
$257$	8.53007	0.532091	0.266046	−	0.963960i	$-0.414283\pi$
$257$	8.53007	0.532091	0.266046	+	0.963960i	$0.414283\pi$
$258$	0	0
$259$	2.76561	0.171847
$260$	0.826214	0.0512396
$261$	0	0
$262$	−13.4938	−0.833647
$263$	−26.2738	−1.62011	−0.810055	−	0.586354i	$-0.800563\pi$
$263$	−26.2738	−1.62011	−0.810055	+	0.586354i	$0.800563\pi$
$264$	0	0
$265$	31.5594	1.93868
$266$	−1.42484	−0.0873625
$267$	0	0
$268$	−0.935293	−0.0571321
$269$	−18.7971	−1.14608	−0.573039	−	0.819528i	$-0.694236\pi$
$269$	−18.7971	−1.14608	−0.573039	+	0.819528i	$0.694236\pi$
$270$	0	0
$271$	−10.2748	−0.624148	−0.312074	−	0.950058i	$-0.601024\pi$
$271$	−10.2748	−0.624148	−0.312074	+	0.950058i	$0.601024\pi$
$272$	12.4940	0.757558
$273$	0	0
$274$	12.2399	0.739437
$275$	9.17389	0.553206
$276$	0	0
$277$	7.58224	0.455573	0.227786	−	0.973711i	$-0.426851\pi$
$277$	7.58224	0.455573	0.227786	+	0.973711i	$0.426851\pi$
$278$	−15.4391	−0.925977
$279$	0	0
$280$	10.7350	0.641542
$281$	−3.86078	−0.230315	−0.115158	−	0.993347i	$-0.536737\pi$
$281$	−3.86078	−0.230315	−0.115158	+	0.993347i	$0.536737\pi$
$282$	0	0
$283$	5.64450	0.335531	0.167765	−	0.985827i	$-0.446345\pi$
$283$	5.64450	0.335531	0.167765	+	0.985827i	$0.446345\pi$
$284$	−0.307291	−0.0182344
$285$	0	0
$286$	−5.18519	−0.306607
$287$	1.60212	0.0945704
$288$	0	0
$289$	−6.52763	−0.383979
$290$	−25.6135	−1.50407
$291$	0	0
$292$	0.995438	0.0582536
$293$	15.5019	0.905629	0.452815	−	0.891605i	$-0.350420\pi$
$293$	15.5019	0.905629	0.452815	+	0.891605i	$0.350420\pi$
$294$	0	0
$295$	−21.6061	−1.25796
$296$	−7.75520	−0.450762
$297$	0	0
$298$	13.1361	0.760954
$299$	−7.55899	−0.437147
$300$	0	0
$301$	−5.35435	−0.308620
$302$	11.0806	0.637618
$303$	0	0
$304$	3.86081	0.221432
$305$	−38.0513	−2.17881
$306$	0	0
$307$	−13.8181	−0.788638	−0.394319	−	0.918974i	$-0.629020\pi$
$307$	−13.8181	−0.788638	−0.394319	+	0.918974i	$0.629020\pi$
$308$	0.0764405	0.00435560
$309$	0	0
$310$	14.9600	0.849670
$311$	−11.4595	−0.649806	−0.324903	−	0.945747i	$-0.605332\pi$
$311$	−11.4595	−0.649806	−0.324903	+	0.945747i	$0.605332\pi$
$312$	0	0
$313$	−9.59659	−0.542432	−0.271216	−	0.962519i	$-0.587426\pi$
$313$	−9.59659	−0.542432	−0.271216	+	0.962519i	$0.587426\pi$
$314$	−4.74094	−0.267547
$315$	0	0
$316$	0.262301	0.0147556
$317$	−18.2495	−1.02499	−0.512497	−	0.858689i	$-0.671279\pi$
$317$	−18.2495	−1.02499	−0.512497	+	0.858689i	$0.671279\pi$
$318$	0	0
$319$	−5.60003	−0.313541
$320$	−30.0697	−1.68095
$321$	0	0
$322$	−3.19869	−0.178256
$323$	3.23610	0.180062
$324$	0	0
$325$	27.8857	1.54682
$326$	−1.65765	−0.0918085
$327$	0	0
$328$	−4.49260	−0.248062
$329$	−1.02491	−0.0565053
$330$	0	0
$331$	−24.4384	−1.34325	−0.671627	−	0.740890i	$-0.734404\pi$
$331$	−24.4384	−1.34325	−0.671627	+	0.740890i	$0.734404\pi$
$332$	−1.09492	−0.0600914
$333$	0	0
$334$	−34.8676	−1.90787
$335$	−50.6255	−2.76596
$336$	0	0
$337$	6.89875	0.375799	0.187899	−	0.982188i	$-0.439832\pi$
$337$	6.89875	0.375799	0.187899	+	0.982188i	$0.439832\pi$
$338$	2.31135	0.125721
$339$	0	0
$340$	−0.794068	−0.0430644
$341$	3.27079	0.177123
$342$	0	0
$343$	13.2722	0.716630
$344$	15.0144	0.809523
$345$	0	0
$346$	−30.4023	−1.63444
$347$	28.9480	1.55401	0.777004	−	0.629496i	$-0.216738\pi$
$347$	28.9480	1.55401	0.777004	+	0.629496i	$0.216738\pi$
$348$	0	0
$349$	−31.8465	−1.70470	−0.852352	−	0.522968i	$-0.824825\pi$
$349$	−31.8465	−1.70470	−0.852352	+	0.522968i	$0.824825\pi$
$350$	11.8002	0.630748
$351$	0	0
$352$	−0.421720	−0.0224778
$353$	−18.8271	−1.00207	−0.501034	−	0.865428i	$-0.667047\pi$
$353$	−18.8271	−1.00207	−0.501034	+	0.865428i	$0.667047\pi$
$354$	0	0
$355$	−16.6330	−0.882790
$356$	0.0114948	0.000609221 0
$357$	0	0
$358$	−33.7400	−1.78322
$359$	−12.3715	−0.652941	−0.326470	−	0.945207i	$-0.605859\pi$
$359$	−12.3715	−0.652941	−0.326470	+	0.945207i	$0.605859\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−31.5789	−1.65975
$363$	0	0
$364$	0.232354	0.0121787
$365$	53.8810	2.82026
$366$	0	0
$367$	9.24205	0.482431	0.241216	−	0.970472i	$-0.422454\pi$
$367$	9.24205	0.482431	0.241216	+	0.970472i	$0.422454\pi$
$368$	8.66732	0.451815
$369$	0	0
$370$	−13.6714	−0.710740
$371$	8.87539	0.460787
$372$	0	0
$373$	15.8891	0.822706	0.411353	−	0.911476i	$-0.365056\pi$
$373$	15.8891	0.822706	0.411353	+	0.911476i	$0.365056\pi$
$374$	4.98345	0.257688
$375$	0	0
$376$	2.87401	0.148216
$377$	−17.0223	−0.876692
$378$	0	0
$379$	−26.1549	−1.34349	−0.671745	−	0.740783i	$-0.734455\pi$
$379$	−26.1549	−1.34349	−0.671745	+	0.740783i	$0.734455\pi$
$380$	−0.245378	−0.0125876
$381$	0	0
$382$	−27.0068	−1.38179
$383$	17.5775	0.898169	0.449085	−	0.893489i	$-0.351750\pi$
$383$	17.5775	0.898169	0.449085	+	0.893489i	$0.351750\pi$
$384$	0	0
$385$	4.13756	0.210870
$386$	1.29864	0.0660992
$387$	0	0
$388$	0.724629	0.0367875
$389$	−35.4154	−1.79563	−0.897817	−	0.440369i	$-0.854848\pi$
$389$	−35.4154	−1.79563	−0.897817	+	0.440369i	$0.854848\pi$
$390$	0	0
$391$	7.26489	0.367401
$392$	−17.0991	−0.863634
$393$	0	0
$394$	−2.64068	−0.133036
$395$	14.1978	0.714368
$396$	0	0
$397$	−28.1430	−1.41246	−0.706229	−	0.707984i	$-0.749605\pi$
$397$	−28.1430	−1.41246	−0.706229	+	0.707984i	$0.749605\pi$
$398$	2.81304	0.141005
$399$	0	0
$400$	−31.9744	−1.59872
$401$	−11.8807	−0.593293	−0.296647	−	0.954987i	$-0.595868\pi$
$401$	−11.8807	−0.593293	−0.296647	+	0.954987i	$0.595868\pi$
$402$	0	0
$403$	9.94216	0.495254
$404$	−0.0540137	−0.00268728
$405$	0	0
$406$	−7.20322	−0.357490
$407$	−2.98905	−0.148162
$408$	0	0
$409$	−9.09852	−0.449893	−0.224946	−	0.974371i	$-0.572221\pi$
$409$	−9.09852	−0.449893	−0.224946	+	0.974371i	$0.572221\pi$
$410$	−7.91984	−0.391133
$411$	0	0
$412$	0.0109446	0.000539201 0
$413$	−6.07624	−0.298992
$414$	0	0
$415$	−59.2656	−2.90923
$416$	−1.28189	−0.0628500
$417$	0	0
$418$	1.53995	0.0753217
$419$	−6.96035	−0.340036	−0.170018	−	0.985441i	$-0.554383\pi$
$419$	−6.96035	−0.340036	−0.170018	+	0.985441i	$0.554383\pi$
$420$	0	0
$421$	19.2779	0.939546	0.469773	−	0.882787i	$-0.344336\pi$
$421$	19.2779	0.939546	0.469773	+	0.882787i	$0.344336\pi$
$422$	25.9759	1.26449
$423$	0	0
$424$	−24.8880	−1.20867
$425$	−26.8007	−1.30003
$426$	0	0
$427$	−10.7011	−0.517863
$428$	0.170570	0.00824483
$429$	0	0
$430$	26.4684	1.27642
$431$	−12.0542	−0.580631	−0.290315	−	0.956931i	$-0.593760\pi$
$431$	−12.0542	−0.580631	−0.290315	+	0.956931i	$0.593760\pi$
$432$	0	0
$433$	1.90534	0.0915649	0.0457825	−	0.998951i	$-0.485422\pi$
$433$	1.90534	0.0915649	0.0457825	+	0.998951i	$0.485422\pi$
$434$	4.20717	0.201950
$435$	0	0
$436$	−0.364416	−0.0174524
$437$	2.24495	0.107391
$438$	0	0
$439$	−16.4116	−0.783283	−0.391641	−	0.920118i	$-0.628093\pi$
$439$	−16.4116	−0.783283	−0.391641	+	0.920118i	$0.628093\pi$
$440$	−11.6024	−0.553120
$441$	0	0
$442$	15.1481	0.720521
$443$	−12.5028	−0.594028	−0.297014	−	0.954873i	$-0.595991\pi$
$443$	−12.5028	−0.594028	−0.297014	+	0.954873i	$0.595991\pi$
$444$	0	0
$445$	0.622188	0.0294945
$446$	−35.2183	−1.66763
$447$	0	0
$448$	−8.45643	−0.399529
$449$	29.6951	1.40140	0.700700	−	0.713456i	$-0.252871\pi$
$449$	29.6951	1.40140	0.700700	+	0.713456i	$0.252871\pi$
$450$	0	0
$451$	−1.73156	−0.0815362
$452$	−0.495000	−0.0232829
$453$	0	0
$454$	38.1442	1.79020
$455$	12.5769	0.589612
$456$	0	0
$457$	−4.89861	−0.229147	−0.114574	−	0.993415i	$-0.536550\pi$
$457$	−4.89861	−0.229147	−0.114574	+	0.993415i	$0.536550\pi$
$458$	1.85325	0.0865966
$459$	0	0
$460$	−0.550861	−0.0256840
$461$	−23.4383	−1.09163	−0.545815	−	0.837906i	$-0.683780\pi$
$461$	−23.4383	−1.09163	−0.545815	+	0.837906i	$0.683780\pi$
$462$	0	0
$463$	−26.7194	−1.24175	−0.620877	−	0.783908i	$-0.713223\pi$
$463$	−26.7194	−1.24175	−0.620877	+	0.783908i	$0.713223\pi$
$464$	19.5182	0.906108
$465$	0	0
$466$	11.9708	0.554539
$467$	−39.3344	−1.82018	−0.910090	−	0.414410i	$-0.863988\pi$
$467$	−39.3344	−1.82018	−0.910090	+	0.414410i	$0.863988\pi$
$468$	0	0
$469$	−14.2373	−0.657417
$470$	5.06649	0.233700
$471$	0	0
$472$	17.0387	0.784270
$473$	5.78694	0.266084
$474$	0	0
$475$	−8.28179	−0.379995
$476$	−0.223314	−0.0102356
$477$	0	0
$478$	−25.2700	−1.15582
$479$	34.9057	1.59488	0.797442	−	0.603396i	$-0.206186\pi$
$479$	34.9057	1.59488	0.797442	+	0.603396i	$0.206186\pi$
$480$	0	0
$481$	−9.08576	−0.414275
$482$	−10.7264	−0.488572
$483$	0	0
$484$	0.658011	0.0299096
$485$	39.2226	1.78101
$486$	0	0
$487$	−26.0576	−1.18078	−0.590392	−	0.807117i	$-0.701027\pi$
$487$	−26.0576	−1.18078	−0.590392	+	0.807117i	$0.701027\pi$
$488$	30.0075	1.35838
$489$	0	0
$490$	−30.1434	−1.36174
$491$	−15.0991	−0.681412	−0.340706	−	0.940170i	$-0.610666\pi$
$491$	−15.0991	−0.681412	−0.340706	+	0.940170i	$0.610666\pi$
$492$	0	0
$493$	16.3600	0.736817
$494$	4.68097	0.210607
$495$	0	0
$496$	−11.3999	−0.511872
$497$	−4.67768	−0.209822
$498$	0	0
$499$	14.1658	0.634146	0.317073	−	0.948401i	$-0.397300\pi$
$499$	14.1658	0.634146	0.317073	+	0.948401i	$0.397300\pi$
$500$	0.805279	0.0360132
$501$	0	0
$502$	−5.75805	−0.256994
$503$	17.6324	0.786189	0.393095	−	0.919498i	$-0.371404\pi$
$503$	17.6324	0.786189	0.393095	+	0.919498i	$0.371404\pi$
$504$	0	0
$505$	−2.92365	−0.130101
$506$	3.45712	0.153688
$507$	0	0
$508$	0.131769	0.00584628
$509$	24.2243	1.07373	0.536863	−	0.843670i	$-0.319609\pi$
$509$	24.2243	1.07373	0.536863	+	0.843670i	$0.319609\pi$
$510$	0	0
$511$	15.1528	0.670322
$512$	23.6619	1.04572
$513$	0	0
$514$	−11.8586	−0.523058
$515$	0.592407	0.0261046
$516$	0	0
$517$	1.10772	0.0487174
$518$	−3.84477	−0.168930
$519$	0	0
$520$	−35.2674	−1.54658
$521$	17.3060	0.758192	0.379096	−	0.925357i	$-0.376235\pi$
$521$	17.3060	0.758192	0.379096	+	0.925357i	$0.376235\pi$
$522$	0	0
$523$	−24.3922	−1.06660	−0.533298	−	0.845927i	$-0.679048\pi$
$523$	−24.3922	−1.06660	−0.533298	+	0.845927i	$0.679048\pi$
$524$	−0.653523	−0.0285493
$525$	0	0
$526$	36.5259	1.59261
$527$	−9.55534	−0.416237
$528$	0	0
$529$	−17.9602	−0.780878
$530$	−43.8741	−1.90577
$531$	0	0
$532$	−0.0690072	−0.00299184
$533$	−5.26340	−0.227983
$534$	0	0
$535$	9.23261	0.399161
$536$	39.9236	1.72444
$537$	0	0
$538$	26.1318	1.12662
$539$	−6.59043	−0.283870
$540$	0	0
$541$	19.9174	0.856315	0.428157	−	0.903704i	$-0.359163\pi$
$541$	19.9174	0.856315	0.428157	+	0.903704i	$0.359163\pi$
$542$	14.2840	0.613552
$543$	0	0
$544$	1.23202	0.0528224
$545$	−19.7251	−0.844930
$546$	0	0
$547$	15.6323	0.668389	0.334195	−	0.942504i	$-0.391536\pi$
$547$	15.6323	0.668389	0.334195	+	0.942504i	$0.391536\pi$
$548$	0.592795	0.0253230
$549$	0	0
$550$	−12.7536	−0.543815
$551$	5.05546	0.215370
$552$	0	0
$553$	3.99282	0.169792
$554$	−10.5409	−0.447839
$555$	0	0
$556$	−0.747741	−0.0317113
$557$	−25.6450	−1.08661	−0.543306	−	0.839535i	$-0.682828\pi$
$557$	−25.6450	−1.08661	−0.543306	+	0.839535i	$0.682828\pi$
$558$	0	0
$559$	17.5904	0.743997
$560$	−14.4209	−0.609396
$561$	0	0
$562$	5.36728	0.226405
$563$	18.1626	0.765463	0.382731	−	0.923860i	$-0.374983\pi$
$563$	18.1626	0.765463	0.382731	+	0.923860i	$0.374983\pi$
$564$	0	0
$565$	−26.7933	−1.12720
$566$	−7.84701	−0.329835
$567$	0	0
$568$	13.1169	0.550374
$569$	−2.61888	−0.109789	−0.0548945	−	0.998492i	$-0.517482\pi$
$569$	−2.61888	−0.109789	−0.0548945	+	0.998492i	$0.517482\pi$
$570$	0	0
$571$	−17.8098	−0.745316	−0.372658	−	0.927969i	$-0.621554\pi$
$571$	−17.8098	−0.745316	−0.372658	+	0.927969i	$0.621554\pi$
$572$	−0.251127	−0.0105001
$573$	0	0
$574$	−2.22728	−0.0929650
$575$	−18.5922	−0.775349
$576$	0	0
$577$	−41.5845	−1.73119	−0.865594	−	0.500747i	$-0.833059\pi$
$577$	−41.5845	−1.73119	−0.865594	+	0.500747i	$0.833059\pi$
$578$	9.07475	0.377460
$579$	0	0
$580$	−1.24050	−0.0515089
$581$	−16.6672	−0.691470
$582$	0	0
$583$	−9.59245	−0.397279
$584$	−42.4909	−1.75828
$585$	0	0
$586$	−21.5508	−0.890255
$587$	−22.8786	−0.944300	−0.472150	−	0.881518i	$-0.656522\pi$
$587$	−22.8786	−0.944300	−0.472150	+	0.881518i	$0.656522\pi$
$588$	0	0
$589$	−2.95273	−0.121665
$590$	30.0369	1.23660
$591$	0	0
$592$	10.4180	0.428176
$593$	−35.4185	−1.45446	−0.727232	−	0.686391i	$-0.759194\pi$
$593$	−35.4185	−1.45446	−0.727232	+	0.686391i	$0.759194\pi$
$594$	0	0
$595$	−12.0875	−0.495541
$596$	0.636202	0.0260599
$597$	0	0
$598$	10.5085	0.429726
$599$	35.3667	1.44505	0.722523	−	0.691347i	$-0.242982\pi$
$599$	35.3667	1.44505	0.722523	+	0.691347i	$0.242982\pi$
$600$	0	0
$601$	−13.8766	−0.566038	−0.283019	−	0.959114i	$-0.591336\pi$
$601$	−13.8766	−0.566038	−0.283019	+	0.959114i	$0.591336\pi$
$602$	7.44365	0.303380
$603$	0	0
$604$	0.536652	0.0218360
$605$	35.6168	1.44803
$606$	0	0
$607$	−18.5871	−0.754428	−0.377214	−	0.926126i	$-0.623118\pi$
$607$	−18.5871	−0.754428	−0.377214	+	0.926126i	$0.623118\pi$
$608$	0.380711	0.0154399
$609$	0	0
$610$	52.8992	2.14182
$611$	3.36711	0.136219
$612$	0	0
$613$	−22.0621	−0.891079	−0.445539	−	0.895262i	$-0.646988\pi$
$613$	−22.0621	−0.891079	−0.445539	+	0.895262i	$0.646988\pi$
$614$	19.2099	0.775250
$615$	0	0
$616$	−3.26291	−0.131466
$617$	7.92832	0.319182	0.159591	−	0.987183i	$-0.448982\pi$
$617$	7.92832	0.319182	0.159591	+	0.987183i	$0.448982\pi$
$618$	0	0
$619$	−29.5327	−1.18702	−0.593510	−	0.804827i	$-0.702258\pi$
$619$	−29.5327	−1.18702	−0.593510	+	0.804827i	$0.702258\pi$
$620$	0.724535	0.0290980
$621$	0	0
$622$	15.9310	0.638775
$623$	0.174977	0.00701029
$624$	0	0
$625$	2.17911	0.0871642
$626$	13.3412	0.533223
$627$	0	0
$628$	−0.229611	−0.00916248
$629$	8.73227	0.348178
$630$	0	0
$631$	25.7928	1.02680	0.513398	−	0.858150i	$-0.328386\pi$
$631$	25.7928	1.02680	0.513398	+	0.858150i	$0.328386\pi$
$632$	−11.1965	−0.445371
$633$	0	0
$634$	25.3705	1.00759
$635$	7.13235	0.283039
$636$	0	0
$637$	−20.0328	−0.793728
$638$	7.78518	0.308218
$639$	0	0
$640$	39.0281	1.54272
$641$	15.6059	0.616397	0.308198	−	0.951322i	$-0.400274\pi$
$641$	15.6059	0.616397	0.308198	+	0.951322i	$0.400274\pi$
$642$	0	0
$643$	−2.42098	−0.0954742	−0.0477371	−	0.998860i	$-0.515201\pi$
$643$	−2.42098	−0.0954742	−0.0477371	+	0.998860i	$0.515201\pi$
$644$	−0.154918	−0.00610461
$645$	0	0
$646$	−4.49885	−0.177005
$647$	−14.5497	−0.572007	−0.286004	−	0.958229i	$-0.592327\pi$
$647$	−14.5497	−0.572007	−0.286004	+	0.958229i	$0.592327\pi$
$648$	0	0
$649$	6.56715	0.257783
$650$	−38.7668	−1.52056
$651$	0	0
$652$	−0.0802823	−0.00314410
$653$	38.3476	1.50066	0.750329	−	0.661064i	$-0.229895\pi$
$653$	38.3476	1.50066	0.750329	+	0.661064i	$0.229895\pi$
$654$	0	0
$655$	−35.3738	−1.38217
$656$	6.03514	0.235633
$657$	0	0
$658$	1.42484	0.0555460
$659$	26.0619	1.01523	0.507613	−	0.861585i	$-0.330528\pi$
$659$	26.0619	1.01523	0.507613	+	0.861585i	$0.330528\pi$
$660$	0	0
$661$	−36.5495	−1.42161	−0.710806	−	0.703388i	$-0.751669\pi$
$661$	−36.5495	−1.42161	−0.710806	+	0.703388i	$0.751669\pi$
$662$	33.9743	1.32045
$663$	0	0
$664$	46.7373	1.81376
$665$	−3.73521	−0.144845
$666$	0	0
$667$	11.3493	0.439445
$668$	−1.68869	−0.0653375
$669$	0	0
$670$	70.3798	2.71901
$671$	11.5657	0.446488
$672$	0	0
$673$	28.0954	1.08300	0.541500	−	0.840701i	$-0.317857\pi$
$673$	28.0954	1.08300	0.541500	+	0.840701i	$0.317857\pi$
$674$	−9.59067	−0.369419
$675$	0	0
$676$	0.111942	0.00430547
$677$	−21.7972	−0.837735	−0.418868	−	0.908047i	$-0.637573\pi$
$677$	−21.7972	−0.837735	−0.418868	+	0.908047i	$0.637573\pi$
$678$	0	0
$679$	11.0305	0.423312
$680$	33.8953	1.29983
$681$	0	0
$682$	−4.54707	−0.174116
$683$	−21.6109	−0.826920	−0.413460	−	0.910522i	$-0.635680\pi$
$683$	−21.6109	−0.826920	−0.413460	+	0.910522i	$0.635680\pi$
$684$	0	0
$685$	32.0868	1.22597
$686$	−18.4510	−0.704464
$687$	0	0
$688$	−20.1696	−0.768960
$689$	−29.1580	−1.11083
$690$	0	0
$691$	10.0624	0.382790	0.191395	−	0.981513i	$-0.438699\pi$
$691$	10.0624	0.382790	0.191395	+	0.981513i	$0.438699\pi$
$692$	−1.47243	−0.0559734
$693$	0	0
$694$	−40.2436	−1.52763
$695$	−40.4736	−1.53525
$696$	0	0
$697$	5.05862	0.191609
$698$	44.2732	1.67576
$699$	0	0
$700$	0.571503	0.0216008
$701$	−12.6782	−0.478850	−0.239425	−	0.970915i	$-0.576959\pi$
$701$	−12.6782	−0.478850	−0.239425	+	0.970915i	$0.576959\pi$
$702$	0	0
$703$	2.69839	0.101772
$704$	9.13965	0.344463
$705$	0	0
$706$	26.1736	0.985056
$707$	−0.822211	−0.0309224
$708$	0	0
$709$	29.7837	1.11855	0.559275	−	0.828982i	$-0.311080\pi$
$709$	29.7837	1.11855	0.559275	+	0.828982i	$0.311080\pi$
$710$	23.1233	0.867803
$711$	0	0
$712$	−0.490661	−0.0183883
$713$	−6.62873	−0.248248
$714$	0	0
$715$	−13.5930	−0.508348
$716$	−1.63408	−0.0610685
$717$	0	0
$718$	17.1989	0.641856
$719$	−41.2951	−1.54005	−0.770024	−	0.638015i	$-0.779756\pi$
$719$	−41.2951	−1.54005	−0.770024	+	0.638015i	$0.779756\pi$
$720$	0	0
$721$	0.166602	0.00620457
$722$	−1.39021	−0.0517381
$723$	0	0
$724$	−1.52942	−0.0568403
$725$	−41.8683	−1.55495
$726$	0	0
$727$	12.1465	0.450488	0.225244	−	0.974302i	$-0.427682\pi$
$727$	12.1465	0.450488	0.225244	+	0.974302i	$0.427682\pi$
$728$	−9.91819	−0.367593
$729$	0	0
$730$	−74.9056	−2.77238
$731$	−16.9061	−0.625293
$732$	0	0
$733$	17.6996	0.653750	0.326875	−	0.945068i	$-0.394004\pi$
$733$	17.6996	0.653750	0.326875	+	0.945068i	$0.394004\pi$
$734$	−12.8484	−0.474241
$735$	0	0
$736$	0.854677	0.0315038
$737$	15.3876	0.566808
$738$	0	0
$739$	23.9128	0.879647	0.439824	−	0.898084i	$-0.355041\pi$
$739$	23.9128	0.879647	0.439824	+	0.898084i	$0.355041\pi$
$740$	−0.662125	−0.0243402
$741$	0	0
$742$	−12.3386	−0.452965
$743$	6.71168	0.246228	0.123114	−	0.992393i	$-0.460712\pi$
$743$	6.71168	0.246228	0.123114	+	0.992393i	$0.460712\pi$
$744$	0	0
$745$	34.4363	1.26165
$746$	−22.0891	−0.808739
$747$	0	0
$748$	0.241356	0.00882486
$749$	2.59647	0.0948729
$750$	0	0
$751$	18.1977	0.664043	0.332021	−	0.943272i	$-0.392269\pi$
$751$	18.1977	0.664043	0.332021	+	0.943272i	$0.392269\pi$
$752$	−3.86081	−0.140789
$753$	0	0
$754$	23.6645	0.861809
$755$	29.0478	1.05716
$756$	0	0
$757$	−11.6433	−0.423182	−0.211591	−	0.977358i	$-0.567864\pi$
$757$	−11.6433	−0.423182	−0.211591	+	0.977358i	$0.567864\pi$
$758$	36.3607	1.32068
$759$	0	0
$760$	10.4741	0.379936
$761$	−27.5735	−0.999538	−0.499769	−	0.866159i	$-0.666582\pi$
$761$	−27.5735	−0.999538	−0.499769	+	0.866159i	$0.666582\pi$
$762$	0	0
$763$	−5.54725	−0.200824
$764$	−1.30798	−0.0473211
$765$	0	0
$766$	−24.4364	−0.882921
$767$	19.9620	0.720787
$768$	0	0
$769$	−5.02958	−0.181371	−0.0906857	−	0.995880i	$-0.528906\pi$
$769$	−5.02958	−0.181371	−0.0906857	+	0.995880i	$0.528906\pi$
$770$	−5.75206	−0.207290
$771$	0	0
$772$	0.0628953	0.00226365
$773$	1.97455	0.0710195	0.0355098	−	0.999369i	$-0.488695\pi$
$773$	1.97455	0.0710195	0.0355098	+	0.999369i	$0.488695\pi$
$774$	0	0
$775$	24.4539	0.878410
$776$	−30.9312	−1.11037
$777$	0	0
$778$	49.2347	1.76515
$779$	1.56318	0.0560068
$780$	0	0
$781$	5.05560	0.180903
$782$	−10.0997	−0.361164
$783$	0	0
$784$	22.9701	0.820360
$785$	−12.4284	−0.443587
$786$	0	0
$787$	−6.84511	−0.244002	−0.122001	−	0.992530i	$-0.538931\pi$
$787$	−6.84511	−0.244002	−0.122001	+	0.992530i	$0.538931\pi$
$788$	−0.127892	−0.00455598
$789$	0	0
$790$	−19.7378	−0.702241
$791$	−7.53503	−0.267915
$792$	0	0
$793$	35.1559	1.24842
$794$	39.1246	1.38848
$795$	0	0
$796$	0.136240	0.00482889
$797$	−38.2638	−1.35537	−0.677687	−	0.735350i	$-0.737018\pi$
$797$	−38.2638	−1.35537	−0.677687	+	0.735350i	$0.737018\pi$
$798$	0	0
$799$	−3.23610	−0.114485
$800$	−3.15297	−0.111474
$801$	0	0
$802$	16.5166	0.583221
$803$	−16.3771	−0.577934
$804$	0	0
$805$	−8.38537	−0.295545
$806$	−13.8216	−0.486846
$807$	0	0
$808$	2.30561	0.0811110
$809$	12.0203	0.422610	0.211305	−	0.977420i	$-0.432229\pi$
$809$	12.0203	0.422610	0.211305	+	0.977420i	$0.432229\pi$
$810$	0	0
$811$	6.94590	0.243904	0.121952	−	0.992536i	$-0.461085\pi$
$811$	6.94590	0.243904	0.121952	+	0.992536i	$0.461085\pi$
$812$	−0.348863	−0.0122427
$813$	0	0
$814$	4.15540	0.145647
$815$	−4.34551	−0.152217
$816$	0	0
$817$	−5.22420	−0.182772
$818$	12.6488	0.442255
$819$	0	0
$820$	−0.383570	−0.0133949
$821$	28.6889	1.00125	0.500625	−	0.865664i	$-0.333104\pi$
$821$	28.6889	1.00125	0.500625	+	0.865664i	$0.333104\pi$
$822$	0	0
$823$	16.9137	0.589575	0.294787	−	0.955563i	$-0.404751\pi$
$823$	16.9137	0.589575	0.294787	+	0.955563i	$0.404751\pi$
$824$	−0.467176	−0.0162749
$825$	0	0
$826$	8.44722	0.293916
$827$	2.43315	0.0846090	0.0423045	−	0.999105i	$-0.486530\pi$
$827$	2.43315	0.0846090	0.0423045	+	0.999105i	$0.486530\pi$
$828$	0	0
$829$	30.4290	1.05684	0.528422	−	0.848982i	$-0.322784\pi$
$829$	30.4290	1.05684	0.528422	+	0.848982i	$0.322784\pi$
$830$	82.3914	2.85985
$831$	0	0
$832$	27.7816	0.963153
$833$	19.2534	0.667090
$834$	0	0
$835$	−91.4054	−3.16321
$836$	0.0745824	0.00257949
$837$	0	0
$838$	9.67632	0.334263
$839$	3.35684	0.115891	0.0579455	−	0.998320i	$-0.481545\pi$
$839$	3.35684	0.115891	0.0579455	+	0.998320i	$0.481545\pi$
$840$	0	0
$841$	−3.44229	−0.118700
$842$	−26.8002	−0.923596
$843$	0	0
$844$	1.25805	0.0433039
$845$	6.05920	0.208443
$846$	0	0
$847$	10.0164	0.344169
$848$	33.4333	1.14810
$849$	0	0
$850$	37.2585	1.27796
$851$	6.05775	0.207657
$852$	0	0
$853$	20.7164	0.709316	0.354658	−	0.934996i	$-0.384597\pi$
$853$	20.7164	0.709316	0.354658	+	0.934996i	$0.384597\pi$
$854$	14.8767	0.509071
$855$	0	0
$856$	−7.28090	−0.248856
$857$	−10.8765	−0.371534	−0.185767	−	0.982594i	$-0.559477\pi$
$857$	−10.8765	−0.371534	−0.185767	+	0.982594i	$0.559477\pi$
$858$	0	0
$859$	26.7914	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$859$	26.7914	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$860$	1.28190	0.0437126
$861$	0	0
$862$	16.7578	0.570773
$863$	−12.9550	−0.440994	−0.220497	−	0.975388i	$-0.570768\pi$
$863$	−12.9550	−0.440994	−0.220497	+	0.975388i	$0.570768\pi$
$864$	0	0
$865$	−79.6996	−2.70987
$866$	−2.64882	−0.0900105
$867$	0	0
$868$	0.203760	0.00691605
$869$	−4.31540	−0.146390
$870$	0	0
$871$	46.7732	1.58485
$872$	15.5553	0.526770
$873$	0	0
$874$	−3.12094	−0.105567
$875$	12.2582	0.414402
$876$	0	0
$877$	0.509928	0.0172190	0.00860952	−	0.999963i	$-0.497259\pi$
$877$	0.509928	0.0172190	0.00860952	+	0.999963i	$0.497259\pi$
$878$	22.8155	0.769985
$879$	0	0
$880$	15.5860	0.525405
$881$	−20.3989	−0.687256	−0.343628	−	0.939106i	$-0.611656\pi$
$881$	−20.3989	−0.687256	−0.343628	+	0.939106i	$0.611656\pi$
$882$	0	0
$883$	26.8989	0.905220	0.452610	−	0.891709i	$-0.350493\pi$
$883$	26.8989	0.905220	0.452610	+	0.891709i	$0.350493\pi$
$884$	0.733646	0.0246752
$885$	0	0
$886$	17.3815	0.583943
$887$	8.88246	0.298244	0.149122	−	0.988819i	$-0.452355\pi$
$887$	8.88246	0.298244	0.149122	+	0.988819i	$0.452355\pi$
$888$	0	0
$889$	2.00582	0.0672730
$890$	−0.864968	−0.0289938
$891$	0	0
$892$	−1.70568	−0.0571102
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−88.4494	−2.95654
$896$	10.9758	0.366675
$897$	0	0
$898$	−41.2823	−1.37761
$899$	−14.9274	−0.497857
$900$	0	0
$901$	28.0235	0.933600
$902$	2.40723	0.0801519
$903$	0	0
$904$	21.1294	0.702753
$905$	−82.7841	−2.75184
$906$	0	0
$907$	1.12312	0.0372925	0.0186462	−	0.999826i	$-0.494064\pi$
$907$	1.12312	0.0372925	0.0186462	+	0.999826i	$0.494064\pi$
$908$	1.84738	0.0613076
$909$	0	0
$910$	−17.4844	−0.579603
$911$	−37.8695	−1.25467	−0.627337	−	0.778748i	$-0.715855\pi$
$911$	−37.8695	−1.25467	−0.627337	+	0.778748i	$0.715855\pi$
$912$	0	0
$913$	18.0137	0.596167
$914$	6.81008	0.225257
$915$	0	0
$916$	0.0897557	0.00296561
$917$	−9.94812	−0.328516
$918$	0	0
$919$	−22.5469	−0.743754	−0.371877	−	0.928282i	$-0.621286\pi$
$919$	−22.5469	−0.743754	−0.371877	+	0.928282i	$0.621286\pi$
$920$	23.5138	0.775229
$921$	0	0
$922$	32.5840	1.07310
$923$	15.3674	0.505824
$924$	0	0
$925$	−22.3475	−0.734781
$926$	37.1454	1.22067
$927$	0	0
$928$	1.92467	0.0631804
$929$	37.3263	1.22464	0.612319	−	0.790611i	$-0.290237\pi$
$929$	37.3263	1.22464	0.612319	+	0.790611i	$0.290237\pi$
$930$	0	0
$931$	5.94955	0.194989
$932$	0.579766	0.0189909
$933$	0	0
$934$	54.6829	1.78928
$935$	13.0641	0.427242
$936$	0	0
$937$	−7.77109	−0.253871	−0.126935	−	0.991911i	$-0.540514\pi$
$937$	−7.77109	−0.253871	−0.126935	+	0.991911i	$0.540514\pi$
$938$	19.7928	0.646257
$939$	0	0
$940$	0.245378	0.00800335
$941$	−21.9348	−0.715054	−0.357527	−	0.933903i	$-0.616380\pi$
$941$	−21.9348	−0.715054	−0.357527	+	0.933903i	$0.616380\pi$
$942$	0	0
$943$	3.50926	0.114277
$944$	−22.8890	−0.744972
$945$	0	0
$946$	−8.04504	−0.261567
$947$	−10.1569	−0.330055	−0.165027	−	0.986289i	$-0.552771\pi$
$947$	−10.1569	−0.330055	−0.165027	+	0.986289i	$0.552771\pi$
$948$	0	0
$949$	−49.7810	−1.61596
$950$	11.5134	0.373544
$951$	0	0
$952$	9.53231	0.308944
$953$	23.6904	0.767408	0.383704	−	0.923456i	$-0.374648\pi$
$953$	23.6904	0.767408	0.383704	+	0.923456i	$0.374648\pi$
$954$	0	0
$955$	−70.7982	−2.29098
$956$	−1.22386	−0.0395826
$957$	0	0
$958$	−48.5261	−1.56781
$959$	9.02370	0.291390
$960$	0	0
$961$	−22.2814	−0.718754
$962$	12.6311	0.407242
$963$	0	0
$964$	−0.519494	−0.0167318
$965$	3.40439	0.109591
$966$	0	0
$967$	−8.62134	−0.277244	−0.138622	−	0.990345i	$-0.544267\pi$
$967$	−8.62134	−0.277244	−0.138622	+	0.990345i	$0.544267\pi$
$968$	−28.0876	−0.902770
$969$	0	0
$970$	−54.5275	−1.75077
$971$	−57.8120	−1.85528	−0.927638	−	0.373481i	$-0.878164\pi$
$971$	−57.8120	−1.85528	−0.927638	+	0.373481i	$0.878164\pi$
$972$	0	0
$973$	−11.3823	−0.364900
$974$	36.2254	1.16074
$975$	0	0
$976$	−40.3106	−1.29031
$977$	−3.21341	−0.102806	−0.0514030	−	0.998678i	$-0.516369\pi$
$977$	−3.21341	−0.102806	−0.0514030	+	0.998678i	$0.516369\pi$
$978$	0	0
$979$	−0.189113	−0.00604409
$980$	−1.45989	−0.0466344
$981$	0	0
$982$	20.9908	0.669844
$983$	19.2889	0.615221	0.307611	−	0.951512i	$-0.400471\pi$
$983$	19.2889	0.615221	0.307611	+	0.951512i	$0.400471\pi$
$984$	0	0
$985$	−6.92254	−0.220571
$986$	−22.7438	−0.724309
$987$	0	0
$988$	0.226707	0.00721249
$989$	−11.7281	−0.372931
$990$	0	0
$991$	42.8655	1.36167	0.680834	−	0.732438i	$-0.261618\pi$
$991$	42.8655	1.36167	0.680834	+	0.732438i	$0.261618\pi$
$992$	−1.12414	−0.0356914
$993$	0	0
$994$	6.50293	0.206260
$995$	7.37437	0.233783
$996$	0	0
$997$	25.3269	0.802109	0.401055	−	0.916054i	$-0.368644\pi$
$997$	25.3269	0.802109	0.401055	+	0.916054i	$0.368644\pi$
$998$	−19.6933	−0.623381
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.s.1.9		24
3.2	odd	2		2679.2.a.p.1.16	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.p.1.16	✓	24	3.2	odd	2
8037.2.a.s.1.9		24	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)

\(f(q)\)	\(=\)	\(q-1.39021 q^{2} -0.0673298 q^{4} -3.64442 q^{5} -1.02491 q^{7} +2.87401 q^{8} +O(q^{10})\)	\(q-1.39021 q^{2} -0.0673298 q^{4} -3.64442 q^{5} -1.02491 q^{7} +2.87401 q^{8} +5.06649 q^{10} +1.10772 q^{11} +3.36711 q^{13} +1.42484 q^{14} -3.86081 q^{16} -3.23610 q^{17} -1.00000 q^{19} +0.245378 q^{20} -1.53995 q^{22} -2.24495 q^{23} +8.28179 q^{25} -4.68097 q^{26} +0.0690072 q^{28} -5.05546 q^{29} +2.95273 q^{31} -0.380711 q^{32} +4.49885 q^{34} +3.73521 q^{35} -2.69839 q^{37} +1.39021 q^{38} -10.4741 q^{40} -1.56318 q^{41} +5.22420 q^{43} -0.0745824 q^{44} +3.12094 q^{46} +1.00000 q^{47} -5.94955 q^{49} -11.5134 q^{50} -0.226707 q^{52} -8.65966 q^{53} -4.03699 q^{55} -2.94561 q^{56} +7.02813 q^{58} +5.92854 q^{59} +10.4410 q^{61} -4.10490 q^{62} +8.25088 q^{64} -12.2711 q^{65} +13.8912 q^{67} +0.217886 q^{68} -5.19271 q^{70} +4.56398 q^{71} -14.7845 q^{73} +3.75131 q^{74} +0.0673298 q^{76} -1.13531 q^{77} -3.89576 q^{79} +14.0704 q^{80} +2.17314 q^{82} +16.2620 q^{83} +11.7937 q^{85} -7.26271 q^{86} +3.18359 q^{88} -0.170723 q^{89} -3.45099 q^{91} +0.151152 q^{92} -1.39021 q^{94} +3.64442 q^{95} -10.7624 q^{97} +8.27110 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} + 32 q^{4} - 10 q^{5} + 6 q^{7} - 15 q^{8}+O(q^{10}) \) 24 * q - 6 * q^2 + 32 * q^4 - 10 * q^5 + 6 * q^7 - 15 * q^8	\( 24 q - 6 q^{2} + 32 q^{4} - 10 q^{5} + 6 q^{7} - 15 q^{8} + 5 q^{10} - 15 q^{11} - 3 q^{13} - 27 q^{14} + 48 q^{16} - 22 q^{17} - 24 q^{19} - 29 q^{20} - 9 q^{22} - 15 q^{23} + 58 q^{25} - q^{26} + 18 q^{28} - 36 q^{29} - 16 q^{31} - 26 q^{32} + 10 q^{34} - 37 q^{35} + 18 q^{37} + 6 q^{38} + 7 q^{40} - 24 q^{41} + 2 q^{43} - 57 q^{44} - 23 q^{46} + 24 q^{47} + 64 q^{49} - 4 q^{50} - 40 q^{52} - 42 q^{53} + 8 q^{55} - 38 q^{56} - 5 q^{58} - 39 q^{59} + 6 q^{61} + 14 q^{62} + 53 q^{64} - 47 q^{65} - 26 q^{67} - 80 q^{68} - 46 q^{70} - 6 q^{71} + 3 q^{73} - 62 q^{74} - 32 q^{76} - 13 q^{77} + 10 q^{79} - 17 q^{80} + 15 q^{82} - 44 q^{83} + 21 q^{85} + 22 q^{86} - 88 q^{88} - 32 q^{89} - 17 q^{91} + 37 q^{92} - 6 q^{94} + 10 q^{95} + 28 q^{97} - 44 q^{98}+O(q^{100}) \) 24 * q - 6 * q^2 + 32 * q^4 - 10 * q^5 + 6 * q^7 - 15 * q^8 + 5 * q^10 - 15 * q^11 - 3 * q^13 - 27 * q^14 + 48 * q^16 - 22 * q^17 - 24 * q^19 - 29 * q^20 - 9 * q^22 - 15 * q^23 + 58 * q^25 - q^26 + 18 * q^28 - 36 * q^29 - 16 * q^31 - 26 * q^32 + 10 * q^34 - 37 * q^35 + 18 * q^37 + 6 * q^38 + 7 * q^40 - 24 * q^41 + 2 * q^43 - 57 * q^44 - 23 * q^46 + 24 * q^47 + 64 * q^49 - 4 * q^50 - 40 * q^52 - 42 * q^53 + 8 * q^55 - 38 * q^56 - 5 * q^58 - 39 * q^59 + 6 * q^61 + 14 * q^62 + 53 * q^64 - 47 * q^65 - 26 * q^67 - 80 * q^68 - 46 * q^70 - 6 * q^71 + 3 * q^73 - 62 * q^74 - 32 * q^76 - 13 * q^77 + 10 * q^79 - 17 * q^80 + 15 * q^82 - 44 * q^83 + 21 * q^85 + 22 * q^86 - 88 * q^88 - 32 * q^89 - 17 * q^91 + 37 * q^92 - 6 * q^94 + 10 * q^95 + 28 * q^97 - 44 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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