LMFDB - Embedded newform 8036.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.287349.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} + 7x - 1 $ x^5 - 6x^3 + 7x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.14203$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.695770 q^{3} -1.38288 q^{5} -2.51590 q^{9} +O(q^{10})$	$q+0.695770 q^{3} -1.38288 q^{5} -2.51590 q^{9} +0.420713 q^{11} +1.63239 q^{13} -0.962164 q^{15} -4.89878 q^{17} -3.98771 q^{19} +6.81774 q^{23} -3.08765 q^{25} -3.83780 q^{27} +0.394048 q^{29} -5.18524 q^{31} +0.292720 q^{33} +3.85546 q^{37} +1.13577 q^{39} +1.00000 q^{41} +4.97445 q^{43} +3.47919 q^{45} +3.46041 q^{47} -3.40842 q^{51} +1.26091 q^{53} -0.581795 q^{55} -2.77453 q^{57} -14.8049 q^{59} +9.77191 q^{61} -2.25739 q^{65} -10.9655 q^{67} +4.74357 q^{69} +16.3205 q^{71} -2.14740 q^{73} -2.14829 q^{75} +7.49774 q^{79} +4.87749 q^{81} +4.29395 q^{83} +6.77442 q^{85} +0.274167 q^{87} +3.54474 q^{89} -3.60773 q^{93} +5.51452 q^{95} -17.4675 q^{97} -1.05847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 3 q^{5} + q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 3 * q^5 + q^9	$ 5 q - 2 q^{3} + 3 q^{5} + q^{9} + 6 q^{11} - 7 q^{13} + 9 q^{15} - q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 8 q^{27} + 11 q^{29} - 13 q^{31} - 11 q^{33} + 5 q^{37} + 23 q^{39} + 5 q^{41} + 29 q^{43} - 11 q^{45} - 7 q^{47} - 3 q^{51} + 21 q^{53} - 19 q^{55} + 9 q^{57} - 3 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} - 10 q^{69} + 22 q^{71} + 16 q^{73} + 18 q^{75} + 4 q^{79} - 15 q^{81} + 6 q^{83} + 13 q^{85} - 6 q^{87} + 20 q^{89} - 5 q^{93} - 7 q^{95} + 24 q^{97} + 27 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 3 * q^5 + q^9 + 6 * q^11 - 7 * q^13 + 9 * q^15 - q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 - 8 * q^27 + 11 * q^29 - 13 * q^31 - 11 * q^33 + 5 * q^37 + 23 * q^39 + 5 * q^41 + 29 * q^43 - 11 * q^45 - 7 * q^47 - 3 * q^51 + 21 * q^53 - 19 * q^55 + 9 * q^57 - 3 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 - 10 * q^69 + 22 * q^71 + 16 * q^73 + 18 * q^75 + 4 * q^79 - 15 * q^81 + 6 * q^83 + 13 * q^85 - 6 * q^87 + 20 * q^89 - 5 * q^93 - 7 * q^95 + 24 * q^97 + 27 * 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
$3$	0.695770	0.401703	0.200851	−	0.979622i	$-0.435629\pi$
$3$	0.695770	0.401703	0.200851	+	0.979622i	$0.435629\pi$
$4$	0	0
$5$	−1.38288	−0.618442	−0.309221	−	0.950990i	$-0.600068\pi$
$5$	−1.38288	−0.618442	−0.309221	+	0.950990i	$0.600068\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.51590	−0.838635
$10$	0	0
$11$	0.420713	0.126850	0.0634249	−	0.997987i	$-0.479798\pi$
$11$	0.420713	0.126850	0.0634249	+	0.997987i	$0.479798\pi$
$12$	0	0
$13$	1.63239	0.452743	0.226371	−	0.974041i	$-0.427314\pi$
$13$	1.63239	0.452743	0.226371	+	0.974041i	$0.427314\pi$
$14$	0	0
$15$	−0.962164	−0.248430
$16$	0	0
$17$	−4.89878	−1.18813	−0.594065	−	0.804417i	$-0.702478\pi$
$17$	−4.89878	−1.18813	−0.594065	+	0.804417i	$0.702478\pi$
$18$	0	0
$19$	−3.98771	−0.914843	−0.457422	−	0.889250i	$-0.651227\pi$
$19$	−3.98771	−0.914843	−0.457422	+	0.889250i	$0.651227\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.81774	1.42160	0.710798	−	0.703396i	$-0.248334\pi$
$23$	6.81774	1.42160	0.710798	+	0.703396i	$0.248334\pi$
$24$	0	0
$25$	−3.08765	−0.617530
$26$	0	0
$27$	−3.83780	−0.738585
$28$	0	0
$29$	0.394048	0.0731729	0.0365864	−	0.999330i	$-0.488352\pi$
$29$	0.394048	0.0731729	0.0365864	+	0.999330i	$0.488352\pi$
$30$	0	0
$31$	−5.18524	−0.931296	−0.465648	−	0.884970i	$-0.654179\pi$
$31$	−5.18524	−0.931296	−0.465648	+	0.884970i	$0.654179\pi$
$32$	0	0
$33$	0.292720	0.0509559
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.85546	0.633834	0.316917	−	0.948453i	$-0.397352\pi$
$37$	3.85546	0.633834	0.316917	+	0.948453i	$0.397352\pi$
$38$	0	0
$39$	1.13577	0.181868
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	4.97445	0.758597	0.379299	−	0.925274i	$-0.376165\pi$
$43$	4.97445	0.758597	0.379299	+	0.925274i	$0.376165\pi$
$44$	0	0
$45$	3.47919	0.518647
$46$	0	0
$47$	3.46041	0.504752	0.252376	−	0.967629i	$-0.418788\pi$
$47$	3.46041	0.504752	0.252376	+	0.967629i	$0.418788\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.40842	−0.477275
$52$	0	0
$53$	1.26091	0.173199	0.0865997	−	0.996243i	$-0.472400\pi$
$53$	1.26091	0.173199	0.0865997	+	0.996243i	$0.472400\pi$
$54$	0	0
$55$	−0.581795	−0.0784493
$56$	0	0
$57$	−2.77453	−0.367495
$58$	0	0
$59$	−14.8049	−1.92744	−0.963720	−	0.266915i	$-0.913996\pi$
$59$	−14.8049	−1.92744	−0.963720	+	0.266915i	$0.913996\pi$
$60$	0	0
$61$	9.77191	1.25116	0.625582	−	0.780158i	$-0.284861\pi$
$61$	9.77191	1.25116	0.625582	+	0.780158i	$0.284861\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.25739	−0.279995
$66$	0	0
$67$	−10.9655	−1.33965	−0.669823	−	0.742521i	$-0.733630\pi$
$67$	−10.9655	−1.33965	−0.669823	+	0.742521i	$0.733630\pi$
$68$	0	0
$69$	4.74357	0.571059
$70$	0	0
$71$	16.3205	1.93688	0.968441	−	0.249242i	$-0.0801814\pi$
$71$	16.3205	1.93688	0.968441	+	0.249242i	$0.0801814\pi$
$72$	0	0
$73$	−2.14740	−0.251334	−0.125667	−	0.992072i	$-0.540107\pi$
$73$	−2.14740	−0.251334	−0.125667	+	0.992072i	$0.540107\pi$
$74$	0	0
$75$	−2.14829	−0.248063
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.49774	0.843562	0.421781	−	0.906698i	$-0.361405\pi$
$79$	7.49774	0.843562	0.421781	+	0.906698i	$0.361405\pi$
$80$	0	0
$81$	4.87749	0.541943
$82$	0	0
$83$	4.29395	0.471322	0.235661	−	0.971835i	$-0.424274\pi$
$83$	4.29395	0.471322	0.235661	+	0.971835i	$0.424274\pi$
$84$	0	0
$85$	6.77442	0.734789
$86$	0	0
$87$	0.274167	0.0293937
$88$	0	0
$89$	3.54474	0.375742	0.187871	−	0.982194i	$-0.439841\pi$
$89$	3.54474	0.375742	0.187871	+	0.982194i	$0.439841\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.60773	−0.374104
$94$	0	0
$95$	5.51452	0.565777
$96$	0	0
$97$	−17.4675	−1.77355	−0.886777	−	0.462198i	$-0.847061\pi$
$97$	−17.4675	−1.77355	−0.886777	+	0.462198i	$0.847061\pi$
$98$	0	0
$99$	−1.05847	−0.106381
$100$	0	0
$101$	−1.60684	−0.159887	−0.0799434	−	0.996799i	$-0.525474\pi$
$101$	−1.60684	−0.159887	−0.0799434	+	0.996799i	$0.525474\pi$
$102$	0	0
$103$	16.5926	1.63492	0.817460	−	0.575986i	$-0.195382\pi$
$103$	16.5926	1.63492	0.817460	+	0.575986i	$0.195382\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.7025	1.51802	0.759009	−	0.651080i	$-0.225684\pi$
$107$	15.7025	1.51802	0.759009	+	0.651080i	$0.225684\pi$
$108$	0	0
$109$	17.3795	1.66465	0.832326	−	0.554286i	$-0.187009\pi$
$109$	17.3795	1.66465	0.832326	+	0.554286i	$0.187009\pi$
$110$	0	0
$111$	2.68251	0.254613
$112$	0	0
$113$	−14.6674	−1.37979	−0.689897	−	0.723908i	$-0.742344\pi$
$113$	−14.6674	−1.37979	−0.689897	+	0.723908i	$0.742344\pi$
$114$	0	0
$115$	−9.42810	−0.879175
$116$	0	0
$117$	−4.10693	−0.379686
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.8230	−0.983909
$122$	0	0
$123$	0.695770	0.0627354
$124$	0	0
$125$	11.1842	1.00035
$126$	0	0
$127$	9.47509	0.840779	0.420389	−	0.907344i	$-0.361894\pi$
$127$	9.47509	0.840779	0.420389	+	0.907344i	$0.361894\pi$
$128$	0	0
$129$	3.46107	0.304731
$130$	0	0
$131$	8.50836	0.743379	0.371690	−	0.928357i	$-0.378779\pi$
$131$	8.50836	0.743379	0.371690	+	0.928357i	$0.378779\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.30721	0.456772
$136$	0	0
$137$	2.07954	0.177667	0.0888334	−	0.996046i	$-0.471686\pi$
$137$	2.07954	0.177667	0.0888334	+	0.996046i	$0.471686\pi$
$138$	0	0
$139$	0.826739	0.0701231	0.0350615	−	0.999385i	$-0.488837\pi$
$139$	0.826739	0.0701231	0.0350615	+	0.999385i	$0.488837\pi$
$140$	0	0
$141$	2.40764	0.202760
$142$	0	0
$143$	0.686768	0.0574304
$144$	0	0
$145$	−0.544920	−0.0452531
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.8316	−1.54275	−0.771373	−	0.636383i	$-0.780430\pi$
$149$	−18.8316	−1.54275	−0.771373	+	0.636383i	$0.780430\pi$
$150$	0	0
$151$	15.8939	1.29343	0.646713	−	0.762734i	$-0.276143\pi$
$151$	15.8939	1.29343	0.646713	+	0.762734i	$0.276143\pi$
$152$	0	0
$153$	12.3249	0.996407
$154$	0	0
$155$	7.17055	0.575953
$156$	0	0
$157$	−15.7782	−1.25924	−0.629618	−	0.776905i	$-0.716788\pi$
$157$	−15.7782	−1.25924	−0.629618	+	0.776905i	$0.716788\pi$
$158$	0	0
$159$	0.877303	0.0695747
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.94463	0.778923	0.389462	−	0.921043i	$-0.372661\pi$
$163$	9.94463	0.778923	0.389462	+	0.921043i	$0.372661\pi$
$164$	0	0
$165$	−0.404795	−0.0315133
$166$	0	0
$167$	11.0363	0.854014	0.427007	−	0.904248i	$-0.359568\pi$
$167$	11.0363	0.854014	0.427007	+	0.904248i	$0.359568\pi$
$168$	0	0
$169$	−10.3353	−0.795024
$170$	0	0
$171$	10.0327	0.767220
$172$	0	0
$173$	1.56128	0.118702	0.0593511	−	0.998237i	$-0.481097\pi$
$173$	1.56128	0.118702	0.0593511	+	0.998237i	$0.481097\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.3008	−0.774258
$178$	0	0
$179$	−1.95255	−0.145941	−0.0729703	−	0.997334i	$-0.523248\pi$
$179$	−1.95255	−0.145941	−0.0729703	+	0.997334i	$0.523248\pi$
$180$	0	0
$181$	20.0896	1.49325	0.746625	−	0.665245i	$-0.231673\pi$
$181$	20.0896	1.49325	0.746625	+	0.665245i	$0.231673\pi$
$182$	0	0
$183$	6.79900	0.502596
$184$	0	0
$185$	−5.33163	−0.391989
$186$	0	0
$187$	−2.06098	−0.150714
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.6167	−1.92591	−0.962957	−	0.269656i	$-0.913090\pi$
$191$	−26.6167	−1.92591	−0.962957	+	0.269656i	$0.913090\pi$
$192$	0	0
$193$	9.44751	0.680046	0.340023	−	0.940417i	$-0.389565\pi$
$193$	9.44751	0.680046	0.340023	+	0.940417i	$0.389565\pi$
$194$	0	0
$195$	−1.57063	−0.112475
$196$	0	0
$197$	5.41959	0.386130	0.193065	−	0.981186i	$-0.438157\pi$
$197$	5.41959	0.386130	0.193065	+	0.981186i	$0.438157\pi$
$198$	0	0
$199$	−1.31780	−0.0934163	−0.0467081	−	0.998909i	$-0.514873\pi$
$199$	−1.31780	−0.0934163	−0.0467081	+	0.998909i	$0.514873\pi$
$200$	0	0
$201$	−7.62945	−0.538140
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.38288	−0.0965844
$206$	0	0
$207$	−17.1528	−1.19220
$208$	0	0
$209$	−1.67768	−0.116048
$210$	0	0
$211$	−13.9135	−0.957845	−0.478923	−	0.877857i	$-0.658973\pi$
$211$	−13.9135	−0.957845	−0.478923	+	0.877857i	$0.658973\pi$
$212$	0	0
$213$	11.3553	0.778051
$214$	0	0
$215$	−6.87906	−0.469148
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.49410	−0.100962
$220$	0	0
$221$	−7.99671	−0.537917
$222$	0	0
$223$	−1.95601	−0.130984	−0.0654921	−	0.997853i	$-0.520862\pi$
$223$	−1.95601	−0.130984	−0.0654921	+	0.997853i	$0.520862\pi$
$224$	0	0
$225$	7.76823	0.517882
$226$	0	0
$227$	10.6085	0.704113	0.352057	−	0.935979i	$-0.385482\pi$
$227$	10.6085	0.704113	0.352057	+	0.935979i	$0.385482\pi$
$228$	0	0
$229$	17.4305	1.15184	0.575921	−	0.817506i	$-0.304644\pi$
$229$	17.4305	1.15184	0.575921	+	0.817506i	$0.304644\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.20588	0.603097	0.301549	−	0.953451i	$-0.402496\pi$
$233$	9.20588	0.603097	0.301549	+	0.953451i	$0.402496\pi$
$234$	0	0
$235$	−4.78532	−0.312160
$236$	0	0
$237$	5.21670	0.338861
$238$	0	0
$239$	−4.95559	−0.320550	−0.160275	−	0.987072i	$-0.551238\pi$
$239$	−4.95559	−0.320550	−0.160275	+	0.987072i	$0.551238\pi$
$240$	0	0
$241$	28.7875	1.85436	0.927182	−	0.374611i	$-0.122224\pi$
$241$	28.7875	1.85436	0.927182	+	0.374611i	$0.122224\pi$
$242$	0	0
$243$	14.9070	0.956285
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.50949	−0.414189
$248$	0	0
$249$	2.98760	0.189331
$250$	0	0
$251$	10.4891	0.662068	0.331034	−	0.943619i	$-0.392603\pi$
$251$	10.4891	0.662068	0.331034	+	0.943619i	$0.392603\pi$
$252$	0	0
$253$	2.86831	0.180329
$254$	0	0
$255$	4.71343	0.295167
$256$	0	0
$257$	14.5685	0.908755	0.454378	−	0.890809i	$-0.349862\pi$
$257$	14.5685	0.908755	0.454378	+	0.890809i	$0.349862\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.991387	−0.0613653
$262$	0	0
$263$	11.9276	0.735490	0.367745	−	0.929927i	$-0.380130\pi$
$263$	11.9276	0.735490	0.367745	+	0.929927i	$0.380130\pi$
$264$	0	0
$265$	−1.74368	−0.107114
$266$	0	0
$267$	2.46632	0.150936
$268$	0	0
$269$	−24.2502	−1.47856	−0.739281	−	0.673397i	$-0.764834\pi$
$269$	−24.2502	−1.47856	−0.739281	+	0.673397i	$0.764834\pi$
$270$	0	0
$271$	5.79169	0.351820	0.175910	−	0.984406i	$-0.443713\pi$
$271$	5.79169	0.351820	0.175910	+	0.984406i	$0.443713\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.29902	−0.0783336
$276$	0	0
$277$	8.11985	0.487874	0.243937	−	0.969791i	$-0.421561\pi$
$277$	8.11985	0.487874	0.243937	+	0.969791i	$0.421561\pi$
$278$	0	0
$279$	13.0456	0.781018
$280$	0	0
$281$	18.1448	1.08243	0.541215	−	0.840884i	$-0.317964\pi$
$281$	18.1448	1.08243	0.541215	+	0.840884i	$0.317964\pi$
$282$	0	0
$283$	−18.8697	−1.12169	−0.560844	−	0.827921i	$-0.689523\pi$
$283$	−18.8697	−1.12169	−0.560844	+	0.827921i	$0.689523\pi$
$284$	0	0
$285$	3.83683	0.227274
$286$	0	0
$287$	0	0
$288$	0	0
$289$	6.99807	0.411651
$290$	0	0
$291$	−12.1533	−0.712441
$292$	0	0
$293$	5.97434	0.349025	0.174512	−	0.984655i	$-0.444165\pi$
$293$	5.97434	0.349025	0.174512	+	0.984655i	$0.444165\pi$
$294$	0	0
$295$	20.4734	1.19201
$296$	0	0
$297$	−1.61461	−0.0936894
$298$	0	0
$299$	11.1292	0.643618
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.11799	−0.0642269
$304$	0	0
$305$	−13.5134	−0.773772
$306$	0	0
$307$	1.46018	0.0833366	0.0416683	−	0.999131i	$-0.486733\pi$
$307$	1.46018	0.0833366	0.0416683	+	0.999131i	$0.486733\pi$
$308$	0	0
$309$	11.5446	0.656752
$310$	0	0
$311$	2.92567	0.165899	0.0829497	−	0.996554i	$-0.473566\pi$
$311$	2.92567	0.165899	0.0829497	+	0.996554i	$0.473566\pi$
$312$	0	0
$313$	−3.96660	−0.224205	−0.112103	−	0.993697i	$-0.535759\pi$
$313$	−3.96660	−0.224205	−0.112103	+	0.993697i	$0.535759\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.30492	−0.466451	−0.233225	−	0.972423i	$-0.574928\pi$
$317$	−8.30492	−0.466451	−0.233225	+	0.972423i	$0.574928\pi$
$318$	0	0
$319$	0.165781	0.00928197
$320$	0	0
$321$	10.9253	0.609792
$322$	0	0
$323$	19.5349	1.08695
$324$	0	0
$325$	−5.04024	−0.279582
$326$	0	0
$327$	12.0921	0.668696
$328$	0	0
$329$	0	0
$330$	0	0
$331$	24.9693	1.37244	0.686220	−	0.727394i	$-0.259269\pi$
$331$	24.9693	1.37244	0.686220	+	0.727394i	$0.259269\pi$
$332$	0	0
$333$	−9.69998	−0.531555
$334$	0	0
$335$	15.1639	0.828493
$336$	0	0
$337$	25.0653	1.36539	0.682697	−	0.730701i	$-0.260807\pi$
$337$	25.0653	1.36539	0.682697	+	0.730701i	$0.260807\pi$
$338$	0	0
$339$	−10.2051	−0.554267
$340$	0	0
$341$	−2.18150	−0.118135
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6.55978	−0.353167
$346$	0	0
$347$	25.0863	1.34670	0.673351	−	0.739323i	$-0.264854\pi$
$347$	25.0863	1.34670	0.673351	+	0.739323i	$0.264854\pi$
$348$	0	0
$349$	−6.17198	−0.330379	−0.165189	−	0.986262i	$-0.552824\pi$
$349$	−6.17198	−0.330379	−0.165189	+	0.986262i	$0.552824\pi$
$350$	0	0
$351$	−6.26478	−0.334389
$352$	0	0
$353$	30.5006	1.62339	0.811693	−	0.584085i	$-0.198547\pi$
$353$	30.5006	1.62339	0.811693	+	0.584085i	$0.198547\pi$
$354$	0	0
$355$	−22.5692	−1.19785
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−34.9395	−1.84404	−0.922019	−	0.387145i	$-0.873461\pi$
$359$	−34.9395	−1.84404	−0.922019	+	0.387145i	$0.873461\pi$
$360$	0	0
$361$	−3.09817	−0.163061
$362$	0	0
$363$	−7.53031	−0.395239
$364$	0	0
$365$	2.96960	0.155436
$366$	0	0
$367$	−4.21943	−0.220252	−0.110126	−	0.993918i	$-0.535126\pi$
$367$	−4.21943	−0.220252	−0.110126	+	0.993918i	$0.535126\pi$
$368$	0	0
$369$	−2.51590	−0.130973
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7.79843	0.403787	0.201894	−	0.979407i	$-0.435290\pi$
$373$	7.79843	0.403787	0.201894	+	0.979407i	$0.435290\pi$
$374$	0	0
$375$	7.78165	0.401843
$376$	0	0
$377$	0.643239	0.0331285
$378$	0	0
$379$	13.2064	0.678366	0.339183	−	0.940720i	$-0.389849\pi$
$379$	13.2064	0.678366	0.339183	+	0.940720i	$0.389849\pi$
$380$	0	0
$381$	6.59248	0.337743
$382$	0	0
$383$	25.6058	1.30839	0.654197	−	0.756325i	$-0.273007\pi$
$383$	25.6058	1.30839	0.654197	+	0.756325i	$0.273007\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.5153	−0.636186
$388$	0	0
$389$	−11.4055	−0.578283	−0.289142	−	0.957286i	$-0.593370\pi$
$389$	−11.4055	−0.578283	−0.289142	+	0.957286i	$0.593370\pi$
$390$	0	0
$391$	−33.3986	−1.68904
$392$	0	0
$393$	5.91986	0.298617
$394$	0	0
$395$	−10.3685	−0.521694
$396$	0	0
$397$	−19.2085	−0.964048	−0.482024	−	0.876158i	$-0.660098\pi$
$397$	−19.2085	−0.964048	−0.482024	+	0.876158i	$0.660098\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.5679	−1.32674	−0.663368	−	0.748293i	$-0.730874\pi$
$401$	−26.5679	−1.32674	−0.663368	+	0.748293i	$0.730874\pi$
$402$	0	0
$403$	−8.46432	−0.421638
$404$	0	0
$405$	−6.74497	−0.335160
$406$	0	0
$407$	1.62204	0.0804018
$408$	0	0
$409$	34.1132	1.68679	0.843395	−	0.537293i	$-0.180553\pi$
$409$	34.1132	1.68679	0.843395	+	0.537293i	$0.180553\pi$
$410$	0	0
$411$	1.44688	0.0713693
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.93801	−0.291485
$416$	0	0
$417$	0.575220	0.0281686
$418$	0	0
$419$	−27.4936	−1.34315	−0.671576	−	0.740935i	$-0.734382\pi$
$419$	−27.4936	−1.34315	−0.671576	+	0.740935i	$0.734382\pi$
$420$	0	0
$421$	17.7465	0.864910	0.432455	−	0.901655i	$-0.357647\pi$
$421$	17.7465	0.864910	0.432455	+	0.901655i	$0.357647\pi$
$422$	0	0
$423$	−8.70605	−0.423303
$424$	0	0
$425$	15.1257	0.733705
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.477832	0.0230699
$430$	0	0
$431$	−1.48717	−0.0716344	−0.0358172	−	0.999358i	$-0.511403\pi$
$431$	−1.48717	−0.0716344	−0.0358172	+	0.999358i	$0.511403\pi$
$432$	0	0
$433$	9.68672	0.465514	0.232757	−	0.972535i	$-0.425225\pi$
$433$	9.68672	0.465514	0.232757	+	0.972535i	$0.425225\pi$
$434$	0	0
$435$	−0.379139	−0.0181783
$436$	0	0
$437$	−27.1872	−1.30054
$438$	0	0
$439$	17.6885	0.844224	0.422112	−	0.906544i	$-0.361289\pi$
$439$	17.6885	0.844224	0.422112	+	0.906544i	$0.361289\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.201513	−0.00957417	−0.00478709	−	0.999989i	$-0.501524\pi$
$443$	−0.201513	−0.00957417	−0.00478709	+	0.999989i	$0.501524\pi$
$444$	0	0
$445$	−4.90194	−0.232374
$446$	0	0
$447$	−13.1025	−0.619725
$448$	0	0
$449$	−7.15899	−0.337854	−0.168927	−	0.985629i	$-0.554030\pi$
$449$	−7.15899	−0.337854	−0.168927	+	0.985629i	$0.554030\pi$
$450$	0	0
$451$	0.420713	0.0198106
$452$	0	0
$453$	11.0585	0.519572
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.68197	−0.312569	−0.156285	−	0.987712i	$-0.549952\pi$
$457$	−6.68197	−0.312569	−0.156285	+	0.987712i	$0.549952\pi$
$458$	0	0
$459$	18.8005	0.877534
$460$	0	0
$461$	17.5515	0.817453	0.408727	−	0.912657i	$-0.365973\pi$
$461$	17.5515	0.817453	0.408727	+	0.912657i	$0.365973\pi$
$462$	0	0
$463$	31.9646	1.48552	0.742760	−	0.669557i	$-0.233516\pi$
$463$	31.9646	1.48552	0.742760	+	0.669557i	$0.233516\pi$
$464$	0	0
$465$	4.98905	0.231362
$466$	0	0
$467$	18.4627	0.854351	0.427175	−	0.904169i	$-0.359509\pi$
$467$	18.4627	0.854351	0.427175	+	0.904169i	$0.359509\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10.9780	−0.505838
$472$	0	0
$473$	2.09282	0.0962280
$474$	0	0
$475$	12.3127	0.564943
$476$	0	0
$477$	−3.17233	−0.145251
$478$	0	0
$479$	−26.0091	−1.18839	−0.594193	−	0.804323i	$-0.702528\pi$
$479$	−26.0091	−1.18839	−0.594193	+	0.804323i	$0.702528\pi$
$480$	0	0
$481$	6.29361	0.286964
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.1554	1.09684
$486$	0	0
$487$	23.7538	1.07639	0.538194	−	0.842821i	$-0.319107\pi$
$487$	23.7538	1.07639	0.538194	+	0.842821i	$0.319107\pi$
$488$	0	0
$489$	6.91917	0.312896
$490$	0	0
$491$	−5.82937	−0.263076	−0.131538	−	0.991311i	$-0.541991\pi$
$491$	−5.82937	−0.263076	−0.131538	+	0.991311i	$0.541991\pi$
$492$	0	0
$493$	−1.93035	−0.0869388
$494$	0	0
$495$	1.46374	0.0657903
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.24655	−0.145336	−0.0726678	−	0.997356i	$-0.523151\pi$
$499$	−3.24655	−0.145336	−0.0726678	+	0.997356i	$0.523151\pi$
$500$	0	0
$501$	7.67872	0.343060
$502$	0	0
$503$	3.57448	0.159378	0.0796890	−	0.996820i	$-0.474607\pi$
$503$	3.57448	0.159378	0.0796890	+	0.996820i	$0.474607\pi$
$504$	0	0
$505$	2.22207	0.0988806
$506$	0	0
$507$	−7.19099	−0.319363
$508$	0	0
$509$	−38.0193	−1.68517	−0.842587	−	0.538560i	$-0.818968\pi$
$509$	−38.0193	−1.68517	−0.842587	+	0.538560i	$0.818968\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	15.3040	0.675689
$514$	0	0
$515$	−22.9456	−1.01110
$516$	0	0
$517$	1.45584	0.0640277
$518$	0	0
$519$	1.08629	0.0476830
$520$	0	0
$521$	−8.32370	−0.364668	−0.182334	−	0.983237i	$-0.558365\pi$
$521$	−8.32370	−0.364668	−0.182334	+	0.983237i	$0.558365\pi$
$522$	0	0
$523$	9.09347	0.397630	0.198815	−	0.980037i	$-0.436291\pi$
$523$	9.09347	0.397630	0.198815	+	0.980037i	$0.436291\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.4014	1.10650
$528$	0	0
$529$	23.4815	1.02094
$530$	0	0
$531$	37.2478	1.61642
$532$	0	0
$533$	1.63239	0.0707066
$534$	0	0
$535$	−21.7146	−0.938806
$536$	0	0
$537$	−1.35853	−0.0586247
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3.08992	0.132846	0.0664230	−	0.997792i	$-0.478841\pi$
$541$	3.08992	0.132846	0.0664230	+	0.997792i	$0.478841\pi$
$542$	0	0
$543$	13.9777	0.599842
$544$	0	0
$545$	−24.0337	−1.02949
$546$	0	0
$547$	30.0026	1.28282	0.641410	−	0.767198i	$-0.278350\pi$
$547$	30.0026	1.28282	0.641410	+	0.767198i	$0.278350\pi$
$548$	0	0
$549$	−24.5852	−1.04927
$550$	0	0
$551$	−1.57135	−0.0669417
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.70959	−0.157463
$556$	0	0
$557$	26.7538	1.13359	0.566797	−	0.823858i	$-0.308182\pi$
$557$	26.7538	1.13359	0.566797	+	0.823858i	$0.308182\pi$
$558$	0	0
$559$	8.12024	0.343449
$560$	0	0
$561$	−1.43397	−0.0605423
$562$	0	0
$563$	37.9386	1.59892	0.799461	−	0.600717i	$-0.205118\pi$
$563$	37.9386	1.59892	0.799461	+	0.600717i	$0.205118\pi$
$564$	0	0
$565$	20.2832	0.853322
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−34.1146	−1.43016	−0.715080	−	0.699043i	$-0.753610\pi$
$569$	−34.1146	−1.43016	−0.715080	+	0.699043i	$0.753610\pi$
$570$	0	0
$571$	−43.9693	−1.84006	−0.920029	−	0.391849i	$-0.871835\pi$
$571$	−43.9693	−1.84006	−0.920029	+	0.391849i	$0.871835\pi$
$572$	0	0
$573$	−18.5191	−0.773645
$574$	0	0
$575$	−21.0508	−0.877878
$576$	0	0
$577$	−20.9560	−0.872410	−0.436205	−	0.899847i	$-0.643678\pi$
$577$	−20.9560	−0.872410	−0.436205	+	0.899847i	$0.643678\pi$
$578$	0	0
$579$	6.57329	0.273176
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.530482	0.0219703
$584$	0	0
$585$	5.67938	0.234814
$586$	0	0
$587$	0.514473	0.0212346	0.0106173	−	0.999944i	$-0.496620\pi$
$587$	0.514473	0.0212346	0.0106173	+	0.999944i	$0.496620\pi$
$588$	0	0
$589$	20.6772	0.851990
$590$	0	0
$591$	3.77079	0.155110
$592$	0	0
$593$	11.0733	0.454727	0.227363	−	0.973810i	$-0.426990\pi$
$593$	11.0733	0.454727	0.227363	+	0.973810i	$0.426990\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.916884	−0.0375256
$598$	0	0
$599$	−0.478008	−0.0195309	−0.00976543	−	0.999952i	$-0.503108\pi$
$599$	−0.478008	−0.0195309	−0.00976543	+	0.999952i	$0.503108\pi$
$600$	0	0
$601$	46.5241	1.89776	0.948878	−	0.315643i	$-0.102220\pi$
$601$	46.5241	1.89776	0.948878	+	0.315643i	$0.102220\pi$
$602$	0	0
$603$	27.5881	1.12347
$604$	0	0
$605$	14.9669	0.608490
$606$	0	0
$607$	2.83076	0.114897	0.0574484	−	0.998348i	$-0.481704\pi$
$607$	2.83076	0.114897	0.0574484	+	0.998348i	$0.481704\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.64872	0.228523
$612$	0	0
$613$	41.9804	1.69557	0.847786	−	0.530339i	$-0.177935\pi$
$613$	41.9804	1.69557	0.847786	+	0.530339i	$0.177935\pi$
$614$	0	0
$615$	−0.962164	−0.0387982
$616$	0	0
$617$	−29.9759	−1.20678	−0.603392	−	0.797445i	$-0.706185\pi$
$617$	−29.9759	−1.20678	−0.603392	+	0.797445i	$0.706185\pi$
$618$	0	0
$619$	−40.6317	−1.63312	−0.816562	−	0.577258i	$-0.804123\pi$
$619$	−40.6317	−1.63312	−0.816562	+	0.577258i	$0.804123\pi$
$620$	0	0
$621$	−26.1651	−1.04997
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−0.0281751	−0.00112700
$626$	0	0
$627$	−1.16728	−0.0466167
$628$	0	0
$629$	−18.8871	−0.753077
$630$	0	0
$631$	22.4789	0.894869	0.447435	−	0.894317i	$-0.352338\pi$
$631$	22.4789	0.894869	0.447435	+	0.894317i	$0.352338\pi$
$632$	0	0
$633$	−9.68059	−0.384769
$634$	0	0
$635$	−13.1029	−0.519973
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−41.0607	−1.62434
$640$	0	0
$641$	−20.0003	−0.789964	−0.394982	−	0.918689i	$-0.629249\pi$
$641$	−20.0003	−0.789964	−0.394982	+	0.918689i	$0.629249\pi$
$642$	0	0
$643$	−24.3509	−0.960306	−0.480153	−	0.877185i	$-0.659419\pi$
$643$	−24.3509	−0.960306	−0.480153	+	0.877185i	$0.659419\pi$
$644$	0	0
$645$	−4.78624	−0.188458
$646$	0	0
$647$	10.0915	0.396738	0.198369	−	0.980127i	$-0.436436\pi$
$647$	10.0915	0.396738	0.198369	+	0.980127i	$0.436436\pi$
$648$	0	0
$649$	−6.22864	−0.244496
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20.0767	0.785661	0.392830	−	0.919611i	$-0.371496\pi$
$653$	20.0767	0.785661	0.392830	+	0.919611i	$0.371496\pi$
$654$	0	0
$655$	−11.7660	−0.459737
$656$	0	0
$657$	5.40266	0.210778
$658$	0	0
$659$	−1.19452	−0.0465318	−0.0232659	−	0.999729i	$-0.507406\pi$
$659$	−1.19452	−0.0465318	−0.0232659	+	0.999729i	$0.507406\pi$
$660$	0	0
$661$	1.62930	0.0633725	0.0316862	−	0.999498i	$-0.489912\pi$
$661$	1.62930	0.0633725	0.0316862	+	0.999498i	$0.489912\pi$
$662$	0	0
$663$	−5.56387	−0.216083
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.68651	0.104022
$668$	0	0
$669$	−1.36093	−0.0526167
$670$	0	0
$671$	4.11117	0.158710
$672$	0	0
$673$	0.884185	0.0340828	0.0170414	−	0.999855i	$-0.494575\pi$
$673$	0.884185	0.0340828	0.0170414	+	0.999855i	$0.494575\pi$
$674$	0	0
$675$	11.8498	0.456098
$676$	0	0
$677$	−49.0401	−1.88476	−0.942381	−	0.334540i	$-0.891419\pi$
$677$	−49.0401	−1.88476	−0.942381	+	0.334540i	$0.891419\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	7.38110	0.282844
$682$	0	0
$683$	−26.5105	−1.01440	−0.507199	−	0.861829i	$-0.669319\pi$
$683$	−26.5105	−1.01440	−0.507199	+	0.861829i	$0.669319\pi$
$684$	0	0
$685$	−2.87575	−0.109877
$686$	0	0
$687$	12.1276	0.462698
$688$	0	0
$689$	2.05829	0.0784148
$690$	0	0
$691$	−44.7577	−1.70266	−0.851331	−	0.524629i	$-0.824204\pi$
$691$	−44.7577	−1.70266	−0.851331	+	0.524629i	$0.824204\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.14328	−0.0433670
$696$	0	0
$697$	−4.89878	−0.185555
$698$	0	0
$699$	6.40517	0.242266
$700$	0	0
$701$	18.0394	0.681340	0.340670	−	0.940183i	$-0.389346\pi$
$701$	18.0394	0.681340	0.340670	+	0.940183i	$0.389346\pi$
$702$	0	0
$703$	−15.3745	−0.579859
$704$	0	0
$705$	−3.32948	−0.125395
$706$	0	0
$707$	0	0
$708$	0	0
$709$	29.2148	1.09719	0.548593	−	0.836089i	$-0.315164\pi$
$709$	29.2148	1.09719	0.548593	+	0.836089i	$0.315164\pi$
$710$	0	0
$711$	−18.8636	−0.707440
$712$	0	0
$713$	−35.3516	−1.32393
$714$	0	0
$715$	−0.949715	−0.0355173
$716$	0	0
$717$	−3.44795	−0.128766
$718$	0	0
$719$	−23.1604	−0.863738	−0.431869	−	0.901936i	$-0.642146\pi$
$719$	−23.1604	−0.863738	−0.431869	+	0.901936i	$0.642146\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	20.0294	0.744903
$724$	0	0
$725$	−1.21668	−0.0451864
$726$	0	0
$727$	−24.9452	−0.925167	−0.462584	−	0.886576i	$-0.653078\pi$
$727$	−24.9452	−0.925167	−0.462584	+	0.886576i	$0.653078\pi$
$728$	0	0
$729$	−4.26063	−0.157801
$730$	0	0
$731$	−24.3688	−0.901311
$732$	0	0
$733$	5.87755	0.217092	0.108546	−	0.994091i	$-0.465380\pi$
$733$	5.87755	0.217092	0.108546	+	0.994091i	$0.465380\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.61333	−0.169934
$738$	0	0
$739$	−38.4992	−1.41622	−0.708108	−	0.706105i	$-0.750451\pi$
$739$	−38.4992	−1.41622	−0.708108	+	0.706105i	$0.750451\pi$
$740$	0	0
$741$	−4.52910	−0.166381
$742$	0	0
$743$	−28.7849	−1.05601	−0.528007	−	0.849240i	$-0.677060\pi$
$743$	−28.7849	−1.05601	−0.528007	+	0.849240i	$0.677060\pi$
$744$	0	0
$745$	26.0418	0.954098
$746$	0	0
$747$	−10.8032	−0.395267
$748$	0	0
$749$	0	0
$750$	0	0
$751$	35.5927	1.29880	0.649399	−	0.760448i	$-0.275021\pi$
$751$	35.5927	1.29880	0.649399	+	0.760448i	$0.275021\pi$
$752$	0	0
$753$	7.29802	0.265955
$754$	0	0
$755$	−21.9793	−0.799908
$756$	0	0
$757$	17.0120	0.618313	0.309156	−	0.951011i	$-0.399953\pi$
$757$	17.0120	0.618313	0.309156	+	0.951011i	$0.399953\pi$
$758$	0	0
$759$	1.99569	0.0724388
$760$	0	0
$761$	5.23201	0.189660	0.0948300	−	0.995493i	$-0.469769\pi$
$761$	5.23201	0.189660	0.0948300	+	0.995493i	$0.469769\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−17.0438	−0.616219
$766$	0	0
$767$	−24.1674	−0.872635
$768$	0	0
$769$	39.8347	1.43648	0.718239	−	0.695797i	$-0.244949\pi$
$769$	39.8347	1.43648	0.718239	+	0.695797i	$0.244949\pi$
$770$	0	0
$771$	10.1363	0.365050
$772$	0	0
$773$	−13.7087	−0.493067	−0.246533	−	0.969134i	$-0.579292\pi$
$773$	−13.7087	−0.493067	−0.246533	+	0.969134i	$0.579292\pi$
$774$	0	0
$775$	16.0102	0.575103
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.98771	−0.142875
$780$	0	0
$781$	6.86624	0.245693
$782$	0	0
$783$	−1.51228	−0.0540444
$784$	0	0
$785$	21.8193	0.778764
$786$	0	0
$787$	44.5046	1.58642	0.793208	−	0.608950i	$-0.208409\pi$
$787$	44.5046	1.58642	0.793208	+	0.608950i	$0.208409\pi$
$788$	0	0
$789$	8.29889	0.295448
$790$	0	0
$791$	0	0
$792$	0	0
$793$	15.9515	0.566456
$794$	0	0
$795$	−1.21320	−0.0430279
$796$	0	0
$797$	14.0505	0.497694	0.248847	−	0.968543i	$-0.419948\pi$
$797$	14.0505	0.497694	0.248847	+	0.968543i	$0.419948\pi$
$798$	0	0
$799$	−16.9518	−0.599711
$800$	0	0
$801$	−8.91822	−0.315110
$802$	0	0
$803$	−0.903441	−0.0318818
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16.8726	−0.593942
$808$	0	0
$809$	41.4259	1.45646	0.728229	−	0.685334i	$-0.240344\pi$
$809$	41.4259	1.45646	0.728229	+	0.685334i	$0.240344\pi$
$810$	0	0
$811$	2.40261	0.0843671	0.0421836	−	0.999110i	$-0.486569\pi$
$811$	2.40261	0.0843671	0.0421836	+	0.999110i	$0.486569\pi$
$812$	0	0
$813$	4.02968	0.141327
$814$	0	0
$815$	−13.7522	−0.481719
$816$	0	0
$817$	−19.8367	−0.693998
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11.6142	−0.405339	−0.202670	−	0.979247i	$-0.564962\pi$
$821$	−11.6142	−0.405339	−0.202670	+	0.979247i	$0.564962\pi$
$822$	0	0
$823$	−0.363199	−0.0126603	−0.00633016	−	0.999980i	$-0.502015\pi$
$823$	−0.363199	−0.0126603	−0.00633016	+	0.999980i	$0.502015\pi$
$824$	0	0
$825$	−0.903816	−0.0314668
$826$	0	0
$827$	−19.9556	−0.693923	−0.346961	−	0.937879i	$-0.612787\pi$
$827$	−19.9556	−0.693923	−0.346961	+	0.937879i	$0.612787\pi$
$828$	0	0
$829$	5.68101	0.197310	0.0986548	−	0.995122i	$-0.468546\pi$
$829$	5.68101	0.197310	0.0986548	+	0.995122i	$0.468546\pi$
$830$	0	0
$831$	5.64954	0.195980
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−15.2618	−0.528158
$836$	0	0
$837$	19.8999	0.687841
$838$	0	0
$839$	27.6700	0.955274	0.477637	−	0.878557i	$-0.341493\pi$
$839$	27.6700	0.955274	0.477637	+	0.878557i	$0.341493\pi$
$840$	0	0
$841$	−28.8447	−0.994646
$842$	0	0
$843$	12.6246	0.434815
$844$	0	0
$845$	14.2925	0.491676
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−13.1290	−0.450585
$850$	0	0
$851$	26.2855	0.901056
$852$	0	0
$853$	−40.9819	−1.40319	−0.701596	−	0.712575i	$-0.747529\pi$
$853$	−40.9819	−1.40319	−0.701596	+	0.712575i	$0.747529\pi$
$854$	0	0
$855$	−13.8740	−0.474481
$856$	0	0
$857$	−49.3668	−1.68634	−0.843169	−	0.537649i	$-0.819313\pi$
$857$	−49.3668	−1.68634	−0.843169	+	0.537649i	$0.819313\pi$
$858$	0	0
$859$	3.10517	0.105947	0.0529736	−	0.998596i	$-0.483130\pi$
$859$	3.10517	0.105947	0.0529736	+	0.998596i	$0.483130\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41.0979	−1.39899	−0.699494	−	0.714639i	$-0.746591\pi$
$863$	−41.0979	−1.39899	−0.699494	+	0.714639i	$0.746591\pi$
$864$	0	0
$865$	−2.15906	−0.0734103
$866$	0	0
$867$	4.86904	0.165361
$868$	0	0
$869$	3.15440	0.107006
$870$	0	0
$871$	−17.8999	−0.606516
$872$	0	0
$873$	43.9465	1.48736
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33.7652	1.14017	0.570085	−	0.821586i	$-0.306910\pi$
$877$	33.7652	1.14017	0.570085	+	0.821586i	$0.306910\pi$
$878$	0	0
$879$	4.15677	0.140204
$880$	0	0
$881$	33.3472	1.12349	0.561747	−	0.827309i	$-0.310129\pi$
$881$	33.3472	1.12349	0.561747	+	0.827309i	$0.310129\pi$
$882$	0	0
$883$	−0.302689	−0.0101863	−0.00509315	−	0.999987i	$-0.501621\pi$
$883$	−0.302689	−0.0101863	−0.00509315	+	0.999987i	$0.501621\pi$
$884$	0	0
$885$	14.2448	0.478833
$886$	0	0
$887$	25.0623	0.841508	0.420754	−	0.907175i	$-0.361765\pi$
$887$	25.0623	0.841508	0.420754	+	0.907175i	$0.361765\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.05203	0.0687455
$892$	0	0
$893$	−13.7991	−0.461769
$894$	0	0
$895$	2.70014	0.0902557
$896$	0	0
$897$	7.74335	0.258543
$898$	0	0
$899$	−2.04323	−0.0681456
$900$	0	0
$901$	−6.17692	−0.205783
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−27.7815	−0.923488
$906$	0	0
$907$	−2.15692	−0.0716193	−0.0358096	−	0.999359i	$-0.511401\pi$
$907$	−2.15692	−0.0716193	−0.0358096	+	0.999359i	$0.511401\pi$
$908$	0	0
$909$	4.04266	0.134087
$910$	0	0
$911$	11.4663	0.379895	0.189948	−	0.981794i	$-0.439168\pi$
$911$	11.4663	0.379895	0.189948	+	0.981794i	$0.439168\pi$
$912$	0	0
$913$	1.80652	0.0597872
$914$	0	0
$915$	−9.40218	−0.310826
$916$	0	0
$917$	0	0
$918$	0	0
$919$	22.5518	0.743915	0.371958	−	0.928250i	$-0.378687\pi$
$919$	22.5518	0.743915	0.371958	+	0.928250i	$0.378687\pi$
$920$	0	0
$921$	1.01595	0.0334765
$922$	0	0
$923$	26.6413	0.876910
$924$	0	0
$925$	−11.9043	−0.391411
$926$	0	0
$927$	−41.7455	−1.37110
$928$	0	0
$929$	−9.86617	−0.323698	−0.161849	−	0.986815i	$-0.551746\pi$
$929$	−9.86617	−0.323698	−0.161849	+	0.986815i	$0.551746\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.03559	0.0666423
$934$	0	0
$935$	2.85009	0.0932079
$936$	0	0
$937$	14.4978	0.473622	0.236811	−	0.971556i	$-0.423898\pi$
$937$	14.4978	0.473622	0.236811	+	0.971556i	$0.423898\pi$
$938$	0	0
$939$	−2.75984	−0.0900640
$940$	0	0
$941$	40.5898	1.32319	0.661594	−	0.749862i	$-0.269880\pi$
$941$	40.5898	1.32319	0.661594	+	0.749862i	$0.269880\pi$
$942$	0	0
$943$	6.81774	0.222016
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.8863	−1.58859	−0.794295	−	0.607532i	$-0.792160\pi$
$947$	−48.8863	−1.58859	−0.794295	+	0.607532i	$0.792160\pi$
$948$	0	0
$949$	−3.50539	−0.113790
$950$	0	0
$951$	−5.77831	−0.187375
$952$	0	0
$953$	−33.9550	−1.09991	−0.549955	−	0.835194i	$-0.685355\pi$
$953$	−33.9550	−1.09991	−0.549955	+	0.835194i	$0.685355\pi$
$954$	0	0
$955$	36.8076	1.19107
$956$	0	0
$957$	0.115346	0.00372859
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−4.11330	−0.132687
$962$	0	0
$963$	−39.5060	−1.27306
$964$	0	0
$965$	−13.0647	−0.420569
$966$	0	0
$967$	47.8869	1.53994	0.769970	−	0.638080i	$-0.220271\pi$
$967$	47.8869	1.53994	0.769970	+	0.638080i	$0.220271\pi$
$968$	0	0
$969$	13.5918	0.436632
$970$	0	0
$971$	12.6049	0.404511	0.202255	−	0.979333i	$-0.435173\pi$
$971$	12.6049	0.404511	0.202255	+	0.979333i	$0.435173\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.50685	−0.112309
$976$	0	0
$977$	30.3763	0.971824	0.485912	−	0.874008i	$-0.338488\pi$
$977$	30.3763	0.971824	0.485912	+	0.874008i	$0.338488\pi$
$978$	0	0
$979$	1.49132	0.0476628
$980$	0	0
$981$	−43.7251	−1.39604
$982$	0	0
$983$	38.6361	1.23230	0.616151	−	0.787628i	$-0.288691\pi$
$983$	38.6361	1.23230	0.616151	+	0.787628i	$0.288691\pi$
$984$	0	0
$985$	−7.49464	−0.238799
$986$	0	0
$987$	0	0
$988$	0	0
$989$	33.9145	1.07842
$990$	0	0
$991$	−47.9244	−1.52237	−0.761184	−	0.648535i	$-0.775382\pi$
$991$	−47.9244	−1.52237	−0.761184	+	0.648535i	$0.775382\pi$
$992$	0	0
$993$	17.3729	0.551313
$994$	0	0
$995$	1.82235	0.0577725
$996$	0	0
$997$	−13.0226	−0.412429	−0.206215	−	0.978507i	$-0.566115\pi$
$997$	−13.0226	−0.412429	−0.206215	+	0.978507i	$0.566115\pi$
$998$	0	0
$999$	−14.7965	−0.468140

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.k.1.4		5
7.6	odd	2		1148.2.a.d.1.2	✓	5
28.27	even	2		4592.2.a.bc.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.d.1.2	✓	5	7.6	odd	2
4592.2.a.bc.1.4		5	28.27	even	2
8036.2.a.k.1.4		5	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+0.695770 q^{3} -1.38288 q^{5} -2.51590 q^{9} +O(q^{10})\)	\(q+0.695770 q^{3} -1.38288 q^{5} -2.51590 q^{9} +0.420713 q^{11} +1.63239 q^{13} -0.962164 q^{15} -4.89878 q^{17} -3.98771 q^{19} +6.81774 q^{23} -3.08765 q^{25} -3.83780 q^{27} +0.394048 q^{29} -5.18524 q^{31} +0.292720 q^{33} +3.85546 q^{37} +1.13577 q^{39} +1.00000 q^{41} +4.97445 q^{43} +3.47919 q^{45} +3.46041 q^{47} -3.40842 q^{51} +1.26091 q^{53} -0.581795 q^{55} -2.77453 q^{57} -14.8049 q^{59} +9.77191 q^{61} -2.25739 q^{65} -10.9655 q^{67} +4.74357 q^{69} +16.3205 q^{71} -2.14740 q^{73} -2.14829 q^{75} +7.49774 q^{79} +4.87749 q^{81} +4.29395 q^{83} +6.77442 q^{85} +0.274167 q^{87} +3.54474 q^{89} -3.60773 q^{93} +5.51452 q^{95} -17.4675 q^{97} -1.05847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 3 q^{5} + q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 3 * q^5 + q^9	\( 5 q - 2 q^{3} + 3 q^{5} + q^{9} + 6 q^{11} - 7 q^{13} + 9 q^{15} - q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 8 q^{27} + 11 q^{29} - 13 q^{31} - 11 q^{33} + 5 q^{37} + 23 q^{39} + 5 q^{41} + 29 q^{43} - 11 q^{45} - 7 q^{47} - 3 q^{51} + 21 q^{53} - 19 q^{55} + 9 q^{57} - 3 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} - 10 q^{69} + 22 q^{71} + 16 q^{73} + 18 q^{75} + 4 q^{79} - 15 q^{81} + 6 q^{83} + 13 q^{85} - 6 q^{87} + 20 q^{89} - 5 q^{93} - 7 q^{95} + 24 q^{97} + 27 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 3 * q^5 + q^9 + 6 * q^11 - 7 * q^13 + 9 * q^15 - q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 - 8 * q^27 + 11 * q^29 - 13 * q^31 - 11 * q^33 + 5 * q^37 + 23 * q^39 + 5 * q^41 + 29 * q^43 - 11 * q^45 - 7 * q^47 - 3 * q^51 + 21 * q^53 - 19 * q^55 + 9 * q^57 - 3 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 - 10 * q^69 + 22 * q^71 + 16 * q^73 + 18 * q^75 + 4 * q^79 - 15 * q^81 + 6 * q^83 + 13 * q^85 - 6 * q^87 + 20 * q^89 - 5 * q^93 - 7 * q^95 + 24 * q^97 + 27 * q^99

Introduction

L-functions

Modular forms

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Database

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