LMFDB - Embedded newform 8037.2.a.w.1.17

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:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q+0.0583690 q^{2} -1.99659 q^{4} -3.92229 q^{5} -2.69203 q^{7} -0.233277 q^{8} +O(q^{10})$	\(q+0.0583690 q^{2} -1.99659 q^{4} -3.92229 q^{5} -2.69203 q^{7} -0.233277 q^{8} -0.228940 q^{10} +3.25421 q^{11} +0.958204 q^{13} -0.157131 q^{14} +3.97957 q^{16} +1.23868 q^{17} -1.00000 q^{19} +7.83122 q^{20} +0.189945 q^{22} -3.54584 q^{23} +10.3844 q^{25} +0.0559294 q^{26} +5.37489 q^{28} -4.36399 q^{29} -9.98657 q^{31} +0.698837 q^{32} +0.0723005 q^{34} +10.5589 q^{35} -7.14715 q^{37} -0.0583690 q^{38} +0.914980 q^{40} -5.14995 q^{41} +2.11808 q^{43} -6.49732 q^{44} -0.206967 q^{46} +1.00000 q^{47} +0.247029 q^{49} +0.606125 q^{50} -1.91314 q^{52} -7.53771 q^{53} -12.7639 q^{55} +0.627989 q^{56} -0.254722 q^{58} -9.08586 q^{59} -2.24461 q^{61} -0.582906 q^{62} -7.91835 q^{64} -3.75835 q^{65} -5.22314 q^{67} -2.47314 q^{68} +0.616314 q^{70} +7.28964 q^{71} +3.19834 q^{73} -0.417172 q^{74} +1.99659 q^{76} -8.76042 q^{77} -4.17752 q^{79} -15.6090 q^{80} -0.300597 q^{82} -5.94677 q^{83} -4.85847 q^{85} +0.123630 q^{86} -0.759131 q^{88} +2.41841 q^{89} -2.57951 q^{91} +7.07961 q^{92} +0.0583690 q^{94} +3.92229 q^{95} -1.85603 q^{97} +0.0144188 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.0583690	0.0412731	0.0206365	−	0.999787i	$-0.493431\pi$
$2$	0.0583690	0.0412731	0.0206365	+	0.999787i	$0.493431\pi$
$3$	0	0
$3$	0	0
$4$	−1.99659	−0.998297
$5$	−3.92229	−1.75410	−0.877051	−	0.480398i	$-0.840492\pi$
$5$	−3.92229	−1.75410	−0.877051	+	0.480398i	$0.840492\pi$
$6$	0	0
$7$	−2.69203	−1.01749	−0.508746	−	0.860917i	$-0.669891\pi$
$7$	−2.69203	−1.01749	−0.508746	+	0.860917i	$0.669891\pi$
$8$	−0.233277	−0.0824759
$9$	0	0
$10$	−0.228940	−0.0723972
$11$	3.25421	0.981180	0.490590	−	0.871391i	$-0.336781\pi$
$11$	3.25421	0.981180	0.490590	+	0.871391i	$0.336781\pi$
$12$	0	0
$13$	0.958204	0.265758	0.132879	−	0.991132i	$-0.457578\pi$
$13$	0.958204	0.265758	0.132879	+	0.991132i	$0.457578\pi$
$14$	−0.157131	−0.0419950
$15$	0	0
$16$	3.97957	0.994892
$17$	1.23868	0.300424	0.150212	−	0.988654i	$-0.452004\pi$
$17$	1.23868	0.300424	0.150212	+	0.988654i	$0.452004\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	7.83122	1.75111
$21$	0	0
$22$	0.189945	0.0404963
$23$	−3.54584	−0.739360	−0.369680	−	0.929159i	$-0.620533\pi$
$23$	−3.54584	−0.739360	−0.369680	+	0.929159i	$0.620533\pi$
$24$	0	0
$25$	10.3844	2.07687
$26$	0.0559294	0.0109687
$27$	0	0
$28$	5.37489	1.01576
$29$	−4.36399	−0.810373	−0.405186	−	0.914234i	$-0.632793\pi$
$29$	−4.36399	−0.810373	−0.405186	+	0.914234i	$0.632793\pi$
$30$	0	0
$31$	−9.98657	−1.79364	−0.896820	−	0.442395i	$-0.854129\pi$
$31$	−9.98657	−1.79364	−0.896820	+	0.442395i	$0.854129\pi$
$32$	0.698837	0.123538
$33$	0	0
$34$	0.0723005	0.0123994
$35$	10.5589	1.78478
$36$	0	0
$37$	−7.14715	−1.17498	−0.587492	−	0.809230i	$-0.699885\pi$
$37$	−7.14715	−1.17498	−0.587492	+	0.809230i	$0.699885\pi$
$38$	−0.0583690	−0.00946870
$39$	0	0
$40$	0.914980	0.144671
$41$	−5.14995	−0.804287	−0.402143	−	0.915577i	$-0.631735\pi$
$41$	−5.14995	−0.804287	−0.402143	+	0.915577i	$0.631735\pi$
$42$	0	0
$43$	2.11808	0.323005	0.161502	−	0.986872i	$-0.448366\pi$
$43$	2.11808	0.323005	0.161502	+	0.986872i	$0.448366\pi$
$44$	−6.49732	−0.979509
$45$	0	0
$46$	−0.206967	−0.0305157
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	0.247029	0.0352899
$50$	0.606125	0.0857190
$51$	0	0
$52$	−1.91314	−0.265305
$53$	−7.53771	−1.03538	−0.517692	−	0.855567i	$-0.673209\pi$
$53$	−7.53771	−1.03538	−0.517692	+	0.855567i	$0.673209\pi$
$54$	0	0
$55$	−12.7639	−1.72109
$56$	0.627989	0.0839185
$57$	0	0
$58$	−0.254722	−0.0334466
$59$	−9.08586	−1.18288	−0.591439	−	0.806350i	$-0.701440\pi$
$59$	−9.08586	−1.18288	−0.591439	+	0.806350i	$0.701440\pi$
$60$	0	0
$61$	−2.24461	−0.287393	−0.143696	−	0.989622i	$-0.545899\pi$
$61$	−2.24461	−0.287393	−0.143696	+	0.989622i	$0.545899\pi$
$62$	−0.582906	−0.0740291
$63$	0	0
$64$	−7.91835	−0.989794
$65$	−3.75835	−0.466166
$66$	0	0
$67$	−5.22314	−0.638108	−0.319054	−	0.947737i	$-0.603365\pi$
$67$	−5.22314	−0.638108	−0.319054	+	0.947737i	$0.603365\pi$
$68$	−2.47314	−0.299913
$69$	0	0
$70$	0.616314	0.0736636
$71$	7.28964	0.865121	0.432561	−	0.901605i	$-0.357610\pi$
$71$	7.28964	0.865121	0.432561	+	0.901605i	$0.357610\pi$
$72$	0	0
$73$	3.19834	0.374338	0.187169	−	0.982328i	$-0.440069\pi$
$73$	3.19834	0.374338	0.187169	+	0.982328i	$0.440069\pi$
$74$	−0.417172	−0.0484952
$75$	0	0
$76$	1.99659	0.229025
$77$	−8.76042	−0.998343
$78$	0	0
$79$	−4.17752	−0.470007	−0.235004	−	0.971994i	$-0.575510\pi$
$79$	−4.17752	−0.470007	−0.235004	+	0.971994i	$0.575510\pi$
$80$	−15.6090	−1.74514
$81$	0	0
$82$	−0.300597	−0.0331954
$83$	−5.94677	−0.652743	−0.326371	−	0.945242i	$-0.605826\pi$
$83$	−5.94677	−0.652743	−0.326371	+	0.945242i	$0.605826\pi$
$84$	0	0
$85$	−4.85847	−0.526975
$86$	0.123630	0.0133314
$87$	0	0
$88$	−0.759131	−0.0809237
$89$	2.41841	0.256351	0.128175	−	0.991752i	$-0.459088\pi$
$89$	2.41841	0.256351	0.128175	+	0.991752i	$0.459088\pi$
$90$	0	0
$91$	−2.57951	−0.270407
$92$	7.07961	0.738100
$93$	0	0
$94$	0.0583690	0.00602030
$95$	3.92229	0.402419
$96$	0	0
$97$	−1.85603	−0.188451	−0.0942257	−	0.995551i	$-0.530038\pi$
$97$	−1.85603	−0.188451	−0.0942257	+	0.995551i	$0.530038\pi$
$98$	0.0144188	0.00145652
$99$	0	0
$100$	−20.7334	−2.07334
$101$	3.10520	0.308979	0.154489	−	0.987994i	$-0.450627\pi$
$101$	3.10520	0.308979	0.154489	+	0.987994i	$0.450627\pi$
$102$	0	0
$103$	0.809379	0.0797505	0.0398753	−	0.999205i	$-0.487304\pi$
$103$	0.809379	0.0797505	0.0398753	+	0.999205i	$0.487304\pi$
$104$	−0.223527	−0.0219186
$105$	0	0
$106$	−0.439968	−0.0427335
$107$	6.08839	0.588587	0.294293	−	0.955715i	$-0.404916\pi$
$107$	6.08839	0.588587	0.294293	+	0.955715i	$0.404916\pi$
$108$	0	0
$109$	−8.40915	−0.805450	−0.402725	−	0.915321i	$-0.631937\pi$
$109$	−8.40915	−0.805450	−0.402725	+	0.915321i	$0.631937\pi$
$110$	−0.745018	−0.0710347
$111$	0	0
$112$	−10.7131	−1.01230
$113$	−3.91847	−0.368619	−0.184309	−	0.982868i	$-0.559005\pi$
$113$	−3.91847	−0.368619	−0.184309	+	0.982868i	$0.559005\pi$
$114$	0	0
$115$	13.9078	1.29691
$116$	8.71311	0.808992
$117$	0	0
$118$	−0.530332	−0.0488211
$119$	−3.33457	−0.305679
$120$	0	0
$121$	−0.410145	−0.0372859
$122$	−0.131016	−0.0118616
$123$	0	0
$124$	19.9391	1.79058
$125$	−21.1190	−1.88894
$126$	0	0
$127$	−9.28938	−0.824299	−0.412150	−	0.911116i	$-0.635222\pi$
$127$	−9.28938	−0.824299	−0.412150	+	0.911116i	$0.635222\pi$
$128$	−1.85986	−0.164390
$129$	0	0
$130$	−0.219371	−0.0192401
$131$	−5.41389	−0.473014	−0.236507	−	0.971630i	$-0.576003\pi$
$131$	−5.41389	−0.473014	−0.236507	+	0.971630i	$0.576003\pi$
$132$	0	0
$133$	2.69203	0.233429
$134$	−0.304869	−0.0263367
$135$	0	0
$136$	−0.288956	−0.0247778
$137$	8.03656	0.686610	0.343305	−	0.939224i	$-0.388454\pi$
$137$	8.03656	0.686610	0.343305	+	0.939224i	$0.388454\pi$
$138$	0	0
$139$	−8.93300	−0.757687	−0.378843	−	0.925461i	$-0.623678\pi$
$139$	−8.93300	−0.757687	−0.378843	+	0.925461i	$0.623678\pi$
$140$	−21.0819	−1.78174
$141$	0	0
$142$	0.425489	0.0357062
$143$	3.11819	0.260756
$144$	0	0
$145$	17.1168	1.42148
$146$	0.186684	0.0154501
$147$	0	0
$148$	14.2699	1.17298
$149$	20.4156	1.67251	0.836256	−	0.548339i	$-0.184740\pi$
$149$	20.4156	1.67251	0.836256	+	0.548339i	$0.184740\pi$
$150$	0	0
$151$	−5.91334	−0.481221	−0.240611	−	0.970622i	$-0.577348\pi$
$151$	−5.91334	−0.481221	−0.240611	+	0.970622i	$0.577348\pi$
$152$	0.233277	0.0189213
$153$	0	0
$154$	−0.511337	−0.0412047
$155$	39.1702	3.14623
$156$	0	0
$157$	2.07151	0.165325	0.0826624	−	0.996578i	$-0.473658\pi$
$157$	2.07151	0.165325	0.0826624	+	0.996578i	$0.473658\pi$
$158$	−0.243837	−0.0193987
$159$	0	0
$160$	−2.74104	−0.216699
$161$	9.54552	0.752293
$162$	0	0
$163$	2.84625	0.222935	0.111468	−	0.993768i	$-0.464445\pi$
$163$	2.84625	0.222935	0.111468	+	0.993768i	$0.464445\pi$
$164$	10.2823	0.802916
$165$	0	0
$166$	−0.347107	−0.0269407
$167$	5.50373	0.425891	0.212946	−	0.977064i	$-0.431694\pi$
$167$	5.50373	0.425891	0.212946	+	0.977064i	$0.431694\pi$
$168$	0	0
$169$	−12.0818	−0.929373
$170$	−0.283584	−0.0217499
$171$	0	0
$172$	−4.22895	−0.322454
$173$	−8.09191	−0.615217	−0.307608	−	0.951513i	$-0.599529\pi$
$173$	−8.09191	−0.615217	−0.307608	+	0.951513i	$0.599529\pi$
$174$	0	0
$175$	−27.9550	−2.11320
$176$	12.9503	0.976169
$177$	0	0
$178$	0.141160	0.0105804
$179$	7.18726	0.537201	0.268600	−	0.963252i	$-0.413439\pi$
$179$	7.18726	0.537201	0.268600	+	0.963252i	$0.413439\pi$
$180$	0	0
$181$	15.7425	1.17013	0.585065	−	0.810987i	$-0.301069\pi$
$181$	15.7425	1.17013	0.585065	+	0.810987i	$0.301069\pi$
$182$	−0.150564	−0.0111605
$183$	0	0
$184$	0.827164	0.0609793
$185$	28.0332	2.06104
$186$	0	0
$187$	4.03092	0.294770
$188$	−1.99659	−0.145617
$189$	0	0
$190$	0.228940	0.0166091
$191$	−0.497019	−0.0359630	−0.0179815	−	0.999838i	$-0.505724\pi$
$191$	−0.497019	−0.0359630	−0.0179815	+	0.999838i	$0.505724\pi$
$192$	0	0
$193$	−22.9777	−1.65397	−0.826986	−	0.562222i	$-0.809947\pi$
$193$	−22.9777	−1.65397	−0.826986	+	0.562222i	$0.809947\pi$
$194$	−0.108335	−0.00777797
$195$	0	0
$196$	−0.493217	−0.0352298
$197$	−9.38137	−0.668395	−0.334197	−	0.942503i	$-0.608465\pi$
$197$	−9.38137	−0.668395	−0.334197	+	0.942503i	$0.608465\pi$
$198$	0	0
$199$	−22.0877	−1.56576	−0.782878	−	0.622175i	$-0.786249\pi$
$199$	−22.0877	−1.56576	−0.782878	+	0.622175i	$0.786249\pi$
$200$	−2.42243	−0.171292
$201$	0	0
$202$	0.181247	0.0127525
$203$	11.7480	0.824548
$204$	0	0
$205$	20.1996	1.41080
$206$	0.0472426	0.00329155
$207$	0	0
$208$	3.81324	0.264401
$209$	−3.25421	−0.225098
$210$	0	0
$211$	−7.54583	−0.519476	−0.259738	−	0.965679i	$-0.583636\pi$
$211$	−7.54583	−0.519476	−0.259738	+	0.965679i	$0.583636\pi$
$212$	15.0497	1.03362
$213$	0	0
$214$	0.355373	0.0242928
$215$	−8.30774	−0.566583
$216$	0	0
$217$	26.8841	1.82501
$218$	−0.490833	−0.0332434
$219$	0	0
$220$	25.4844	1.71816
$221$	1.18691	0.0798401
$222$	0	0
$223$	5.36960	0.359575	0.179787	−	0.983705i	$-0.442459\pi$
$223$	5.36960	0.359575	0.179787	+	0.983705i	$0.442459\pi$
$224$	−1.88129	−0.125699
$225$	0	0
$226$	−0.228717	−0.0152140
$227$	25.1124	1.66677	0.833383	−	0.552695i	$-0.186401\pi$
$227$	25.1124	1.66677	0.833383	+	0.552695i	$0.186401\pi$
$228$	0	0
$229$	6.02719	0.398288	0.199144	−	0.979970i	$-0.436184\pi$
$229$	6.02719	0.398288	0.199144	+	0.979970i	$0.436184\pi$
$230$	0.811786	0.0535276
$231$	0	0
$232$	1.01802	0.0668362
$233$	−18.6378	−1.22100	−0.610501	−	0.792015i	$-0.709032\pi$
$233$	−18.6378	−1.22100	−0.610501	+	0.792015i	$0.709032\pi$
$234$	0	0
$235$	−3.92229	−0.255862
$236$	18.1408	1.18086
$237$	0	0
$238$	−0.194635	−0.0126163
$239$	−12.6020	−0.815153	−0.407577	−	0.913171i	$-0.633626\pi$
$239$	−12.6020	−0.815153	−0.407577	+	0.913171i	$0.633626\pi$
$240$	0	0
$241$	15.8802	1.02293	0.511466	−	0.859304i	$-0.329103\pi$
$241$	15.8802	1.02293	0.511466	+	0.859304i	$0.329103\pi$
$242$	−0.0239398	−0.00153891
$243$	0	0
$244$	4.48157	0.286903
$245$	−0.968921	−0.0619021
$246$	0	0
$247$	−0.958204	−0.0609690
$248$	2.32964	0.147932
$249$	0	0
$250$	−1.23270	−0.0779626
$251$	6.13072	0.386968	0.193484	−	0.981103i	$-0.438021\pi$
$251$	6.13072	0.386968	0.193484	+	0.981103i	$0.438021\pi$
$252$	0	0
$253$	−11.5389	−0.725445
$254$	−0.542211	−0.0340214
$255$	0	0
$256$	15.7281	0.983009
$257$	16.8892	1.05352	0.526758	−	0.850015i	$-0.323407\pi$
$257$	16.8892	1.05352	0.526758	+	0.850015i	$0.323407\pi$
$258$	0	0
$259$	19.2403	1.19554
$260$	7.50390	0.465372
$261$	0	0
$262$	−0.316003	−0.0195227
$263$	9.36502	0.577472	0.288736	−	0.957409i	$-0.406765\pi$
$263$	9.36502	0.577472	0.288736	+	0.957409i	$0.406765\pi$
$264$	0	0
$265$	29.5651	1.81617
$266$	0.157131	0.00963432
$267$	0	0
$268$	10.4285	0.637021
$269$	17.6549	1.07644	0.538220	−	0.842804i	$-0.319097\pi$
$269$	17.6549	1.07644	0.538220	+	0.842804i	$0.319097\pi$
$270$	0	0
$271$	6.90409	0.419394	0.209697	−	0.977766i	$-0.432752\pi$
$271$	6.90409	0.419394	0.209697	+	0.977766i	$0.432752\pi$
$272$	4.92942	0.298890
$273$	0	0
$274$	0.469086	0.0283385
$275$	33.7929	2.03779
$276$	0	0
$277$	−26.3984	−1.58612	−0.793062	−	0.609141i	$-0.791514\pi$
$277$	−26.3984	−1.58612	−0.793062	+	0.609141i	$0.791514\pi$
$278$	−0.521410	−0.0312721
$279$	0	0
$280$	−2.46315	−0.147202
$281$	15.6314	0.932489	0.466244	−	0.884656i	$-0.345607\pi$
$281$	15.6314	0.932489	0.466244	+	0.884656i	$0.345607\pi$
$282$	0	0
$283$	−11.1569	−0.663210	−0.331605	−	0.943418i	$-0.607590\pi$
$283$	−11.1569	−0.663210	−0.331605	+	0.943418i	$0.607590\pi$
$284$	−14.5544	−0.863648
$285$	0	0
$286$	0.182006	0.0107622
$287$	13.8638	0.818355
$288$	0	0
$289$	−15.4657	−0.909745
$290$	0.999092	0.0586687
$291$	0	0
$292$	−6.38579	−0.373700
$293$	−25.8187	−1.50834	−0.754172	−	0.656677i	$-0.771962\pi$
$293$	−25.8187	−1.50834	−0.754172	+	0.656677i	$0.771962\pi$
$294$	0	0
$295$	35.6374	2.07489
$296$	1.66726	0.0969078
$297$	0	0
$298$	1.19164	0.0690297
$299$	−3.39764	−0.196491
$300$	0	0
$301$	−5.70195	−0.328655
$302$	−0.345156	−0.0198615
$303$	0	0
$304$	−3.97957	−0.228244
$305$	8.80401	0.504116
$306$	0	0
$307$	−17.0836	−0.975012	−0.487506	−	0.873120i	$-0.662093\pi$
$307$	−17.0836	−0.975012	−0.487506	+	0.873120i	$0.662093\pi$
$308$	17.4910	0.996642
$309$	0	0
$310$	2.28633	0.129855
$311$	2.54260	0.144177	0.0720887	−	0.997398i	$-0.477034\pi$
$311$	2.54260	0.144177	0.0720887	+	0.997398i	$0.477034\pi$
$312$	0	0
$313$	12.7662	0.721589	0.360794	−	0.932645i	$-0.382506\pi$
$313$	12.7662	0.721589	0.360794	+	0.932645i	$0.382506\pi$
$314$	0.120912	0.00682346
$315$	0	0
$316$	8.34080	0.469207
$317$	−11.3675	−0.638461	−0.319231	−	0.947677i	$-0.603424\pi$
$317$	−11.3675	−0.638461	−0.319231	+	0.947677i	$0.603424\pi$
$318$	0	0
$319$	−14.2013	−0.795121
$320$	31.0581	1.73620
$321$	0	0
$322$	0.557162	0.0310494
$323$	−1.23868	−0.0689221
$324$	0	0
$325$	9.95034	0.551945
$326$	0.166132	0.00920123
$327$	0	0
$328$	1.20136	0.0663342
$329$	−2.69203	−0.148416
$330$	0	0
$331$	19.9934	1.09894	0.549468	−	0.835515i	$-0.314831\pi$
$331$	19.9934	1.09894	0.549468	+	0.835515i	$0.314831\pi$
$332$	11.8733	0.651631
$333$	0	0
$334$	0.321247	0.0175779
$335$	20.4867	1.11931
$336$	0	0
$337$	−7.22651	−0.393653	−0.196826	−	0.980438i	$-0.563064\pi$
$337$	−7.22651	−0.393653	−0.196826	+	0.980438i	$0.563064\pi$
$338$	−0.705205	−0.0383581
$339$	0	0
$340$	9.70038	0.526077
$341$	−32.4983	−1.75988
$342$	0	0
$343$	18.1792	0.981585
$344$	−0.494100	−0.0266401
$345$	0	0
$346$	−0.472317	−0.0253919
$347$	0.475341	0.0255177	0.0127588	−	0.999919i	$-0.495939\pi$
$347$	0.475341	0.0255177	0.0127588	+	0.999919i	$0.495939\pi$
$348$	0	0
$349$	27.6274	1.47886	0.739432	−	0.673232i	$-0.235094\pi$
$349$	27.6274	1.47886	0.739432	+	0.673232i	$0.235094\pi$
$350$	−1.63171	−0.0872184
$351$	0	0
$352$	2.27416	0.121213
$353$	7.69911	0.409782	0.204891	−	0.978785i	$-0.434316\pi$
$353$	7.69911	0.409782	0.204891	+	0.978785i	$0.434316\pi$
$354$	0	0
$355$	−28.5921	−1.51751
$356$	−4.82858	−0.255914
$357$	0	0
$358$	0.419513	0.0221719
$359$	10.0283	0.529275	0.264637	−	0.964348i	$-0.414748\pi$
$359$	10.0283	0.529275	0.264637	+	0.964348i	$0.414748\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0.918873	0.0482949
$363$	0	0
$364$	5.15024	0.269946
$365$	−12.5448	−0.656626
$366$	0	0
$367$	15.0725	0.786777	0.393389	−	0.919372i	$-0.371303\pi$
$367$	15.0725	0.786777	0.393389	+	0.919372i	$0.371303\pi$
$368$	−14.1109	−0.735583
$369$	0	0
$370$	1.63627	0.0850655
$371$	20.2917	1.05349
$372$	0	0
$373$	17.2688	0.894147	0.447073	−	0.894497i	$-0.352466\pi$
$373$	17.2688	0.894147	0.447073	+	0.894497i	$0.352466\pi$
$374$	0.235281	0.0121661
$375$	0	0
$376$	−0.233277	−0.0120303
$377$	−4.18159	−0.215363
$378$	0	0
$379$	18.4064	0.945471	0.472735	−	0.881204i	$-0.343267\pi$
$379$	18.4064	0.945471	0.472735	+	0.881204i	$0.343267\pi$
$380$	−7.83122	−0.401733
$381$	0	0
$382$	−0.0290105	−0.00148431
$383$	36.2668	1.85314	0.926572	−	0.376116i	$-0.122741\pi$
$383$	36.2668	1.85314	0.926572	+	0.376116i	$0.122741\pi$
$384$	0	0
$385$	34.3609	1.75119
$386$	−1.34119	−0.0682646
$387$	0	0
$388$	3.70574	0.188130
$389$	−35.6197	−1.80599	−0.902994	−	0.429652i	$-0.858636\pi$
$389$	−35.6197	−1.80599	−0.902994	+	0.429652i	$0.858636\pi$
$390$	0	0
$391$	−4.39217	−0.222122
$392$	−0.0576263	−0.00291057
$393$	0	0
$394$	−0.547581	−0.0275867
$395$	16.3854	0.824441
$396$	0	0
$397$	−27.9551	−1.40303	−0.701514	−	0.712656i	$-0.747492\pi$
$397$	−27.9551	−1.40303	−0.701514	+	0.712656i	$0.747492\pi$
$398$	−1.28924	−0.0646236
$399$	0	0
$400$	41.3253	2.06627
$401$	4.55086	0.227259	0.113629	−	0.993523i	$-0.463752\pi$
$401$	4.55086	0.227259	0.113629	+	0.993523i	$0.463752\pi$
$402$	0	0
$403$	−9.56917	−0.476674
$404$	−6.19981	−0.308452
$405$	0	0
$406$	0.685718	0.0340316
$407$	−23.2583	−1.15287
$408$	0	0
$409$	−20.1574	−0.996720	−0.498360	−	0.866970i	$-0.666064\pi$
$409$	−20.1574	−0.996720	−0.498360	+	0.866970i	$0.666064\pi$
$410$	1.17903	0.0582281
$411$	0	0
$412$	−1.61600	−0.0796147
$413$	24.4594	1.20357
$414$	0	0
$415$	23.3250	1.14498
$416$	0.669629	0.0328312
$417$	0	0
$418$	−0.189945	−0.00929050
$419$	19.0593	0.931109	0.465554	−	0.885019i	$-0.345855\pi$
$419$	19.0593	0.931109	0.465554	+	0.885019i	$0.345855\pi$
$420$	0	0
$421$	2.35006	0.114535	0.0572675	−	0.998359i	$-0.481761\pi$
$421$	2.35006	0.114535	0.0572675	+	0.998359i	$0.481761\pi$
$422$	−0.440442	−0.0214404
$423$	0	0
$424$	1.75837	0.0853942
$425$	12.8629	0.623943
$426$	0	0
$427$	6.04256	0.292420
$428$	−12.1560	−0.587584
$429$	0	0
$430$	−0.484914	−0.0233846
$431$	7.27760	0.350550	0.175275	−	0.984520i	$-0.443919\pi$
$431$	7.27760	0.350550	0.175275	+	0.984520i	$0.443919\pi$
$432$	0	0
$433$	−16.1117	−0.774279	−0.387139	−	0.922021i	$-0.626537\pi$
$433$	−16.1117	−0.774279	−0.387139	+	0.922021i	$0.626537\pi$
$434$	1.56920	0.0753240
$435$	0	0
$436$	16.7896	0.804078
$437$	3.54584	0.169621
$438$	0	0
$439$	−29.4394	−1.40506	−0.702532	−	0.711652i	$-0.747947\pi$
$439$	−29.4394	−1.40506	−0.702532	+	0.711652i	$0.747947\pi$
$440$	2.97753	0.141948
$441$	0	0
$442$	0.0692786	0.00329525
$443$	−9.56237	−0.454322	−0.227161	−	0.973857i	$-0.572944\pi$
$443$	−9.56237	−0.454322	−0.227161	+	0.973857i	$0.572944\pi$
$444$	0	0
$445$	−9.48571	−0.449666
$446$	0.313418	0.0148408
$447$	0	0
$448$	21.3164	1.00711
$449$	7.65861	0.361432	0.180716	−	0.983535i	$-0.442159\pi$
$449$	7.65861	0.361432	0.180716	+	0.983535i	$0.442159\pi$
$450$	0	0
$451$	−16.7590	−0.789150
$452$	7.82359	0.367991
$453$	0	0
$454$	1.46578	0.0687926
$455$	10.1176	0.474321
$456$	0	0
$457$	−20.3519	−0.952021	−0.476010	−	0.879440i	$-0.657918\pi$
$457$	−20.3519	−0.952021	−0.476010	+	0.879440i	$0.657918\pi$
$458$	0.351801	0.0164386
$459$	0	0
$460$	−27.7683	−1.29470
$461$	19.2793	0.897927	0.448963	−	0.893550i	$-0.351793\pi$
$461$	19.2793	0.897927	0.448963	+	0.893550i	$0.351793\pi$
$462$	0	0
$463$	−37.0674	−1.72267	−0.861334	−	0.508040i	$-0.830370\pi$
$463$	−37.0674	−1.72267	−0.861334	+	0.508040i	$0.830370\pi$
$464$	−17.3668	−0.806234
$465$	0	0
$466$	−1.08787	−0.0503946
$467$	23.8239	1.10244	0.551219	−	0.834361i	$-0.314163\pi$
$467$	23.8239	1.10244	0.551219	+	0.834361i	$0.314163\pi$
$468$	0	0
$469$	14.0609	0.649270
$470$	−0.228940	−0.0105602
$471$	0	0
$472$	2.11952	0.0975589
$473$	6.89268	0.316926
$474$	0	0
$475$	−10.3844	−0.476467
$476$	6.65777	0.305159
$477$	0	0
$478$	−0.735564	−0.0336439
$479$	−37.9773	−1.73523	−0.867614	−	0.497239i	$-0.834347\pi$
$479$	−37.9773	−1.73523	−0.867614	+	0.497239i	$0.834347\pi$
$480$	0	0
$481$	−6.84842	−0.312261
$482$	0.926909	0.0422195
$483$	0	0
$484$	0.818893	0.0372224
$485$	7.27989	0.330563
$486$	0	0
$487$	−4.10774	−0.186139	−0.0930697	−	0.995660i	$-0.529668\pi$
$487$	−4.10774	−0.186139	−0.0930697	+	0.995660i	$0.529668\pi$
$488$	0.523616	0.0237030
$489$	0	0
$490$	−0.0565549	−0.00255489
$491$	37.8308	1.70728	0.853639	−	0.520865i	$-0.174390\pi$
$491$	37.8308	1.70728	0.853639	+	0.520865i	$0.174390\pi$
$492$	0	0
$493$	−5.40559	−0.243456
$494$	−0.0559294	−0.00251638
$495$	0	0
$496$	−39.7422	−1.78448
$497$	−19.6239	−0.880254
$498$	0	0
$499$	12.9781	0.580980	0.290490	−	0.956878i	$-0.406182\pi$
$499$	12.9781	0.580980	0.290490	+	0.956878i	$0.406182\pi$
$500$	42.1661	1.88573
$501$	0	0
$502$	0.357844	0.0159714
$503$	22.9837	1.02479	0.512397	−	0.858749i	$-0.328758\pi$
$503$	22.9837	1.02479	0.512397	+	0.858749i	$0.328758\pi$
$504$	0	0
$505$	−12.1795	−0.541980
$506$	−0.673514	−0.0299414
$507$	0	0
$508$	18.5471	0.822895
$509$	35.0195	1.55221	0.776107	−	0.630602i	$-0.217192\pi$
$509$	35.0195	1.55221	0.776107	+	0.630602i	$0.217192\pi$
$510$	0	0
$511$	−8.61003	−0.380885
$512$	4.63776	0.204962
$513$	0	0
$514$	0.985803	0.0434819
$515$	−3.17462	−0.139891
$516$	0	0
$517$	3.25421	0.143120
$518$	1.12304	0.0493435
$519$	0	0
$520$	0.876738	0.0384475
$521$	7.24650	0.317475	0.158737	−	0.987321i	$-0.449258\pi$
$521$	7.24650	0.317475	0.158737	+	0.987321i	$0.449258\pi$
$522$	0	0
$523$	30.5056	1.33392	0.666958	−	0.745095i	$-0.267596\pi$
$523$	30.5056	1.33392	0.666958	+	0.745095i	$0.267596\pi$
$524$	10.8093	0.472208
$525$	0	0
$526$	0.546627	0.0238341
$527$	−12.3702	−0.538853
$528$	0	0
$529$	−10.4270	−0.453347
$530$	1.72568	0.0749589
$531$	0	0
$532$	−5.37489	−0.233031
$533$	−4.93470	−0.213746
$534$	0	0
$535$	−23.8804	−1.03244
$536$	1.21844	0.0526285
$537$	0	0
$538$	1.03050	0.0444280
$539$	0.803884	0.0346257
$540$	0	0
$541$	45.3568	1.95004	0.975020	−	0.222116i	$-0.0712965\pi$
$541$	45.3568	1.95004	0.975020	+	0.222116i	$0.0712965\pi$
$542$	0.402985	0.0173097
$543$	0	0
$544$	0.865637	0.0371139
$545$	32.9831	1.41284
$546$	0	0
$547$	27.1052	1.15894	0.579468	−	0.814995i	$-0.303261\pi$
$547$	27.1052	1.15894	0.579468	+	0.814995i	$0.303261\pi$
$548$	−16.0457	−0.685440
$549$	0	0
$550$	1.97245	0.0841057
$551$	4.36399	0.185912
$552$	0	0
$553$	11.2460	0.478229
$554$	−1.54084	−0.0654642
$555$	0	0
$556$	17.8356	0.756396
$557$	−29.3689	−1.24440	−0.622200	−	0.782858i	$-0.713761\pi$
$557$	−29.3689	−1.24440	−0.622200	+	0.782858i	$0.713761\pi$
$558$	0	0
$559$	2.02956	0.0858411
$560$	42.0200	1.77567
$561$	0	0
$562$	0.912386	0.0384867
$563$	6.25956	0.263809	0.131905	−	0.991262i	$-0.457891\pi$
$563$	6.25956	0.263809	0.131905	+	0.991262i	$0.457891\pi$
$564$	0	0
$565$	15.3694	0.646595
$566$	−0.651218	−0.0273727
$567$	0	0
$568$	−1.70051	−0.0713516
$569$	30.8993	1.29537	0.647683	−	0.761910i	$-0.275738\pi$
$569$	30.8993	1.29537	0.647683	+	0.761910i	$0.275738\pi$
$570$	0	0
$571$	−45.6550	−1.91060	−0.955302	−	0.295633i	$-0.904470\pi$
$571$	−45.6550	−1.91060	−0.955302	+	0.295633i	$0.904470\pi$
$572$	−6.22576	−0.260312
$573$	0	0
$574$	0.809217	0.0337760
$575$	−36.8213	−1.53556
$576$	0	0
$577$	31.9192	1.32882	0.664408	−	0.747370i	$-0.268684\pi$
$577$	31.9192	1.32882	0.664408	+	0.747370i	$0.268684\pi$
$578$	−0.902715	−0.0375480
$579$	0	0
$580$	−34.1754	−1.41905
$581$	16.0089	0.664160
$582$	0	0
$583$	−24.5292	−1.01590
$584$	−0.746100	−0.0308738
$585$	0	0
$586$	−1.50701	−0.0622540
$587$	−25.4273	−1.04950	−0.524749	−	0.851257i	$-0.675841\pi$
$587$	−25.4273	−1.04950	−0.524749	+	0.851257i	$0.675841\pi$
$588$	0	0
$589$	9.98657	0.411489
$590$	2.08012	0.0856371
$591$	0	0
$592$	−28.4426	−1.16898
$593$	−20.2428	−0.831274	−0.415637	−	0.909531i	$-0.636441\pi$
$593$	−20.2428	−0.831274	−0.415637	+	0.909531i	$0.636441\pi$
$594$	0	0
$595$	13.0791	0.536193
$596$	−40.7617	−1.66966
$597$	0	0
$598$	−0.198317	−0.00810978
$599$	4.93393	0.201595	0.100797	−	0.994907i	$-0.467861\pi$
$599$	4.93393	0.201595	0.100797	+	0.994907i	$0.467861\pi$
$600$	0	0
$601$	17.5697	0.716683	0.358341	−	0.933591i	$-0.383342\pi$
$601$	17.5697	0.716683	0.358341	+	0.933591i	$0.383342\pi$
$602$	−0.332817	−0.0135646
$603$	0	0
$604$	11.8065	0.480401
$605$	1.60871	0.0654033
$606$	0	0
$607$	12.6569	0.513726	0.256863	−	0.966448i	$-0.417311\pi$
$607$	12.6569	0.513726	0.256863	+	0.966448i	$0.417311\pi$
$608$	−0.698837	−0.0283416
$609$	0	0
$610$	0.513881	0.0208064
$611$	0.958204	0.0387648
$612$	0	0
$613$	−0.891915	−0.0360241	−0.0180121	−	0.999838i	$-0.505734\pi$
$613$	−0.891915	−0.0360241	−0.0180121	+	0.999838i	$0.505734\pi$
$614$	−0.997151	−0.0402417
$615$	0	0
$616$	2.04360	0.0823392
$617$	16.7698	0.675125	0.337563	−	0.941303i	$-0.390397\pi$
$617$	16.7698	0.675125	0.337563	+	0.941303i	$0.390397\pi$
$618$	0	0
$619$	−13.9870	−0.562186	−0.281093	−	0.959681i	$-0.590697\pi$
$619$	−13.9870	−0.562186	−0.281093	+	0.959681i	$0.590697\pi$
$620$	−78.2070	−3.14087
$621$	0	0
$622$	0.148409	0.00595065
$623$	−6.51043	−0.260835
$624$	0	0
$625$	30.9132	1.23653
$626$	0.745150	0.0297822
$627$	0	0
$628$	−4.13597	−0.165043
$629$	−8.85303	−0.352994
$630$	0	0
$631$	−38.3888	−1.52823	−0.764117	−	0.645077i	$-0.776825\pi$
$631$	−38.3888	−1.52823	−0.764117	+	0.645077i	$0.776825\pi$
$632$	0.974519	0.0387643
$633$	0	0
$634$	−0.663508	−0.0263513
$635$	36.4356	1.44590
$636$	0	0
$637$	0.236704	0.00937857
$638$	−0.828916	−0.0328171
$639$	0	0
$640$	7.29491	0.288357
$641$	−11.5873	−0.457669	−0.228834	−	0.973465i	$-0.573491\pi$
$641$	−11.5873	−0.457669	−0.228834	+	0.973465i	$0.573491\pi$
$642$	0	0
$643$	10.0849	0.397708	0.198854	−	0.980029i	$-0.436278\pi$
$643$	10.0849	0.397708	0.198854	+	0.980029i	$0.436278\pi$
$644$	−19.0585	−0.751011
$645$	0	0
$646$	−0.0723005	−0.00284463
$647$	11.0723	0.435296	0.217648	−	0.976027i	$-0.430161\pi$
$647$	11.0723	0.435296	0.217648	+	0.976027i	$0.430161\pi$
$648$	0	0
$649$	−29.5673	−1.16062
$650$	0.580791	0.0227805
$651$	0	0
$652$	−5.68280	−0.222555
$653$	7.85319	0.307319	0.153660	−	0.988124i	$-0.450894\pi$
$653$	7.85319	0.307319	0.153660	+	0.988124i	$0.450894\pi$
$654$	0	0
$655$	21.2348	0.829714
$656$	−20.4946	−0.800179
$657$	0	0
$658$	−0.157131	−0.00612561
$659$	−33.5039	−1.30513	−0.652563	−	0.757735i	$-0.726306\pi$
$659$	−33.5039	−1.30513	−0.652563	+	0.757735i	$0.726306\pi$
$660$	0	0
$661$	−32.2327	−1.25371	−0.626854	−	0.779137i	$-0.715658\pi$
$661$	−32.2327	−1.25371	−0.626854	+	0.779137i	$0.715658\pi$
$662$	1.16699	0.0453565
$663$	0	0
$664$	1.38724	0.0538355
$665$	−10.5589	−0.409458
$666$	0	0
$667$	15.4740	0.599157
$668$	−10.9887	−0.425166
$669$	0	0
$670$	1.19579	0.0461972
$671$	−7.30442	−0.281984
$672$	0	0
$673$	−0.483867	−0.0186517	−0.00932585	−	0.999957i	$-0.502969\pi$
$673$	−0.483867	−0.0186517	−0.00932585	+	0.999957i	$0.502969\pi$
$674$	−0.421804	−0.0162473
$675$	0	0
$676$	24.1225	0.927790
$677$	11.1546	0.428704	0.214352	−	0.976756i	$-0.431236\pi$
$677$	11.1546	0.428704	0.214352	+	0.976756i	$0.431236\pi$
$678$	0	0
$679$	4.99649	0.191748
$680$	1.13337	0.0434627
$681$	0	0
$682$	−1.89689	−0.0726358
$683$	1.40004	0.0535711	0.0267856	−	0.999641i	$-0.491473\pi$
$683$	1.40004	0.0535711	0.0267856	+	0.999641i	$0.491473\pi$
$684$	0	0
$685$	−31.5217	−1.20438
$686$	1.06110	0.0405130
$687$	0	0
$688$	8.42906	0.321355
$689$	−7.22266	−0.275161
$690$	0	0
$691$	−11.7253	−0.446050	−0.223025	−	0.974813i	$-0.571593\pi$
$691$	−11.7253	−0.446050	−0.223025	+	0.974813i	$0.571593\pi$
$692$	16.1563	0.614169
$693$	0	0
$694$	0.0277452	0.00105319
$695$	35.0378	1.32906
$696$	0	0
$697$	−6.37914	−0.241627
$698$	1.61259	0.0610373
$699$	0	0
$700$	55.8148	2.10960
$701$	2.53781	0.0958517	0.0479259	−	0.998851i	$-0.484739\pi$
$701$	2.53781	0.0958517	0.0479259	+	0.998851i	$0.484739\pi$
$702$	0	0
$703$	7.14715	0.269560
$704$	−25.7679	−0.971166
$705$	0	0
$706$	0.449389	0.0169130
$707$	−8.35928	−0.314383
$708$	0	0
$709$	−28.3299	−1.06395	−0.531976	−	0.846759i	$-0.678550\pi$
$709$	−28.3299	−1.06395	−0.531976	+	0.846759i	$0.678550\pi$
$710$	−1.66889	−0.0626324
$711$	0	0
$712$	−0.564159	−0.0211428
$713$	35.4108	1.32615
$714$	0	0
$715$	−12.2305	−0.457393
$716$	−14.3500	−0.536286
$717$	0	0
$718$	0.585343	0.0218448
$719$	−18.5618	−0.692238	−0.346119	−	0.938191i	$-0.612501\pi$
$719$	−18.5618	−0.692238	−0.346119	+	0.938191i	$0.612501\pi$
$720$	0	0
$721$	−2.17887	−0.0811455
$722$	0.0583690	0.00217227
$723$	0	0
$724$	−31.4313	−1.16814
$725$	−45.3173	−1.68304
$726$	0	0
$727$	27.1041	1.00524	0.502618	−	0.864508i	$-0.332370\pi$
$727$	27.1041	1.00524	0.502618	+	0.864508i	$0.332370\pi$
$728$	0.601741	0.0223020
$729$	0	0
$730$	−0.732228	−0.0271010
$731$	2.62363	0.0970385
$732$	0	0
$733$	4.52229	0.167034	0.0835172	−	0.996506i	$-0.473385\pi$
$733$	4.52229	0.167034	0.0835172	+	0.996506i	$0.473385\pi$
$734$	0.879765	0.0324727
$735$	0	0
$736$	−2.47797	−0.0913392
$737$	−16.9972	−0.626099
$738$	0	0
$739$	−15.8199	−0.581943	−0.290971	−	0.956732i	$-0.593978\pi$
$739$	−15.8199	−0.581943	−0.290971	+	0.956732i	$0.593978\pi$
$740$	−55.9709	−2.05753
$741$	0	0
$742$	1.18441	0.0434810
$743$	0.307529	0.0112821	0.00564106	−	0.999984i	$-0.498204\pi$
$743$	0.307529	0.0112821	0.00564106	+	0.999984i	$0.498204\pi$
$744$	0	0
$745$	−80.0760	−2.93376
$746$	1.00796	0.0369042
$747$	0	0
$748$	−8.04811	−0.294268
$749$	−16.3901	−0.598882
$750$	0	0
$751$	−2.98279	−0.108844	−0.0544218	−	0.998518i	$-0.517332\pi$
$751$	−2.98279	−0.108844	−0.0544218	+	0.998518i	$0.517332\pi$
$752$	3.97957	0.145120
$753$	0	0
$754$	−0.244075	−0.00888869
$755$	23.1939	0.844111
$756$	0	0
$757$	−29.7977	−1.08302	−0.541508	−	0.840696i	$-0.682146\pi$
$757$	−29.7977	−1.08302	−0.541508	+	0.840696i	$0.682146\pi$
$758$	1.07436	0.0390225
$759$	0	0
$760$	−0.914980	−0.0331898
$761$	28.4971	1.03302	0.516510	−	0.856281i	$-0.327231\pi$
$761$	28.4971	1.03302	0.516510	+	0.856281i	$0.327231\pi$
$762$	0	0
$763$	22.6377	0.819539
$764$	0.992345	0.0359018
$765$	0	0
$766$	2.11685	0.0764850
$767$	−8.70611	−0.314359
$768$	0	0
$769$	13.9347	0.502498	0.251249	−	0.967922i	$-0.419159\pi$
$769$	13.9347	0.502498	0.251249	+	0.967922i	$0.419159\pi$
$770$	2.00561	0.0722772
$771$	0	0
$772$	45.8772	1.65116
$773$	−25.4372	−0.914914	−0.457457	−	0.889232i	$-0.651240\pi$
$773$	−25.4372	−0.914914	−0.457457	+	0.889232i	$0.651240\pi$
$774$	0	0
$775$	−103.704	−3.72516
$776$	0.432969	0.0155427
$777$	0	0
$778$	−2.07908	−0.0745387
$779$	5.14995	0.184516
$780$	0	0
$781$	23.7220	0.848840
$782$	−0.256366	−0.00916765
$783$	0	0
$784$	0.983070	0.0351097
$785$	−8.12508	−0.289996
$786$	0	0
$787$	−0.142345	−0.00507404	−0.00253702	−	0.999997i	$-0.500808\pi$
$787$	−0.142345	−0.00507404	−0.00253702	+	0.999997i	$0.500808\pi$
$788$	18.7308	0.667256
$789$	0	0
$790$	0.956401	0.0340272
$791$	10.5486	0.375067
$792$	0	0
$793$	−2.15079	−0.0763769
$794$	−1.63171	−0.0579073
$795$	0	0
$796$	44.1002	1.56309
$797$	48.7263	1.72597	0.862987	−	0.505225i	$-0.168591\pi$
$797$	48.7263	1.72597	0.862987	+	0.505225i	$0.168591\pi$
$798$	0	0
$799$	1.23868	0.0438214
$800$	7.25698	0.256573
$801$	0	0
$802$	0.265629	0.00937968
$803$	10.4081	0.367292
$804$	0	0
$805$	−37.4403	−1.31960
$806$	−0.558542	−0.0196738
$807$	0	0
$808$	−0.724371	−0.0254833
$809$	3.93819	0.138459	0.0692297	−	0.997601i	$-0.477946\pi$
$809$	3.93819	0.138459	0.0692297	+	0.997601i	$0.477946\pi$
$810$	0	0
$811$	−40.2026	−1.41170	−0.705852	−	0.708360i	$-0.749435\pi$
$811$	−40.2026	−1.41170	−0.705852	+	0.708360i	$0.749435\pi$
$812$	−23.4560	−0.823143
$813$	0	0
$814$	−1.35756	−0.0475825
$815$	−11.1638	−0.391051
$816$	0	0
$817$	−2.11808	−0.0741024
$818$	−1.17657	−0.0411377
$819$	0	0
$820$	−40.3304	−1.40840
$821$	11.3535	0.396241	0.198121	−	0.980178i	$-0.436516\pi$
$821$	11.3535	0.396241	0.198121	+	0.980178i	$0.436516\pi$
$822$	0	0
$823$	−2.34396	−0.0817052	−0.0408526	−	0.999165i	$-0.513007\pi$
$823$	−2.34396	−0.0817052	−0.0408526	+	0.999165i	$0.513007\pi$
$824$	−0.188810	−0.00657749
$825$	0	0
$826$	1.42767	0.0496750
$827$	−16.7923	−0.583925	−0.291962	−	0.956430i	$-0.594308\pi$
$827$	−16.7923	−0.583925	−0.291962	+	0.956430i	$0.594308\pi$
$828$	0	0
$829$	20.9544	0.727775	0.363888	−	0.931443i	$-0.381449\pi$
$829$	20.9544	0.727775	0.363888	+	0.931443i	$0.381449\pi$
$830$	1.36145	0.0472567
$831$	0	0
$832$	−7.58739	−0.263046
$833$	0.305990	0.0106019
$834$	0	0
$835$	−21.5872	−0.747057
$836$	6.49732	0.224715
$837$	0	0
$838$	1.11247	0.0384297
$839$	−33.8719	−1.16939	−0.584694	−	0.811254i	$-0.698786\pi$
$839$	−33.8719	−1.16939	−0.584694	+	0.811254i	$0.698786\pi$
$840$	0	0
$841$	−9.95559	−0.343296
$842$	0.137171	0.00472722
$843$	0	0
$844$	15.0660	0.518591
$845$	47.3885	1.63021
$846$	0	0
$847$	1.10412	0.0379381
$848$	−29.9968	−1.03010
$849$	0	0
$850$	0.750795	0.0257521
$851$	25.3427	0.868736
$852$	0	0
$853$	−16.5404	−0.566333	−0.283167	−	0.959071i	$-0.591385\pi$
$853$	−16.5404	−0.566333	−0.283167	+	0.959071i	$0.591385\pi$
$854$	0.352698	0.0120691
$855$	0	0
$856$	−1.42028	−0.0485442
$857$	28.9141	0.987686	0.493843	−	0.869551i	$-0.335592\pi$
$857$	28.9141	0.987686	0.493843	+	0.869551i	$0.335592\pi$
$858$	0	0
$859$	16.2129	0.553175	0.276588	−	0.960989i	$-0.410796\pi$
$859$	16.2129	0.553175	0.276588	+	0.960989i	$0.410796\pi$
$860$	16.5872	0.565618
$861$	0	0
$862$	0.424786	0.0144683
$863$	0.985631	0.0335513	0.0167756	−	0.999859i	$-0.494660\pi$
$863$	0.985631	0.0335513	0.0167756	+	0.999859i	$0.494660\pi$
$864$	0	0
$865$	31.7388	1.07915
$866$	−0.940423	−0.0319569
$867$	0	0
$868$	−53.6767	−1.82191
$869$	−13.5945	−0.461162
$870$	0	0
$871$	−5.00483	−0.169582
$872$	1.96166	0.0664302
$873$	0	0
$874$	0.206967	0.00700077
$875$	56.8531	1.92199
$876$	0	0
$877$	12.9394	0.436932	0.218466	−	0.975845i	$-0.429895\pi$
$877$	12.9394	0.436932	0.218466	+	0.975845i	$0.429895\pi$
$878$	−1.71835	−0.0579914
$879$	0	0
$880$	−50.7950	−1.71230
$881$	30.1050	1.01426	0.507131	−	0.861869i	$-0.330706\pi$
$881$	30.1050	1.01426	0.507131	+	0.861869i	$0.330706\pi$
$882$	0	0
$883$	−28.2647	−0.951183	−0.475591	−	0.879666i	$-0.657766\pi$
$883$	−28.2647	−0.951183	−0.475591	+	0.879666i	$0.657766\pi$
$884$	−2.36977	−0.0797041
$885$	0	0
$886$	−0.558146	−0.0187513
$887$	−20.9074	−0.702003	−0.351001	−	0.936375i	$-0.614159\pi$
$887$	−20.9074	−0.702003	−0.351001	+	0.936375i	$0.614159\pi$
$888$	0	0
$889$	25.0073	0.838718
$890$	−0.553671	−0.0185591
$891$	0	0
$892$	−10.7209	−0.358962
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−28.1905	−0.942305
$896$	5.00680	0.167266
$897$	0	0
$898$	0.447025	0.0149174
$899$	43.5813	1.45352
$900$	0	0
$901$	−9.33681	−0.311054
$902$	−0.978205	−0.0325707
$903$	0	0
$904$	0.914089	0.0304021
$905$	−61.7466	−2.05253
$906$	0	0
$907$	−24.1496	−0.801876	−0.400938	−	0.916105i	$-0.631316\pi$
$907$	−24.1496	−0.801876	−0.400938	+	0.916105i	$0.631316\pi$
$908$	−50.1392	−1.66393
$909$	0	0
$910$	0.590554	0.0195767
$911$	26.3605	0.873364	0.436682	−	0.899616i	$-0.356154\pi$
$911$	26.3605	0.873364	0.436682	+	0.899616i	$0.356154\pi$
$912$	0	0
$913$	−19.3520	−0.640458
$914$	−1.18792	−0.0392928
$915$	0	0
$916$	−12.0338	−0.397609
$917$	14.5744	0.481288
$918$	0	0
$919$	19.8347	0.654288	0.327144	−	0.944975i	$-0.393914\pi$
$919$	19.8347	0.654288	0.327144	+	0.944975i	$0.393914\pi$
$920$	−3.24438	−0.106964
$921$	0	0
$922$	1.12531	0.0370602
$923$	6.98496	0.229913
$924$	0	0
$925$	−74.2186	−2.44029
$926$	−2.16359	−0.0710998
$927$	0	0
$928$	−3.04972	−0.100112
$929$	−41.3461	−1.35652	−0.678261	−	0.734821i	$-0.737266\pi$
$929$	−41.3461	−1.35652	−0.678261	+	0.734821i	$0.737266\pi$
$930$	0	0
$931$	−0.247029	−0.00809606
$932$	37.2121	1.21892
$933$	0	0
$934$	1.39058	0.0455010
$935$	−15.8105	−0.517057
$936$	0	0
$937$	11.3744	0.371585	0.185792	−	0.982589i	$-0.440515\pi$
$937$	11.3744	0.371585	0.185792	+	0.982589i	$0.440515\pi$
$938$	0.820717	0.0267974
$939$	0	0
$940$	7.83122	0.255426
$941$	14.2681	0.465127	0.232564	−	0.972581i	$-0.425289\pi$
$941$	14.2681	0.465127	0.232564	+	0.972581i	$0.425289\pi$
$942$	0	0
$943$	18.2609	0.594657
$944$	−36.1578	−1.17684
$945$	0	0
$946$	0.402319	0.0130805
$947$	52.8824	1.71845	0.859224	−	0.511600i	$-0.170947\pi$
$947$	52.8824	1.71845	0.859224	+	0.511600i	$0.170947\pi$
$948$	0	0
$949$	3.06466	0.0994832
$950$	−0.606125	−0.0196653
$951$	0	0
$952$	0.777878	0.0252112
$953$	35.2479	1.14179	0.570896	−	0.821023i	$-0.306596\pi$
$953$	35.2479	1.14179	0.570896	+	0.821023i	$0.306596\pi$
$954$	0	0
$955$	1.94945	0.0630828
$956$	25.1610	0.813765
$957$	0	0
$958$	−2.21670	−0.0716182
$959$	−21.6347	−0.698620
$960$	0	0
$961$	68.7315	2.21715
$962$	−0.399735	−0.0128880
$963$	0	0
$964$	−31.7062	−1.02119
$965$	90.1253	2.90124
$966$	0	0
$967$	23.1956	0.745922	0.372961	−	0.927847i	$-0.378343\pi$
$967$	23.1956	0.745922	0.372961	+	0.927847i	$0.378343\pi$
$968$	0.0956775	0.00307519
$969$	0	0
$970$	0.424920	0.0136433
$971$	−36.5236	−1.17210	−0.586049	−	0.810276i	$-0.699317\pi$
$971$	−36.5236	−1.17210	−0.586049	+	0.810276i	$0.699317\pi$
$972$	0	0
$973$	24.0479	0.770940
$974$	−0.239764	−0.00768255
$975$	0	0
$976$	−8.93258	−0.285925
$977$	−21.1770	−0.677511	−0.338756	−	0.940874i	$-0.610006\pi$
$977$	−21.1770	−0.677511	−0.338756	+	0.940874i	$0.610006\pi$
$978$	0	0
$979$	7.87000	0.251526
$980$	1.93454	0.0617966
$981$	0	0
$982$	2.20814	0.0704647
$983$	11.6654	0.372068	0.186034	−	0.982543i	$-0.440437\pi$
$983$	11.6654	0.372068	0.186034	+	0.982543i	$0.440437\pi$
$984$	0	0
$985$	36.7964	1.17243
$986$	−0.315519	−0.0100482
$987$	0	0
$988$	1.91314	0.0608652
$989$	−7.51040	−0.238817
$990$	0	0
$991$	−26.8617	−0.853289	−0.426645	−	0.904419i	$-0.640305\pi$
$991$	−26.8617	−0.853289	−0.426645	+	0.904419i	$0.640305\pi$
$992$	−6.97899	−0.221583
$993$	0	0
$994$	−1.14543	−0.0363308
$995$	86.6344	2.74650
$996$	0	0
$997$	−46.2110	−1.46352	−0.731759	−	0.681563i	$-0.761300\pi$
$997$	−46.2110	−1.46352	−0.731759	+	0.681563i	$0.761300\pi$
$998$	0.757519	0.0239788
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.w.1.17	yes	34
3.2	odd	2		8037.2.a.v.1.18	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.18	✓	34	3.2	odd	2
8037.2.a.w.1.17	yes	34	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)

\(f(q)\)	\(=\)	\(q+0.0583690 q^{2} -1.99659 q^{4} -3.92229 q^{5} -2.69203 q^{7} -0.233277 q^{8} +O(q^{10})\)	\(q+0.0583690 q^{2} -1.99659 q^{4} -3.92229 q^{5} -2.69203 q^{7} -0.233277 q^{8} -0.228940 q^{10} +3.25421 q^{11} +0.958204 q^{13} -0.157131 q^{14} +3.97957 q^{16} +1.23868 q^{17} -1.00000 q^{19} +7.83122 q^{20} +0.189945 q^{22} -3.54584 q^{23} +10.3844 q^{25} +0.0559294 q^{26} +5.37489 q^{28} -4.36399 q^{29} -9.98657 q^{31} +0.698837 q^{32} +0.0723005 q^{34} +10.5589 q^{35} -7.14715 q^{37} -0.0583690 q^{38} +0.914980 q^{40} -5.14995 q^{41} +2.11808 q^{43} -6.49732 q^{44} -0.206967 q^{46} +1.00000 q^{47} +0.247029 q^{49} +0.606125 q^{50} -1.91314 q^{52} -7.53771 q^{53} -12.7639 q^{55} +0.627989 q^{56} -0.254722 q^{58} -9.08586 q^{59} -2.24461 q^{61} -0.582906 q^{62} -7.91835 q^{64} -3.75835 q^{65} -5.22314 q^{67} -2.47314 q^{68} +0.616314 q^{70} +7.28964 q^{71} +3.19834 q^{73} -0.417172 q^{74} +1.99659 q^{76} -8.76042 q^{77} -4.17752 q^{79} -15.6090 q^{80} -0.300597 q^{82} -5.94677 q^{83} -4.85847 q^{85} +0.123630 q^{86} -0.759131 q^{88} +2.41841 q^{89} -2.57951 q^{91} +7.07961 q^{92} +0.0583690 q^{94} +3.92229 q^{95} -1.85603 q^{97} +0.0144188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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