LMFDB - Embedded newform 8024.2.a.ba.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.991965 q^{3} +2.29317 q^{5} +3.07900 q^{7} -2.01601 q^{9} +O(q^{10})$	$q+0.991965 q^{3} +2.29317 q^{5} +3.07900 q^{7} -2.01601 q^{9} +2.20043 q^{11} +5.37799 q^{13} +2.27475 q^{15} -1.00000 q^{17} +7.08899 q^{19} +3.05426 q^{21} -0.583087 q^{23} +0.258640 q^{25} -4.97570 q^{27} +4.86614 q^{29} +2.38523 q^{31} +2.18275 q^{33} +7.06068 q^{35} +4.26091 q^{37} +5.33478 q^{39} +6.26582 q^{41} +0.584755 q^{43} -4.62305 q^{45} -7.15305 q^{47} +2.48024 q^{49} -0.991965 q^{51} -0.565808 q^{53} +5.04596 q^{55} +7.03203 q^{57} +1.00000 q^{59} -7.24570 q^{61} -6.20728 q^{63} +12.3327 q^{65} -9.58567 q^{67} -0.578402 q^{69} -14.0773 q^{71} -11.8358 q^{73} +0.256562 q^{75} +6.77512 q^{77} +12.4279 q^{79} +1.11229 q^{81} -6.71800 q^{83} -2.29317 q^{85} +4.82704 q^{87} +10.9393 q^{89} +16.5588 q^{91} +2.36606 q^{93} +16.2563 q^{95} +7.23503 q^{97} -4.43607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * q^99

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	0
$2$	0	0
$3$	0.991965	0.572711	0.286356	−	0.958123i	$-0.407556\pi$
$3$	0.991965	0.572711	0.286356	+	0.958123i	$0.407556\pi$
$4$	0	0
$5$	2.29317	1.02554	0.512769	−	0.858527i	$-0.328620\pi$
$5$	2.29317	1.02554	0.512769	+	0.858527i	$0.328620\pi$
$6$	0	0
$7$	3.07900	1.16375	0.581876	−	0.813277i	$-0.302319\pi$
$7$	3.07900	1.16375	0.581876	+	0.813277i	$0.302319\pi$
$8$	0	0
$9$	−2.01601	−0.672002
$10$	0	0
$11$	2.20043	0.663454	0.331727	−	0.943375i	$-0.392369\pi$
$11$	2.20043	0.663454	0.331727	+	0.943375i	$0.392369\pi$
$12$	0	0
$13$	5.37799	1.49159	0.745793	−	0.666178i	$-0.232071\pi$
$13$	5.37799	1.49159	0.745793	+	0.666178i	$0.232071\pi$
$14$	0	0
$15$	2.27475	0.587337
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	7.08899	1.62633	0.813163	−	0.582037i	$-0.197744\pi$
$19$	7.08899	1.62633	0.813163	+	0.582037i	$0.197744\pi$
$20$	0	0
$21$	3.05426	0.666494
$22$	0	0
$23$	−0.583087	−0.121582	−0.0607910	−	0.998151i	$-0.519362\pi$
$23$	−0.583087	−0.121582	−0.0607910	+	0.998151i	$0.519362\pi$
$24$	0	0
$25$	0.258640	0.0517280
$26$	0	0
$27$	−4.97570	−0.957574
$28$	0	0
$29$	4.86614	0.903619	0.451810	−	0.892114i	$-0.350779\pi$
$29$	4.86614	0.903619	0.451810	+	0.892114i	$0.350779\pi$
$30$	0	0
$31$	2.38523	0.428400	0.214200	−	0.976790i	$-0.431286\pi$
$31$	2.38523	0.428400	0.214200	+	0.976790i	$0.431286\pi$
$32$	0	0
$33$	2.18275	0.379968
$34$	0	0
$35$	7.06068	1.19347
$36$	0	0
$37$	4.26091	0.700490	0.350245	−	0.936658i	$-0.386098\pi$
$37$	4.26091	0.700490	0.350245	+	0.936658i	$0.386098\pi$
$38$	0	0
$39$	5.33478	0.854248
$40$	0	0
$41$	6.26582	0.978557	0.489278	−	0.872128i	$-0.337260\pi$
$41$	6.26582	0.978557	0.489278	+	0.872128i	$0.337260\pi$
$42$	0	0
$43$	0.584755	0.0891744	0.0445872	−	0.999005i	$-0.485803\pi$
$43$	0.584755	0.0891744	0.0445872	+	0.999005i	$0.485803\pi$
$44$	0	0
$45$	−4.62305	−0.689163
$46$	0	0
$47$	−7.15305	−1.04338	−0.521690	−	0.853135i	$-0.674698\pi$
$47$	−7.15305	−1.04338	−0.521690	+	0.853135i	$0.674698\pi$
$48$	0	0
$49$	2.48024	0.354320
$50$	0	0
$51$	−0.991965	−0.138903
$52$	0	0
$53$	−0.565808	−0.0777197	−0.0388599	−	0.999245i	$-0.512373\pi$
$53$	−0.565808	−0.0777197	−0.0388599	+	0.999245i	$0.512373\pi$
$54$	0	0
$55$	5.04596	0.680397
$56$	0	0
$57$	7.03203	0.931415
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−7.24570	−0.927717	−0.463858	−	0.885909i	$-0.653535\pi$
$61$	−7.24570	−0.927717	−0.463858	+	0.885909i	$0.653535\pi$
$62$	0	0
$63$	−6.20728	−0.782044
$64$	0	0
$65$	12.3327	1.52968
$66$	0	0
$67$	−9.58567	−1.17108	−0.585538	−	0.810645i	$-0.699117\pi$
$67$	−9.58567	−1.17108	−0.585538	+	0.810645i	$0.699117\pi$
$68$	0	0
$69$	−0.578402	−0.0696314
$70$	0	0
$71$	−14.0773	−1.67066	−0.835332	−	0.549745i	$-0.814725\pi$
$71$	−14.0773	−1.67066	−0.835332	+	0.549745i	$0.814725\pi$
$72$	0	0
$73$	−11.8358	−1.38528	−0.692640	−	0.721284i	$-0.743552\pi$
$73$	−11.8358	−1.38528	−0.692640	+	0.721284i	$0.743552\pi$
$74$	0	0
$75$	0.256562	0.0296252
$76$	0	0
$77$	6.77512	0.772096
$78$	0	0
$79$	12.4279	1.39824	0.699121	−	0.715003i	$-0.253575\pi$
$79$	12.4279	1.39824	0.699121	+	0.715003i	$0.253575\pi$
$80$	0	0
$81$	1.11229	0.123588
$82$	0	0
$83$	−6.71800	−0.737396	−0.368698	−	0.929549i	$-0.620196\pi$
$83$	−6.71800	−0.737396	−0.368698	+	0.929549i	$0.620196\pi$
$84$	0	0
$85$	−2.29317	−0.248729
$86$	0	0
$87$	4.82704	0.517513
$88$	0	0
$89$	10.9393	1.15956	0.579781	−	0.814772i	$-0.303138\pi$
$89$	10.9393	1.15956	0.579781	+	0.814772i	$0.303138\pi$
$90$	0	0
$91$	16.5588	1.73584
$92$	0	0
$93$	2.36606	0.245349
$94$	0	0
$95$	16.2563	1.66786
$96$	0	0
$97$	7.23503	0.734606	0.367303	−	0.930101i	$-0.380281\pi$
$97$	7.23503	0.734606	0.367303	+	0.930101i	$0.380281\pi$
$98$	0	0
$99$	−4.43607	−0.445842
$100$	0	0
$101$	−6.78029	−0.674664	−0.337332	−	0.941386i	$-0.609525\pi$
$101$	−6.78029	−0.674664	−0.337332	+	0.941386i	$0.609525\pi$
$102$	0	0
$103$	−1.09436	−0.107831	−0.0539154	−	0.998546i	$-0.517170\pi$
$103$	−1.09436	−0.107831	−0.0539154	+	0.998546i	$0.517170\pi$
$104$	0	0
$105$	7.00395	0.683515
$106$	0	0
$107$	−17.1251	−1.65555	−0.827773	−	0.561063i	$-0.810393\pi$
$107$	−17.1251	−1.65555	−0.827773	+	0.561063i	$0.810393\pi$
$108$	0	0
$109$	2.38309	0.228259	0.114129	−	0.993466i	$-0.463592\pi$
$109$	2.38309	0.228259	0.114129	+	0.993466i	$0.463592\pi$
$110$	0	0
$111$	4.22668	0.401178
$112$	0	0
$113$	5.79345	0.545002	0.272501	−	0.962156i	$-0.412149\pi$
$113$	5.79345	0.545002	0.272501	+	0.962156i	$0.412149\pi$
$114$	0	0
$115$	−1.33712	−0.124687
$116$	0	0
$117$	−10.8421	−1.00235
$118$	0	0
$119$	−3.07900	−0.282251
$120$	0	0
$121$	−6.15812	−0.559829
$122$	0	0
$123$	6.21548	0.560431
$124$	0	0
$125$	−10.8728	−0.972489
$126$	0	0
$127$	1.65125	0.146525	0.0732626	−	0.997313i	$-0.476659\pi$
$127$	1.65125	0.146525	0.0732626	+	0.997313i	$0.476659\pi$
$128$	0	0
$129$	0.580057	0.0510712
$130$	0	0
$131$	17.6913	1.54570	0.772849	−	0.634590i	$-0.218831\pi$
$131$	17.6913	1.54570	0.772849	+	0.634590i	$0.218831\pi$
$132$	0	0
$133$	21.8270	1.89264
$134$	0	0
$135$	−11.4101	−0.982029
$136$	0	0
$137$	−9.42494	−0.805227	−0.402614	−	0.915370i	$-0.631898\pi$
$137$	−9.42494	−0.805227	−0.402614	+	0.915370i	$0.631898\pi$
$138$	0	0
$139$	19.2032	1.62879	0.814396	−	0.580310i	$-0.197069\pi$
$139$	19.2032	1.62879	0.814396	+	0.580310i	$0.197069\pi$
$140$	0	0
$141$	−7.09558	−0.597555
$142$	0	0
$143$	11.8339	0.989599
$144$	0	0
$145$	11.1589	0.926696
$146$	0	0
$147$	2.46031	0.202923
$148$	0	0
$149$	11.2156	0.918817	0.459408	−	0.888225i	$-0.348061\pi$
$149$	11.2156	0.918817	0.459408	+	0.888225i	$0.348061\pi$
$150$	0	0
$151$	13.0211	1.05964	0.529820	−	0.848110i	$-0.322259\pi$
$151$	13.0211	1.05964	0.529820	+	0.848110i	$0.322259\pi$
$152$	0	0
$153$	2.01601	0.162984
$154$	0	0
$155$	5.46974	0.439340
$156$	0	0
$157$	−14.2702	−1.13889	−0.569444	−	0.822030i	$-0.692841\pi$
$157$	−14.2702	−1.13889	−0.569444	+	0.822030i	$0.692841\pi$
$158$	0	0
$159$	−0.561262	−0.0445110
$160$	0	0
$161$	−1.79533	−0.141491
$162$	0	0
$163$	−17.7388	−1.38941	−0.694706	−	0.719294i	$-0.744466\pi$
$163$	−17.7388	−1.38941	−0.694706	+	0.719294i	$0.744466\pi$
$164$	0	0
$165$	5.00542	0.389671
$166$	0	0
$167$	−21.1907	−1.63979	−0.819893	−	0.572516i	$-0.805967\pi$
$167$	−21.1907	−1.63979	−0.819893	+	0.572516i	$0.805967\pi$
$168$	0	0
$169$	15.9228	1.22483
$170$	0	0
$171$	−14.2914	−1.09289
$172$	0	0
$173$	−9.37693	−0.712915	−0.356457	−	0.934312i	$-0.616016\pi$
$173$	−9.37693	−0.712915	−0.356457	+	0.934312i	$0.616016\pi$
$174$	0	0
$175$	0.796353	0.0601986
$176$	0	0
$177$	0.991965	0.0745607
$178$	0	0
$179$	1.08922	0.0814120	0.0407060	−	0.999171i	$-0.487039\pi$
$179$	1.08922	0.0814120	0.0407060	+	0.999171i	$0.487039\pi$
$180$	0	0
$181$	−6.07068	−0.451230	−0.225615	−	0.974217i	$-0.572439\pi$
$181$	−6.07068	−0.451230	−0.225615	+	0.974217i	$0.572439\pi$
$182$	0	0
$183$	−7.18748	−0.531314
$184$	0	0
$185$	9.77101	0.718379
$186$	0	0
$187$	−2.20043	−0.160911
$188$	0	0
$189$	−15.3202	−1.11438
$190$	0	0
$191$	10.6753	0.772435	0.386217	−	0.922408i	$-0.373781\pi$
$191$	10.6753	0.772435	0.386217	+	0.922408i	$0.373781\pi$
$192$	0	0
$193$	−10.7716	−0.775354	−0.387677	−	0.921795i	$-0.626722\pi$
$193$	−10.7716	−0.775354	−0.387677	+	0.921795i	$0.626722\pi$
$194$	0	0
$195$	12.2336	0.876064
$196$	0	0
$197$	−22.4496	−1.59946	−0.799732	−	0.600357i	$-0.795025\pi$
$197$	−22.4496	−1.59946	−0.799732	+	0.600357i	$0.795025\pi$
$198$	0	0
$199$	19.1363	1.35654	0.678270	−	0.734813i	$-0.262730\pi$
$199$	19.1363	1.35654	0.678270	+	0.734813i	$0.262730\pi$
$200$	0	0
$201$	−9.50865	−0.670689
$202$	0	0
$203$	14.9828	1.05159
$204$	0	0
$205$	14.3686	1.00355
$206$	0	0
$207$	1.17551	0.0817034
$208$	0	0
$209$	15.5988	1.07899
$210$	0	0
$211$	−15.8626	−1.09202	−0.546012	−	0.837778i	$-0.683855\pi$
$211$	−15.8626	−1.09202	−0.546012	+	0.837778i	$0.683855\pi$
$212$	0	0
$213$	−13.9642	−0.956809
$214$	0	0
$215$	1.34095	0.0914517
$216$	0	0
$217$	7.34412	0.498551
$218$	0	0
$219$	−11.7407	−0.793365
$220$	0	0
$221$	−5.37799	−0.361763
$222$	0	0
$223$	0.444523	0.0297674	0.0148837	−	0.999889i	$-0.495262\pi$
$223$	0.444523	0.0297674	0.0148837	+	0.999889i	$0.495262\pi$
$224$	0	0
$225$	−0.521420	−0.0347613
$226$	0	0
$227$	−12.7545	−0.846548	−0.423274	−	0.906002i	$-0.639119\pi$
$227$	−12.7545	−0.846548	−0.423274	+	0.906002i	$0.639119\pi$
$228$	0	0
$229$	−13.0351	−0.861381	−0.430691	−	0.902500i	$-0.641730\pi$
$229$	−13.0351	−0.861381	−0.430691	+	0.902500i	$0.641730\pi$
$230$	0	0
$231$	6.72068	0.442188
$232$	0	0
$233$	21.3729	1.40018	0.700092	−	0.714053i	$-0.253142\pi$
$233$	21.3729	1.40018	0.700092	+	0.714053i	$0.253142\pi$
$234$	0	0
$235$	−16.4032	−1.07003
$236$	0	0
$237$	12.3280	0.800790
$238$	0	0
$239$	−24.9746	−1.61547	−0.807735	−	0.589545i	$-0.799307\pi$
$239$	−24.9746	−1.61547	−0.807735	+	0.589545i	$0.799307\pi$
$240$	0	0
$241$	−4.52631	−0.291565	−0.145783	−	0.989317i	$-0.546570\pi$
$241$	−4.52631	−0.291565	−0.145783	+	0.989317i	$0.546570\pi$
$242$	0	0
$243$	16.0305	1.02835
$244$	0	0
$245$	5.68762	0.363369
$246$	0	0
$247$	38.1245	2.42580
$248$	0	0
$249$	−6.66402	−0.422315
$250$	0	0
$251$	16.6259	1.04942	0.524708	−	0.851282i	$-0.324175\pi$
$251$	16.6259	1.04942	0.524708	+	0.851282i	$0.324175\pi$
$252$	0	0
$253$	−1.28304	−0.0806641
$254$	0	0
$255$	−2.27475	−0.142450
$256$	0	0
$257$	13.8388	0.863244	0.431622	−	0.902055i	$-0.357942\pi$
$257$	13.8388	0.863244	0.431622	+	0.902055i	$0.357942\pi$
$258$	0	0
$259$	13.1194	0.815197
$260$	0	0
$261$	−9.81016	−0.607234
$262$	0	0
$263$	−24.9340	−1.53750	−0.768749	−	0.639551i	$-0.779120\pi$
$263$	−24.9340	−1.53750	−0.768749	+	0.639551i	$0.779120\pi$
$264$	0	0
$265$	−1.29750	−0.0797045
$266$	0	0
$267$	10.8514	0.664095
$268$	0	0
$269$	−13.6333	−0.831239	−0.415619	−	0.909539i	$-0.636435\pi$
$269$	−13.6333	−0.831239	−0.415619	+	0.909539i	$0.636435\pi$
$270$	0	0
$271$	25.8078	1.56771	0.783855	−	0.620944i	$-0.213250\pi$
$271$	25.8078	1.56771	0.783855	+	0.620944i	$0.213250\pi$
$272$	0	0
$273$	16.4258	0.994134
$274$	0	0
$275$	0.569119	0.0343191
$276$	0	0
$277$	12.5046	0.751327	0.375663	−	0.926756i	$-0.377415\pi$
$277$	12.5046	0.751327	0.375663	+	0.926756i	$0.377415\pi$
$278$	0	0
$279$	−4.80863	−0.287885
$280$	0	0
$281$	27.5430	1.64308	0.821538	−	0.570153i	$-0.193116\pi$
$281$	27.5430	1.64308	0.821538	+	0.570153i	$0.193116\pi$
$282$	0	0
$283$	16.9929	1.01012	0.505062	−	0.863083i	$-0.331470\pi$
$283$	16.9929	1.01012	0.505062	+	0.863083i	$0.331470\pi$
$284$	0	0
$285$	16.1257	0.955201
$286$	0	0
$287$	19.2925	1.13880
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	7.17690	0.420717
$292$	0	0
$293$	−9.49021	−0.554424	−0.277212	−	0.960809i	$-0.589410\pi$
$293$	−9.49021	−0.554424	−0.277212	+	0.960809i	$0.589410\pi$
$294$	0	0
$295$	2.29317	0.133514
$296$	0	0
$297$	−10.9487	−0.635306
$298$	0	0
$299$	−3.13584	−0.181350
$300$	0	0
$301$	1.80046	0.103777
$302$	0	0
$303$	−6.72581	−0.386388
$304$	0	0
$305$	−16.6156	−0.951409
$306$	0	0
$307$	22.8197	1.30239	0.651194	−	0.758912i	$-0.274269\pi$
$307$	22.8197	1.30239	0.651194	+	0.758912i	$0.274269\pi$
$308$	0	0
$309$	−1.08557	−0.0617559
$310$	0	0
$311$	31.8814	1.80783	0.903915	−	0.427713i	$-0.140681\pi$
$311$	31.8814	1.80783	0.903915	+	0.427713i	$0.140681\pi$
$312$	0	0
$313$	26.9332	1.52236	0.761179	−	0.648542i	$-0.224621\pi$
$313$	26.9332	1.52236	0.761179	+	0.648542i	$0.224621\pi$
$314$	0	0
$315$	−14.2344	−0.802016
$316$	0	0
$317$	−23.7016	−1.33121	−0.665607	−	0.746302i	$-0.731827\pi$
$317$	−23.7016	−1.33121	−0.665607	+	0.746302i	$0.731827\pi$
$318$	0	0
$319$	10.7076	0.599510
$320$	0	0
$321$	−16.9875	−0.948150
$322$	0	0
$323$	−7.08899	−0.394442
$324$	0	0
$325$	1.39096	0.0771568
$326$	0	0
$327$	2.36394	0.130726
$328$	0	0
$329$	−22.0242	−1.21424
$330$	0	0
$331$	7.41917	0.407794	0.203897	−	0.978992i	$-0.434639\pi$
$331$	7.41917	0.407794	0.203897	+	0.978992i	$0.434639\pi$
$332$	0	0
$333$	−8.59002	−0.470730
$334$	0	0
$335$	−21.9816	−1.20098
$336$	0	0
$337$	−20.1664	−1.09853	−0.549267	−	0.835647i	$-0.685093\pi$
$337$	−20.1664	−1.09853	−0.549267	+	0.835647i	$0.685093\pi$
$338$	0	0
$339$	5.74690	0.312129
$340$	0	0
$341$	5.24852	0.284223
$342$	0	0
$343$	−13.9163	−0.751411
$344$	0	0
$345$	−1.32638	−0.0714097
$346$	0	0
$347$	14.1529	0.759770	0.379885	−	0.925034i	$-0.375964\pi$
$347$	14.1529	0.759770	0.379885	+	0.925034i	$0.375964\pi$
$348$	0	0
$349$	−11.3689	−0.608565	−0.304283	−	0.952582i	$-0.598417\pi$
$349$	−11.3689	−0.608565	−0.304283	+	0.952582i	$0.598417\pi$
$350$	0	0
$351$	−26.7593	−1.42830
$352$	0	0
$353$	−17.2668	−0.919017	−0.459508	−	0.888173i	$-0.651974\pi$
$353$	−17.2668	−0.919017	−0.459508	+	0.888173i	$0.651974\pi$
$354$	0	0
$355$	−32.2816	−1.71333
$356$	0	0
$357$	−3.05426	−0.161649
$358$	0	0
$359$	3.56341	0.188069	0.0940347	−	0.995569i	$-0.470024\pi$
$359$	3.56341	0.188069	0.0940347	+	0.995569i	$0.470024\pi$
$360$	0	0
$361$	31.2537	1.64493
$362$	0	0
$363$	−6.10864	−0.320620
$364$	0	0
$365$	−27.1416	−1.42066
$366$	0	0
$367$	22.2839	1.16321	0.581605	−	0.813471i	$-0.302425\pi$
$367$	22.2839	1.16321	0.581605	+	0.813471i	$0.302425\pi$
$368$	0	0
$369$	−12.6319	−0.657592
$370$	0	0
$371$	−1.74212	−0.0904465
$372$	0	0
$373$	7.07098	0.366122	0.183061	−	0.983102i	$-0.441399\pi$
$373$	7.07098	0.366122	0.183061	+	0.983102i	$0.441399\pi$
$374$	0	0
$375$	−10.7854	−0.556955
$376$	0	0
$377$	26.1700	1.34783
$378$	0	0
$379$	28.6344	1.47085	0.735424	−	0.677607i	$-0.236983\pi$
$379$	28.6344	1.47085	0.735424	+	0.677607i	$0.236983\pi$
$380$	0	0
$381$	1.63799	0.0839166
$382$	0	0
$383$	−21.2138	−1.08397	−0.541986	−	0.840387i	$-0.682328\pi$
$383$	−21.2138	−1.08397	−0.541986	+	0.840387i	$0.682328\pi$
$384$	0	0
$385$	15.5365	0.791814
$386$	0	0
$387$	−1.17887	−0.0599253
$388$	0	0
$389$	−9.02987	−0.457833	−0.228916	−	0.973446i	$-0.573518\pi$
$389$	−9.02987	−0.457833	−0.228916	+	0.973446i	$0.573518\pi$
$390$	0	0
$391$	0.583087	0.0294880
$392$	0	0
$393$	17.5492	0.885238
$394$	0	0
$395$	28.4992	1.43395
$396$	0	0
$397$	−2.07093	−0.103937	−0.0519685	−	0.998649i	$-0.516550\pi$
$397$	−2.07093	−0.103937	−0.0519685	+	0.998649i	$0.516550\pi$
$398$	0	0
$399$	21.6516	1.08394
$400$	0	0
$401$	−15.0759	−0.752855	−0.376428	−	0.926446i	$-0.622848\pi$
$401$	−15.0759	−0.752855	−0.376428	+	0.926446i	$0.622848\pi$
$402$	0	0
$403$	12.8277	0.638995
$404$	0	0
$405$	2.55068	0.126744
$406$	0	0
$407$	9.37583	0.464743
$408$	0	0
$409$	−5.91935	−0.292693	−0.146346	−	0.989233i	$-0.546751\pi$
$409$	−5.91935	−0.292693	−0.146346	+	0.989233i	$0.546751\pi$
$410$	0	0
$411$	−9.34922	−0.461163
$412$	0	0
$413$	3.07900	0.151508
$414$	0	0
$415$	−15.4055	−0.756228
$416$	0	0
$417$	19.0489	0.932827
$418$	0	0
$419$	23.2458	1.13563	0.567815	−	0.823156i	$-0.307789\pi$
$419$	23.2458	1.13563	0.567815	+	0.823156i	$0.307789\pi$
$420$	0	0
$421$	−5.88745	−0.286937	−0.143468	−	0.989655i	$-0.545826\pi$
$421$	−5.88745	−0.286937	−0.143468	+	0.989655i	$0.545826\pi$
$422$	0	0
$423$	14.4206	0.701153
$424$	0	0
$425$	−0.258640	−0.0125459
$426$	0	0
$427$	−22.3095	−1.07963
$428$	0	0
$429$	11.7388	0.566754
$430$	0	0
$431$	6.39401	0.307989	0.153994	−	0.988072i	$-0.450786\pi$
$431$	6.39401	0.307989	0.153994	+	0.988072i	$0.450786\pi$
$432$	0	0
$433$	−33.1590	−1.59352	−0.796760	−	0.604296i	$-0.793454\pi$
$433$	−33.1590	−1.59352	−0.796760	+	0.604296i	$0.793454\pi$
$434$	0	0
$435$	11.0692	0.530729
$436$	0	0
$437$	−4.13350	−0.197732
$438$	0	0
$439$	−24.8499	−1.18602	−0.593009	−	0.805196i	$-0.702060\pi$
$439$	−24.8499	−1.18602	−0.593009	+	0.805196i	$0.702060\pi$
$440$	0	0
$441$	−5.00018	−0.238104
$442$	0	0
$443$	−5.43411	−0.258182	−0.129091	−	0.991633i	$-0.541206\pi$
$443$	−5.43411	−0.258182	−0.129091	+	0.991633i	$0.541206\pi$
$444$	0	0
$445$	25.0857	1.18918
$446$	0	0
$447$	11.1255	0.526217
$448$	0	0
$449$	−29.0900	−1.37284	−0.686421	−	0.727204i	$-0.740819\pi$
$449$	−29.0900	−1.37284	−0.686421	+	0.727204i	$0.740819\pi$
$450$	0	0
$451$	13.7875	0.649227
$452$	0	0
$453$	12.9165	0.606868
$454$	0	0
$455$	37.9723	1.78017
$456$	0	0
$457$	2.20703	0.103241	0.0516203	−	0.998667i	$-0.483561\pi$
$457$	2.20703	0.103241	0.0516203	+	0.998667i	$0.483561\pi$
$458$	0	0
$459$	4.97570	0.232246
$460$	0	0
$461$	−1.53820	−0.0716413	−0.0358207	−	0.999358i	$-0.511405\pi$
$461$	−1.53820	−0.0716413	−0.0358207	+	0.999358i	$0.511405\pi$
$462$	0	0
$463$	−4.62027	−0.214722	−0.107361	−	0.994220i	$-0.534240\pi$
$463$	−4.62027	−0.214722	−0.107361	+	0.994220i	$0.534240\pi$
$464$	0	0
$465$	5.42579	0.251615
$466$	0	0
$467$	−6.05330	−0.280113	−0.140057	−	0.990143i	$-0.544728\pi$
$467$	−6.05330	−0.280113	−0.140057	+	0.990143i	$0.544728\pi$
$468$	0	0
$469$	−29.5143	−1.36284
$470$	0	0
$471$	−14.1556	−0.652254
$472$	0	0
$473$	1.28671	0.0591631
$474$	0	0
$475$	1.83350	0.0841266
$476$	0	0
$477$	1.14067	0.0522278
$478$	0	0
$479$	13.3808	0.611386	0.305693	−	0.952130i	$-0.401112\pi$
$479$	13.3808	0.611386	0.305693	+	0.952130i	$0.401112\pi$
$480$	0	0
$481$	22.9152	1.04484
$482$	0	0
$483$	−1.78090	−0.0810338
$484$	0	0
$485$	16.5912	0.753366
$486$	0	0
$487$	−13.2248	−0.599273	−0.299636	−	0.954054i	$-0.596865\pi$
$487$	−13.2248	−0.599273	−0.299636	+	0.954054i	$0.596865\pi$
$488$	0	0
$489$	−17.5963	−0.795732
$490$	0	0
$491$	−11.4800	−0.518086	−0.259043	−	0.965866i	$-0.583407\pi$
$491$	−11.4800	−0.518086	−0.259043	+	0.965866i	$0.583407\pi$
$492$	0	0
$493$	−4.86614	−0.219160
$494$	0	0
$495$	−10.1727	−0.457228
$496$	0	0
$497$	−43.3439	−1.94424
$498$	0	0
$499$	7.81046	0.349644	0.174822	−	0.984600i	$-0.444065\pi$
$499$	7.81046	0.349644	0.174822	+	0.984600i	$0.444065\pi$
$500$	0	0
$501$	−21.0205	−0.939124
$502$	0	0
$503$	−5.22414	−0.232933	−0.116466	−	0.993195i	$-0.537157\pi$
$503$	−5.22414	−0.232933	−0.116466	+	0.993195i	$0.537157\pi$
$504$	0	0
$505$	−15.5484	−0.691894
$506$	0	0
$507$	15.7948	0.701474
$508$	0	0
$509$	−3.17274	−0.140629	−0.0703145	−	0.997525i	$-0.522400\pi$
$509$	−3.17274	−0.140629	−0.0703145	+	0.997525i	$0.522400\pi$
$510$	0	0
$511$	−36.4425	−1.61212
$512$	0	0
$513$	−35.2727	−1.55733
$514$	0	0
$515$	−2.50956	−0.110585
$516$	0	0
$517$	−15.7398	−0.692234
$518$	0	0
$519$	−9.30159	−0.408295
$520$	0	0
$521$	−39.8322	−1.74508	−0.872539	−	0.488544i	$-0.837528\pi$
$521$	−39.8322	−1.74508	−0.872539	+	0.488544i	$0.837528\pi$
$522$	0	0
$523$	−30.1016	−1.31625	−0.658125	−	0.752909i	$-0.728650\pi$
$523$	−30.1016	−1.31625	−0.658125	+	0.752909i	$0.728650\pi$
$524$	0	0
$525$	0.789954	0.0344764
$526$	0	0
$527$	−2.38523	−0.103902
$528$	0	0
$529$	−22.6600	−0.985218
$530$	0	0
$531$	−2.01601	−0.0874872
$532$	0	0
$533$	33.6975	1.45960
$534$	0	0
$535$	−39.2708	−1.69783
$536$	0	0
$537$	1.08047	0.0466256
$538$	0	0
$539$	5.45759	0.235075
$540$	0	0
$541$	13.6338	0.586164	0.293082	−	0.956087i	$-0.405319\pi$
$541$	13.6338	0.586164	0.293082	+	0.956087i	$0.405319\pi$
$542$	0	0
$543$	−6.02190	−0.258425
$544$	0	0
$545$	5.46484	0.234088
$546$	0	0
$547$	15.8828	0.679101	0.339550	−	0.940588i	$-0.389725\pi$
$547$	15.8828	0.679101	0.339550	+	0.940588i	$0.389725\pi$
$548$	0	0
$549$	14.6074	0.623427
$550$	0	0
$551$	34.4960	1.46958
$552$	0	0
$553$	38.2654	1.62721
$554$	0	0
$555$	9.69250	0.411424
$556$	0	0
$557$	8.10061	0.343234	0.171617	−	0.985164i	$-0.445101\pi$
$557$	8.10061	0.343234	0.171617	+	0.985164i	$0.445101\pi$
$558$	0	0
$559$	3.14481	0.133011
$560$	0	0
$561$	−2.18275	−0.0921557
$562$	0	0
$563$	−4.12337	−0.173779	−0.0868897	−	0.996218i	$-0.527693\pi$
$563$	−4.12337	−0.173779	−0.0868897	+	0.996218i	$0.527693\pi$
$564$	0	0
$565$	13.2854	0.558920
$566$	0	0
$567$	3.42475	0.143826
$568$	0	0
$569$	39.7443	1.66617	0.833085	−	0.553145i	$-0.186573\pi$
$569$	39.7443	1.66617	0.833085	+	0.553145i	$0.186573\pi$
$570$	0	0
$571$	−27.4986	−1.15078	−0.575390	−	0.817879i	$-0.695150\pi$
$571$	−27.4986	−1.15078	−0.575390	+	0.817879i	$0.695150\pi$
$572$	0	0
$573$	10.5895	0.442382
$574$	0	0
$575$	−0.150810	−0.00628920
$576$	0	0
$577$	36.4792	1.51865	0.759324	−	0.650713i	$-0.225530\pi$
$577$	36.4792	1.51865	0.759324	+	0.650713i	$0.225530\pi$
$578$	0	0
$579$	−10.6850	−0.444054
$580$	0	0
$581$	−20.6847	−0.858147
$582$	0	0
$583$	−1.24502	−0.0515635
$584$	0	0
$585$	−24.8627	−1.02795
$586$	0	0
$587$	−20.2523	−0.835900	−0.417950	−	0.908470i	$-0.637251\pi$
$587$	−20.2523	−0.835900	−0.417950	+	0.908470i	$0.637251\pi$
$588$	0	0
$589$	16.9089	0.696717
$590$	0	0
$591$	−22.2692	−0.916032
$592$	0	0
$593$	−12.1100	−0.497298	−0.248649	−	0.968594i	$-0.579987\pi$
$593$	−12.1100	−0.497298	−0.248649	+	0.968594i	$0.579987\pi$
$594$	0	0
$595$	−7.06068	−0.289460
$596$	0	0
$597$	18.9826	0.776905
$598$	0	0
$599$	28.6064	1.16883	0.584413	−	0.811457i	$-0.301325\pi$
$599$	28.6064	1.16883	0.584413	+	0.811457i	$0.301325\pi$
$600$	0	0
$601$	45.3264	1.84890	0.924451	−	0.381300i	$-0.124523\pi$
$601$	45.3264	1.84890	0.924451	+	0.381300i	$0.124523\pi$
$602$	0	0
$603$	19.3248	0.786965
$604$	0	0
$605$	−14.1216	−0.574126
$606$	0	0
$607$	12.2354	0.496618	0.248309	−	0.968681i	$-0.420125\pi$
$607$	12.2354	0.496618	0.248309	+	0.968681i	$0.420125\pi$
$608$	0	0
$609$	14.8625	0.602257
$610$	0	0
$611$	−38.4690	−1.55629
$612$	0	0
$613$	−37.1601	−1.50088	−0.750441	−	0.660938i	$-0.770159\pi$
$613$	−37.1601	−1.50088	−0.750441	+	0.660938i	$0.770159\pi$
$614$	0	0
$615$	14.2532	0.574743
$616$	0	0
$617$	4.16954	0.167859	0.0839297	−	0.996472i	$-0.473253\pi$
$617$	4.16954	0.167859	0.0839297	+	0.996472i	$0.473253\pi$
$618$	0	0
$619$	16.4463	0.661032	0.330516	−	0.943800i	$-0.392777\pi$
$619$	16.4463	0.661032	0.330516	+	0.943800i	$0.392777\pi$
$620$	0	0
$621$	2.90127	0.116424
$622$	0	0
$623$	33.6821	1.34944
$624$	0	0
$625$	−26.2263	−1.04905
$626$	0	0
$627$	15.4735	0.617951
$628$	0	0
$629$	−4.26091	−0.169894
$630$	0	0
$631$	−5.98538	−0.238274	−0.119137	−	0.992878i	$-0.538013\pi$
$631$	−5.98538	−0.238274	−0.119137	+	0.992878i	$0.538013\pi$
$632$	0	0
$633$	−15.7351	−0.625414
$634$	0	0
$635$	3.78661	0.150267
$636$	0	0
$637$	13.3387	0.528499
$638$	0	0
$639$	28.3798	1.12269
$640$	0	0
$641$	11.3931	0.449999	0.224999	−	0.974359i	$-0.427762\pi$
$641$	11.3931	0.449999	0.224999	+	0.974359i	$0.427762\pi$
$642$	0	0
$643$	−22.3119	−0.879896	−0.439948	−	0.898023i	$-0.645003\pi$
$643$	−22.3119	−0.879896	−0.439948	+	0.898023i	$0.645003\pi$
$644$	0	0
$645$	1.33017	0.0523754
$646$	0	0
$647$	5.08778	0.200021	0.100011	−	0.994986i	$-0.468112\pi$
$647$	5.08778	0.200021	0.100011	+	0.994986i	$0.468112\pi$
$648$	0	0
$649$	2.20043	0.0863743
$650$	0	0
$651$	7.28511	0.285526
$652$	0	0
$653$	24.9253	0.975404	0.487702	−	0.873010i	$-0.337835\pi$
$653$	24.9253	0.975404	0.487702	+	0.873010i	$0.337835\pi$
$654$	0	0
$655$	40.5692	1.58517
$656$	0	0
$657$	23.8611	0.930910
$658$	0	0
$659$	30.1013	1.17258	0.586290	−	0.810101i	$-0.300588\pi$
$659$	30.1013	1.17258	0.586290	+	0.810101i	$0.300588\pi$
$660$	0	0
$661$	18.6275	0.724526	0.362263	−	0.932076i	$-0.382004\pi$
$661$	18.6275	0.724526	0.362263	+	0.932076i	$0.382004\pi$
$662$	0	0
$663$	−5.33478	−0.207186
$664$	0	0
$665$	50.0531	1.94097
$666$	0	0
$667$	−2.83738	−0.109864
$668$	0	0
$669$	0.440951	0.0170482
$670$	0	0
$671$	−15.9436	−0.615497
$672$	0	0
$673$	−48.7279	−1.87832	−0.939162	−	0.343476i	$-0.888396\pi$
$673$	−48.7279	−1.87832	−0.939162	+	0.343476i	$0.888396\pi$
$674$	0	0
$675$	−1.28692	−0.0495334
$676$	0	0
$677$	39.6876	1.52532	0.762660	−	0.646800i	$-0.223893\pi$
$677$	39.6876	1.52532	0.762660	+	0.646800i	$0.223893\pi$
$678$	0	0
$679$	22.2767	0.854900
$680$	0	0
$681$	−12.6521	−0.484828
$682$	0	0
$683$	5.75339	0.220147	0.110074	−	0.993923i	$-0.464891\pi$
$683$	5.75339	0.220147	0.110074	+	0.993923i	$0.464891\pi$
$684$	0	0
$685$	−21.6130	−0.825791
$686$	0	0
$687$	−12.9303	−0.493323
$688$	0	0
$689$	−3.04291	−0.115926
$690$	0	0
$691$	15.4897	0.589255	0.294628	−	0.955612i	$-0.404804\pi$
$691$	15.4897	0.589255	0.294628	+	0.955612i	$0.404804\pi$
$692$	0	0
$693$	−13.6587	−0.518850
$694$	0	0
$695$	44.0362	1.67039
$696$	0	0
$697$	−6.26582	−0.237335
$698$	0	0
$699$	21.2012	0.801901
$700$	0	0
$701$	−7.42275	−0.280353	−0.140177	−	0.990127i	$-0.544767\pi$
$701$	−7.42275	−0.280353	−0.140177	+	0.990127i	$0.544767\pi$
$702$	0	0
$703$	30.2056	1.13922
$704$	0	0
$705$	−16.2714	−0.612816
$706$	0	0
$707$	−20.8765	−0.785142
$708$	0	0
$709$	34.9742	1.31348	0.656742	−	0.754115i	$-0.271934\pi$
$709$	34.9742	1.31348	0.656742	+	0.754115i	$0.271934\pi$
$710$	0	0
$711$	−25.0546	−0.939622
$712$	0	0
$713$	−1.39080	−0.0520857
$714$	0	0
$715$	27.1371	1.01487
$716$	0	0
$717$	−24.7739	−0.925198
$718$	0	0
$719$	−4.66492	−0.173972	−0.0869862	−	0.996210i	$-0.527724\pi$
$719$	−4.66492	−0.173972	−0.0869862	+	0.996210i	$0.527724\pi$
$720$	0	0
$721$	−3.36954	−0.125488
$722$	0	0
$723$	−4.48994	−0.166983
$724$	0	0
$725$	1.25858	0.0467424
$726$	0	0
$727$	42.5842	1.57936	0.789679	−	0.613520i	$-0.210247\pi$
$727$	42.5842	1.57936	0.789679	+	0.613520i	$0.210247\pi$
$728$	0	0
$729$	12.5648	0.465362
$730$	0	0
$731$	−0.584755	−0.0216280
$732$	0	0
$733$	−9.12726	−0.337123	−0.168561	−	0.985691i	$-0.553912\pi$
$733$	−9.12726	−0.337123	−0.168561	+	0.985691i	$0.553912\pi$
$734$	0	0
$735$	5.64192	0.208106
$736$	0	0
$737$	−21.0926	−0.776955
$738$	0	0
$739$	8.98411	0.330486	0.165243	−	0.986253i	$-0.447159\pi$
$739$	8.98411	0.330486	0.165243	+	0.986253i	$0.447159\pi$
$740$	0	0
$741$	37.8182	1.38929
$742$	0	0
$743$	−19.3487	−0.709836	−0.354918	−	0.934898i	$-0.615491\pi$
$743$	−19.3487	−0.709836	−0.354918	+	0.934898i	$0.615491\pi$
$744$	0	0
$745$	25.7193	0.942281
$746$	0	0
$747$	13.5435	0.495531
$748$	0	0
$749$	−52.7282	−1.92665
$750$	0	0
$751$	44.2128	1.61335	0.806674	−	0.590997i	$-0.201266\pi$
$751$	44.2128	1.61335	0.806674	+	0.590997i	$0.201266\pi$
$752$	0	0
$753$	16.4923	0.601012
$754$	0	0
$755$	29.8596	1.08670
$756$	0	0
$757$	7.74986	0.281673	0.140837	−	0.990033i	$-0.455021\pi$
$757$	7.74986	0.281673	0.140837	+	0.990033i	$0.455021\pi$
$758$	0	0
$759$	−1.27273	−0.0461972
$760$	0	0
$761$	−41.7505	−1.51345	−0.756727	−	0.653731i	$-0.773203\pi$
$761$	−41.7505	−1.51345	−0.756727	+	0.653731i	$0.773203\pi$
$762$	0	0
$763$	7.33753	0.265637
$764$	0	0
$765$	4.62305	0.167147
$766$	0	0
$767$	5.37799	0.194188
$768$	0	0
$769$	−43.1753	−1.55694	−0.778470	−	0.627682i	$-0.784004\pi$
$769$	−43.1753	−1.55694	−0.778470	+	0.627682i	$0.784004\pi$
$770$	0	0
$771$	13.7277	0.494389
$772$	0	0
$773$	18.5357	0.666683	0.333342	−	0.942806i	$-0.391824\pi$
$773$	18.5357	0.666683	0.333342	+	0.942806i	$0.391824\pi$
$774$	0	0
$775$	0.616916	0.0221603
$776$	0	0
$777$	13.0139	0.466873
$778$	0	0
$779$	44.4183	1.59145
$780$	0	0
$781$	−30.9760	−1.10841
$782$	0	0
$783$	−24.2125	−0.865283
$784$	0	0
$785$	−32.7241	−1.16797
$786$	0	0
$787$	22.5523	0.803902	0.401951	−	0.915661i	$-0.368332\pi$
$787$	22.5523	0.803902	0.401951	+	0.915661i	$0.368332\pi$
$788$	0	0
$789$	−24.7337	−0.880542
$790$	0	0
$791$	17.8380	0.634248
$792$	0	0
$793$	−38.9673	−1.38377
$794$	0	0
$795$	−1.28707	−0.0456477
$796$	0	0
$797$	−0.826359	−0.0292711	−0.0146356	−	0.999893i	$-0.504659\pi$
$797$	−0.826359	−0.0292711	−0.0146356	+	0.999893i	$0.504659\pi$
$798$	0	0
$799$	7.15305	0.253057
$800$	0	0
$801$	−22.0537	−0.779228
$802$	0	0
$803$	−26.0439	−0.919069
$804$	0	0
$805$	−4.11699	−0.145105
$806$	0	0
$807$	−13.5238	−0.476060
$808$	0	0
$809$	29.5638	1.03941	0.519705	−	0.854346i	$-0.326042\pi$
$809$	29.5638	1.03941	0.519705	+	0.854346i	$0.326042\pi$
$810$	0	0
$811$	−54.1931	−1.90298	−0.951488	−	0.307685i	$-0.900446\pi$
$811$	−54.1931	−1.90298	−0.951488	+	0.307685i	$0.900446\pi$
$812$	0	0
$813$	25.6004	0.897846
$814$	0	0
$815$	−40.6782	−1.42490
$816$	0	0
$817$	4.14532	0.145027
$818$	0	0
$819$	−33.3827	−1.16649
$820$	0	0
$821$	42.4998	1.48325	0.741626	−	0.670814i	$-0.234055\pi$
$821$	42.4998	1.48325	0.741626	+	0.670814i	$0.234055\pi$
$822$	0	0
$823$	16.6844	0.581580	0.290790	−	0.956787i	$-0.406082\pi$
$823$	16.6844	0.581580	0.290790	+	0.956787i	$0.406082\pi$
$824$	0	0
$825$	0.564546	0.0196550
$826$	0	0
$827$	−0.627582	−0.0218232	−0.0109116	−	0.999940i	$-0.503473\pi$
$827$	−0.627582	−0.0218232	−0.0109116	+	0.999940i	$0.503473\pi$
$828$	0	0
$829$	15.4172	0.535462	0.267731	−	0.963494i	$-0.413726\pi$
$829$	15.4172	0.535462	0.267731	+	0.963494i	$0.413726\pi$
$830$	0	0
$831$	12.4041	0.430293
$832$	0	0
$833$	−2.48024	−0.0859353
$834$	0	0
$835$	−48.5940	−1.68166
$836$	0	0
$837$	−11.8682	−0.410225
$838$	0	0
$839$	−44.8691	−1.54905	−0.774526	−	0.632542i	$-0.782012\pi$
$839$	−44.8691	−1.54905	−0.774526	+	0.632542i	$0.782012\pi$
$840$	0	0
$841$	−5.32070	−0.183472
$842$	0	0
$843$	27.3217	0.941009
$844$	0	0
$845$	36.5137	1.25611
$846$	0	0
$847$	−18.9608	−0.651502
$848$	0	0
$849$	16.8564	0.578510
$850$	0	0
$851$	−2.48448	−0.0851670
$852$	0	0
$853$	−20.5883	−0.704930	−0.352465	−	0.935825i	$-0.614657\pi$
$853$	−20.5883	−0.704930	−0.352465	+	0.935825i	$0.614657\pi$
$854$	0	0
$855$	−32.7727	−1.12080
$856$	0	0
$857$	−54.2041	−1.85158	−0.925788	−	0.378043i	$-0.876597\pi$
$857$	−54.2041	−1.85158	−0.925788	+	0.378043i	$0.876597\pi$
$858$	0	0
$859$	19.7855	0.675073	0.337536	−	0.941312i	$-0.390406\pi$
$859$	19.7855	0.675073	0.337536	+	0.941312i	$0.390406\pi$
$860$	0	0
$861$	19.1375	0.652203
$862$	0	0
$863$	39.7565	1.35333	0.676664	−	0.736292i	$-0.263425\pi$
$863$	39.7565	1.35333	0.676664	+	0.736292i	$0.263425\pi$
$864$	0	0
$865$	−21.5029	−0.731121
$866$	0	0
$867$	0.991965	0.0336889
$868$	0	0
$869$	27.3466	0.927670
$870$	0	0
$871$	−51.5516	−1.74676
$872$	0	0
$873$	−14.5859	−0.493657
$874$	0	0
$875$	−33.4772	−1.13174
$876$	0	0
$877$	12.8874	0.435176	0.217588	−	0.976041i	$-0.430181\pi$
$877$	12.8874	0.435176	0.217588	+	0.976041i	$0.430181\pi$
$878$	0	0
$879$	−9.41395	−0.317525
$880$	0	0
$881$	57.9932	1.95384	0.976920	−	0.213607i	$-0.0685212\pi$
$881$	57.9932	1.95384	0.976920	+	0.213607i	$0.0685212\pi$
$882$	0	0
$883$	−8.84117	−0.297529	−0.148765	−	0.988873i	$-0.547530\pi$
$883$	−8.84117	−0.297529	−0.148765	+	0.988873i	$0.547530\pi$
$884$	0	0
$885$	2.27475	0.0764648
$886$	0	0
$887$	32.0820	1.07721	0.538604	−	0.842559i	$-0.318952\pi$
$887$	32.0820	1.07721	0.538604	+	0.842559i	$0.318952\pi$
$888$	0	0
$889$	5.08421	0.170519
$890$	0	0
$891$	2.44752	0.0819949
$892$	0	0
$893$	−50.7079	−1.69687
$894$	0	0
$895$	2.49777	0.0834911
$896$	0	0
$897$	−3.11064	−0.103861
$898$	0	0
$899$	11.6069	0.387110
$900$	0	0
$901$	0.565808	0.0188498
$902$	0	0
$903$	1.78600	0.0594342
$904$	0	0
$905$	−13.9211	−0.462754
$906$	0	0
$907$	15.6291	0.518957	0.259478	−	0.965749i	$-0.416449\pi$
$907$	15.6291	0.518957	0.259478	+	0.965749i	$0.416449\pi$
$908$	0	0
$909$	13.6691	0.453375
$910$	0	0
$911$	−6.44069	−0.213390	−0.106695	−	0.994292i	$-0.534027\pi$
$911$	−6.44069	−0.213390	−0.106695	+	0.994292i	$0.534027\pi$
$912$	0	0
$913$	−14.7825	−0.489228
$914$	0	0
$915$	−16.4821	−0.544883
$916$	0	0
$917$	54.4716	1.79881
$918$	0	0
$919$	14.1042	0.465255	0.232628	−	0.972566i	$-0.425268\pi$
$919$	14.1042	0.465255	0.232628	+	0.972566i	$0.425268\pi$
$920$	0	0
$921$	22.6363	0.745892
$922$	0	0
$923$	−75.7074	−2.49194
$924$	0	0
$925$	1.10204	0.0362349
$926$	0	0
$927$	2.20624	0.0724625
$928$	0	0
$929$	0.535348	0.0175642	0.00878210	−	0.999961i	$-0.497205\pi$
$929$	0.535348	0.0175642	0.00878210	+	0.999961i	$0.497205\pi$
$930$	0	0
$931$	17.5824	0.576240
$932$	0	0
$933$	31.6253	1.03536
$934$	0	0
$935$	−5.04596	−0.165021
$936$	0	0
$937$	24.7981	0.810117	0.405059	−	0.914291i	$-0.367251\pi$
$937$	24.7981	0.810117	0.405059	+	0.914291i	$0.367251\pi$
$938$	0	0
$939$	26.7168	0.871871
$940$	0	0
$941$	−27.3063	−0.890159	−0.445080	−	0.895491i	$-0.646825\pi$
$941$	−27.3063	−0.890159	−0.445080	+	0.895491i	$0.646825\pi$
$942$	0	0
$943$	−3.65352	−0.118975
$944$	0	0
$945$	−35.1318	−1.14284
$946$	0	0
$947$	25.6103	0.832222	0.416111	−	0.909314i	$-0.363393\pi$
$947$	25.6103	0.832222	0.416111	+	0.909314i	$0.363393\pi$
$948$	0	0
$949$	−63.6530	−2.06626
$950$	0	0
$951$	−23.5112	−0.762402
$952$	0	0
$953$	−2.03269	−0.0658453	−0.0329227	−	0.999458i	$-0.510481\pi$
$953$	−2.03269	−0.0658453	−0.0329227	+	0.999458i	$0.510481\pi$
$954$	0	0
$955$	24.4802	0.792161
$956$	0	0
$957$	10.6216	0.343346
$958$	0	0
$959$	−29.0194	−0.937085
$960$	0	0
$961$	−25.3107	−0.816474
$962$	0	0
$963$	34.5243	1.11253
$964$	0	0
$965$	−24.7010	−0.795155
$966$	0	0
$967$	29.7682	0.957282	0.478641	−	0.878011i	$-0.341130\pi$
$967$	29.7682	0.957282	0.478641	+	0.878011i	$0.341130\pi$
$968$	0	0
$969$	−7.03203	−0.225901
$970$	0	0
$971$	41.5037	1.33192	0.665959	−	0.745989i	$-0.268023\pi$
$971$	41.5037	1.33192	0.665959	+	0.745989i	$0.268023\pi$
$972$	0	0
$973$	59.1265	1.89551
$974$	0	0
$975$	1.37979	0.0441886
$976$	0	0
$977$	22.4511	0.718275	0.359138	−	0.933285i	$-0.383071\pi$
$977$	22.4511	0.718275	0.359138	+	0.933285i	$0.383071\pi$
$978$	0	0
$979$	24.0711	0.769316
$980$	0	0
$981$	−4.80432	−0.153390
$982$	0	0
$983$	−29.0760	−0.927381	−0.463690	−	0.885997i	$-0.653475\pi$
$983$	−29.0760	−0.927381	−0.463690	+	0.885997i	$0.653475\pi$
$984$	0	0
$985$	−51.4807	−1.64031
$986$	0	0
$987$	−21.8473	−0.695407
$988$	0	0
$989$	−0.340963	−0.0108420
$990$	0	0
$991$	48.1204	1.52859	0.764297	−	0.644865i	$-0.223086\pi$
$991$	48.1204	1.52859	0.764297	+	0.644865i	$0.223086\pi$
$992$	0	0
$993$	7.35956	0.233549
$994$	0	0
$995$	43.8829	1.39118
$996$	0	0
$997$	−26.4464	−0.837566	−0.418783	−	0.908086i	$-0.637543\pi$
$997$	−26.4464	−0.837566	−0.418783	+	0.908086i	$0.637543\pi$
$998$	0	0
$999$	−21.2010	−0.670771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.ba.1.20	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.ba.1.20	✓	30	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+0.991965 q^{3} +2.29317 q^{5} +3.07900 q^{7} -2.01601 q^{9} +O(q^{10})\)	\(q+0.991965 q^{3} +2.29317 q^{5} +3.07900 q^{7} -2.01601 q^{9} +2.20043 q^{11} +5.37799 q^{13} +2.27475 q^{15} -1.00000 q^{17} +7.08899 q^{19} +3.05426 q^{21} -0.583087 q^{23} +0.258640 q^{25} -4.97570 q^{27} +4.86614 q^{29} +2.38523 q^{31} +2.18275 q^{33} +7.06068 q^{35} +4.26091 q^{37} +5.33478 q^{39} +6.26582 q^{41} +0.584755 q^{43} -4.62305 q^{45} -7.15305 q^{47} +2.48024 q^{49} -0.991965 q^{51} -0.565808 q^{53} +5.04596 q^{55} +7.03203 q^{57} +1.00000 q^{59} -7.24570 q^{61} -6.20728 q^{63} +12.3327 q^{65} -9.58567 q^{67} -0.578402 q^{69} -14.0773 q^{71} -11.8358 q^{73} +0.256562 q^{75} +6.77512 q^{77} +12.4279 q^{79} +1.11229 q^{81} -6.71800 q^{83} -2.29317 q^{85} +4.82704 q^{87} +10.9393 q^{89} +16.5588 q^{91} +2.36606 q^{93} +16.2563 q^{95} +7.23503 q^{97} -4.43607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more