LMFDB - Embedded newform 3808.2.a.m.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3808,2,Mod(1,3808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3808.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2,0,-6,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.80686992.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 15x^{3} + 8x^{2} - 15x + 3 $ x^6 - x^5 - 10x^4 + 15x^3 + 8x^2 - 15x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.243788$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.03321 q^{3} +1.36630 q^{5} -1.00000 q^{7} -1.93247 q^{9} +3.61819 q^{11} -0.433831 q^{13} +1.41168 q^{15} -1.00000 q^{17} -7.98338 q^{19} -1.03321 q^{21} +1.16816 q^{23} -3.13322 q^{25} -5.09629 q^{27} -5.61819 q^{29} -8.77427 q^{31} +3.73836 q^{33} -1.36630 q^{35} -2.03582 q^{37} -0.448240 q^{39} +2.09740 q^{41} -4.28400 q^{43} -2.64034 q^{45} +3.77537 q^{47} +1.00000 q^{49} -1.03321 q^{51} -12.3925 q^{53} +4.94353 q^{55} -8.24854 q^{57} +13.6386 q^{59} -4.23839 q^{61} +1.93247 q^{63} -0.592744 q^{65} +0.392876 q^{67} +1.20696 q^{69} +6.47096 q^{71} -5.13898 q^{73} -3.23729 q^{75} -3.61819 q^{77} +1.81298 q^{79} +0.531853 q^{81} +3.02836 q^{83} -1.36630 q^{85} -5.80478 q^{87} +15.3734 q^{89} +0.433831 q^{91} -9.06569 q^{93} -10.9077 q^{95} +0.986216 q^{97} -6.99204 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 6 q^{5} - 6 q^{7} + 4 q^{9} + 2 q^{11} - 4 q^{13} - 2 q^{15} - 6 q^{17} + 10 q^{19} - 2 q^{21} - 4 q^{23} + 4 q^{25} + 8 q^{27} - 14 q^{29} - 8 q^{31} - 8 q^{33} + 6 q^{35} - 4 q^{37} - 14 q^{39}+ \cdots + 4 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 6 * q^5 - 6 * q^7 + 4 * q^9 + 2 * q^11 - 4 * q^13 - 2 * q^15 - 6 * q^17 + 10 * q^19 - 2 * q^21 - 4 * q^23 + 4 * q^25 + 8 * q^27 - 14 * q^29 - 8 * q^31 - 8 * q^33 + 6 * q^35 - 4 * q^37 - 14 * q^39 - 2 * q^41 + 8 * q^43 - 8 * q^45 + 2 * q^47 + 6 * q^49 - 2 * q^51 - 10 * q^53 - 2 * q^55 - 18 * q^57 + 14 * q^59 - 20 * q^61 - 4 * q^63 - 20 * q^65 + 6 * q^67 - 14 * q^69 - 20 * q^71 - 6 * q^73 + 10 * q^75 - 2 * q^77 - 38 * q^79 - 18 * q^81 + 16 * q^83 + 6 * q^85 + 4 * q^87 - 2 * q^89 + 4 * q^91 - 16 * q^93 - 42 * q^95 - 12 * q^97 + 4 * q^99

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
$3$	1.03321	0.596526	0.298263	−	0.954484i	$-0.403593\pi$
$3$	1.03321	0.596526	0.298263	+	0.954484i	$0.403593\pi$
$4$	0	0
$5$	1.36630	0.611029	0.305514	−	0.952187i	$-0.401172\pi$
$5$	1.36630	0.611029	0.305514	+	0.952187i	$0.401172\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.93247	−0.644157
$10$	0	0
$11$	3.61819	1.09092	0.545462	−	0.838135i	$-0.316354\pi$
$11$	3.61819	1.09092	0.545462	+	0.838135i	$0.316354\pi$
$12$	0	0
$13$	−0.433831	−0.120323	−0.0601615	−	0.998189i	$-0.519162\pi$
$13$	−0.433831	−0.120323	−0.0601615	+	0.998189i	$0.519162\pi$
$14$	0	0
$15$	1.41168	0.364494
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−7.98338	−1.83151	−0.915757	−	0.401733i	$-0.868408\pi$
$19$	−7.98338	−1.83151	−0.915757	+	0.401733i	$0.868408\pi$
$20$	0	0
$21$	−1.03321	−0.225466
$22$	0	0
$23$	1.16816	0.243579	0.121789	−	0.992556i	$-0.461137\pi$
$23$	1.16816	0.243579	0.121789	+	0.992556i	$0.461137\pi$
$24$	0	0
$25$	−3.13322	−0.626644
$26$	0	0
$27$	−5.09629	−0.980782
$28$	0	0
$29$	−5.61819	−1.04327	−0.521636	−	0.853168i	$-0.674678\pi$
$29$	−5.61819	−1.04327	−0.521636	+	0.853168i	$0.674678\pi$
$30$	0	0
$31$	−8.77427	−1.57591	−0.787953	−	0.615736i	$-0.788859\pi$
$31$	−8.77427	−1.57591	−0.787953	+	0.615736i	$0.788859\pi$
$32$	0	0
$33$	3.73836	0.650765
$34$	0	0
$35$	−1.36630	−0.230947
$36$	0	0
$37$	−2.03582	−0.334687	−0.167344	−	0.985899i	$-0.553519\pi$
$37$	−2.03582	−0.334687	−0.167344	+	0.985899i	$0.553519\pi$
$38$	0	0
$39$	−0.448240	−0.0717758
$40$	0	0
$41$	2.09740	0.327558	0.163779	−	0.986497i	$-0.447632\pi$
$41$	2.09740	0.327558	0.163779	+	0.986497i	$0.447632\pi$
$42$	0	0
$43$	−4.28400	−0.653303	−0.326652	−	0.945145i	$-0.605920\pi$
$43$	−4.28400	−0.653303	−0.326652	+	0.945145i	$0.605920\pi$
$44$	0	0
$45$	−2.64034	−0.393598
$46$	0	0
$47$	3.77537	0.550695	0.275347	−	0.961345i	$-0.411207\pi$
$47$	3.77537	0.550695	0.275347	+	0.961345i	$0.411207\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.03321	−0.144679
$52$	0	0
$53$	−12.3925	−1.70223	−0.851117	−	0.524976i	$-0.824074\pi$
$53$	−12.3925	−1.70223	−0.851117	+	0.524976i	$0.824074\pi$
$54$	0	0
$55$	4.94353	0.666586
$56$	0	0
$57$	−8.24854	−1.09255
$58$	0	0
$59$	13.6386	1.77559	0.887796	−	0.460237i	$-0.152235\pi$
$59$	13.6386	1.77559	0.887796	+	0.460237i	$0.152235\pi$
$60$	0	0
$61$	−4.23839	−0.542670	−0.271335	−	0.962485i	$-0.587465\pi$
$61$	−4.23839	−0.542670	−0.271335	+	0.962485i	$0.587465\pi$
$62$	0	0
$63$	1.93247	0.243468
$64$	0	0
$65$	−0.592744	−0.0735208
$66$	0	0
$67$	0.392876	0.0479975	0.0239987	−	0.999712i	$-0.492360\pi$
$67$	0.392876	0.0479975	0.0239987	+	0.999712i	$0.492360\pi$
$68$	0	0
$69$	1.20696	0.145301
$70$	0	0
$71$	6.47096	0.767962	0.383981	−	0.923341i	$-0.374553\pi$
$71$	6.47096	0.767962	0.383981	+	0.923341i	$0.374553\pi$
$72$	0	0
$73$	−5.13898	−0.601472	−0.300736	−	0.953707i	$-0.597232\pi$
$73$	−5.13898	−0.601472	−0.300736	+	0.953707i	$0.597232\pi$
$74$	0	0
$75$	−3.23729	−0.373809
$76$	0	0
$77$	−3.61819	−0.412331
$78$	0	0
$79$	1.81298	0.203976	0.101988	−	0.994786i	$-0.467480\pi$
$79$	1.81298	0.203976	0.101988	+	0.994786i	$0.467480\pi$
$80$	0	0
$81$	0.531853	0.0590947
$82$	0	0
$83$	3.02836	0.332406	0.166203	−	0.986092i	$-0.446849\pi$
$83$	3.02836	0.332406	0.166203	+	0.986092i	$0.446849\pi$
$84$	0	0
$85$	−1.36630	−0.148196
$86$	0	0
$87$	−5.80478	−0.622338
$88$	0	0
$89$	15.3734	1.62958	0.814790	−	0.579756i	$-0.196852\pi$
$89$	15.3734	1.62958	0.814790	+	0.579756i	$0.196852\pi$
$90$	0	0
$91$	0.433831	0.0454779
$92$	0	0
$93$	−9.06569	−0.940068
$94$	0	0
$95$	−10.9077	−1.11911
$96$	0	0
$97$	0.986216	0.100135	0.0500675	−	0.998746i	$-0.484056\pi$
$97$	0.986216	0.100135	0.0500675	+	0.998746i	$0.484056\pi$
$98$	0	0
$99$	−6.99204	−0.702726
$100$	0	0
$101$	−0.882071	−0.0877694	−0.0438847	−	0.999037i	$-0.513973\pi$
$101$	−0.882071	−0.0877694	−0.0438847	+	0.999037i	$0.513973\pi$
$102$	0	0
$103$	−11.4525	−1.12845	−0.564225	−	0.825621i	$-0.690825\pi$
$103$	−11.4525	−1.12845	−0.564225	+	0.825621i	$0.690825\pi$
$104$	0	0
$105$	−1.41168	−0.137766
$106$	0	0
$107$	15.4503	1.49364	0.746819	−	0.665027i	$-0.231580\pi$
$107$	15.4503	1.49364	0.746819	+	0.665027i	$0.231580\pi$
$108$	0	0
$109$	−7.06371	−0.676580	−0.338290	−	0.941042i	$-0.609849\pi$
$109$	−7.06371	−0.676580	−0.338290	+	0.941042i	$0.609849\pi$
$110$	0	0
$111$	−2.10344	−0.199650
$112$	0	0
$113$	−3.29054	−0.309548	−0.154774	−	0.987950i	$-0.549465\pi$
$113$	−3.29054	−0.309548	−0.154774	+	0.987950i	$0.549465\pi$
$114$	0	0
$115$	1.59606	0.148833
$116$	0	0
$117$	0.838366	0.0775069
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	2.09127	0.190116
$122$	0	0
$123$	2.16706	0.195397
$124$	0	0
$125$	−11.1124	−0.993926
$126$	0	0
$127$	2.36854	0.210174	0.105087	−	0.994463i	$-0.466488\pi$
$127$	2.36854	0.210174	0.105087	+	0.994463i	$0.466488\pi$
$128$	0	0
$129$	−4.42628	−0.389712
$130$	0	0
$131$	0.796553	0.0695951	0.0347976	−	0.999394i	$-0.488921\pi$
$131$	0.796553	0.0695951	0.0347976	+	0.999394i	$0.488921\pi$
$132$	0	0
$133$	7.98338	0.692247
$134$	0	0
$135$	−6.96307	−0.599286
$136$	0	0
$137$	−17.7504	−1.51652	−0.758259	−	0.651953i	$-0.773950\pi$
$137$	−17.7504	−1.51652	−0.758259	+	0.651953i	$0.773950\pi$
$138$	0	0
$139$	6.02547	0.511074	0.255537	−	0.966799i	$-0.417748\pi$
$139$	6.02547	0.511074	0.255537	+	0.966799i	$0.417748\pi$
$140$	0	0
$141$	3.90076	0.328504
$142$	0	0
$143$	−1.56968	−0.131263
$144$	0	0
$145$	−7.67614	−0.637468
$146$	0	0
$147$	1.03321	0.0852180
$148$	0	0
$149$	−8.95136	−0.733324	−0.366662	−	0.930354i	$-0.619499\pi$
$149$	−8.95136	−0.733324	−0.366662	+	0.930354i	$0.619499\pi$
$150$	0	0
$151$	−12.0604	−0.981460	−0.490730	−	0.871312i	$-0.663270\pi$
$151$	−12.0604	−0.981460	−0.490730	+	0.871312i	$0.663270\pi$
$152$	0	0
$153$	1.93247	0.156231
$154$	0	0
$155$	−11.9883	−0.962923
$156$	0	0
$157$	−11.2192	−0.895387	−0.447693	−	0.894187i	$-0.647754\pi$
$157$	−11.2192	−0.895387	−0.447693	+	0.894187i	$0.647754\pi$
$158$	0	0
$159$	−12.8040	−1.01543
$160$	0	0
$161$	−1.16816	−0.0920640
$162$	0	0
$163$	−2.89259	−0.226565	−0.113283	−	0.993563i	$-0.536137\pi$
$163$	−2.89259	−0.226565	−0.113283	+	0.993563i	$0.536137\pi$
$164$	0	0
$165$	5.10772	0.397636
$166$	0	0
$167$	10.4106	0.805595	0.402797	−	0.915289i	$-0.368038\pi$
$167$	10.4106	0.805595	0.402797	+	0.915289i	$0.368038\pi$
$168$	0	0
$169$	−12.8118	−0.985522
$170$	0	0
$171$	15.4277	1.17978
$172$	0	0
$173$	8.39337	0.638136	0.319068	−	0.947732i	$-0.396630\pi$
$173$	8.39337	0.638136	0.319068	+	0.947732i	$0.396630\pi$
$174$	0	0
$175$	3.13322	0.236849
$176$	0	0
$177$	14.0916	1.05919
$178$	0	0
$179$	−14.6527	−1.09519	−0.547597	−	0.836742i	$-0.684457\pi$
$179$	−14.6527	−1.09519	−0.547597	+	0.836742i	$0.684457\pi$
$180$	0	0
$181$	0.291311	0.0216530	0.0108265	−	0.999941i	$-0.496554\pi$
$181$	0.291311	0.0216530	0.0108265	+	0.999941i	$0.496554\pi$
$182$	0	0
$183$	−4.37916	−0.323717
$184$	0	0
$185$	−2.78155	−0.204504
$186$	0	0
$187$	−3.61819	−0.264588
$188$	0	0
$189$	5.09629	0.370701
$190$	0	0
$191$	11.4243	0.826634	0.413317	−	0.910587i	$-0.364370\pi$
$191$	11.4243	0.826634	0.413317	+	0.910587i	$0.364370\pi$
$192$	0	0
$193$	−5.99180	−0.431300	−0.215650	−	0.976471i	$-0.569187\pi$
$193$	−5.99180	−0.431300	−0.215650	+	0.976471i	$0.569187\pi$
$194$	0	0
$195$	−0.612431	−0.0438571
$196$	0	0
$197$	20.5297	1.46268	0.731340	−	0.682014i	$-0.238895\pi$
$197$	20.5297	1.46268	0.731340	+	0.682014i	$0.238895\pi$
$198$	0	0
$199$	−11.1808	−0.792585	−0.396293	−	0.918124i	$-0.629703\pi$
$199$	−11.1808	−0.792585	−0.396293	+	0.918124i	$0.629703\pi$
$200$	0	0
$201$	0.405925	0.0286317
$202$	0	0
$203$	5.61819	0.394319
$204$	0	0
$205$	2.86568	0.200148
$206$	0	0
$207$	−2.25744	−0.156903
$208$	0	0
$209$	−28.8854	−1.99804
$210$	0	0
$211$	26.5734	1.82939	0.914694	−	0.404148i	$-0.132432\pi$
$211$	26.5734	1.82939	0.914694	+	0.404148i	$0.132432\pi$
$212$	0	0
$213$	6.68588	0.458109
$214$	0	0
$215$	−5.85323	−0.399187
$216$	0	0
$217$	8.77427	0.595636
$218$	0	0
$219$	−5.30966	−0.358793
$220$	0	0
$221$	0.433831	0.0291826
$222$	0	0
$223$	−7.73836	−0.518199	−0.259099	−	0.965851i	$-0.583426\pi$
$223$	−7.73836	−0.518199	−0.259099	+	0.965851i	$0.583426\pi$
$224$	0	0
$225$	6.05486	0.403657
$226$	0	0
$227$	3.11135	0.206508	0.103254	−	0.994655i	$-0.467075\pi$
$227$	3.11135	0.206508	0.103254	+	0.994655i	$0.467075\pi$
$228$	0	0
$229$	−20.0645	−1.32590	−0.662949	−	0.748665i	$-0.730695\pi$
$229$	−20.0645	−1.32590	−0.662949	+	0.748665i	$0.730695\pi$
$230$	0	0
$231$	−3.73836	−0.245966
$232$	0	0
$233$	20.5285	1.34487	0.672433	−	0.740158i	$-0.265249\pi$
$233$	20.5285	1.34487	0.672433	+	0.740158i	$0.265249\pi$
$234$	0	0
$235$	5.15830	0.336490
$236$	0	0
$237$	1.87320	0.121677
$238$	0	0
$239$	−16.2161	−1.04893	−0.524466	−	0.851431i	$-0.675735\pi$
$239$	−16.2161	−1.04893	−0.524466	+	0.851431i	$0.675735\pi$
$240$	0	0
$241$	−19.2949	−1.24290	−0.621448	−	0.783456i	$-0.713455\pi$
$241$	−19.2949	−1.24290	−0.621448	+	0.783456i	$0.713455\pi$
$242$	0	0
$243$	15.8384	1.01603
$244$	0	0
$245$	1.36630	0.0872898
$246$	0	0
$247$	3.46344	0.220373
$248$	0	0
$249$	3.12894	0.198289
$250$	0	0
$251$	17.5859	1.11001	0.555005	−	0.831847i	$-0.312716\pi$
$251$	17.5859	1.11001	0.555005	+	0.831847i	$0.312716\pi$
$252$	0	0
$253$	4.22663	0.265726
$254$	0	0
$255$	−1.41168	−0.0884029
$256$	0	0
$257$	−7.72685	−0.481987	−0.240994	−	0.970527i	$-0.577473\pi$
$257$	−7.72685	−0.481987	−0.240994	+	0.970527i	$0.577473\pi$
$258$	0	0
$259$	2.03582	0.126500
$260$	0	0
$261$	10.8570	0.672030
$262$	0	0
$263$	9.69908	0.598071	0.299035	−	0.954242i	$-0.403335\pi$
$263$	9.69908	0.598071	0.299035	+	0.954242i	$0.403335\pi$
$264$	0	0
$265$	−16.9318	−1.04011
$266$	0	0
$267$	15.8840	0.972087
$268$	0	0
$269$	−11.5468	−0.704021	−0.352010	−	0.935996i	$-0.614502\pi$
$269$	−11.5468	−0.704021	−0.352010	+	0.935996i	$0.614502\pi$
$270$	0	0
$271$	−11.8003	−0.716817	−0.358409	−	0.933565i	$-0.616681\pi$
$271$	−11.8003	−0.716817	−0.358409	+	0.933565i	$0.616681\pi$
$272$	0	0
$273$	0.448240	0.0271287
$274$	0	0
$275$	−11.3366	−0.683621
$276$	0	0
$277$	10.8177	0.649974	0.324987	−	0.945719i	$-0.394640\pi$
$277$	10.8177	0.649974	0.324987	+	0.945719i	$0.394640\pi$
$278$	0	0
$279$	16.9560	1.01513
$280$	0	0
$281$	−0.770799	−0.0459820	−0.0229910	−	0.999736i	$-0.507319\pi$
$281$	−0.770799	−0.0459820	−0.0229910	+	0.999736i	$0.507319\pi$
$282$	0	0
$283$	24.5012	1.45645	0.728224	−	0.685340i	$-0.240346\pi$
$283$	24.5012	1.45645	0.728224	+	0.685340i	$0.240346\pi$
$284$	0	0
$285$	−11.2700	−0.667577
$286$	0	0
$287$	−2.09740	−0.123805
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	1.01897	0.0597331
$292$	0	0
$293$	−26.8790	−1.57029	−0.785145	−	0.619312i	$-0.787411\pi$
$293$	−26.8790	−1.57029	−0.785145	+	0.619312i	$0.787411\pi$
$294$	0	0
$295$	18.6344	1.08494
$296$	0	0
$297$	−18.4393	−1.06996
$298$	0	0
$299$	−0.506785	−0.0293081
$300$	0	0
$301$	4.28400	0.246925
$302$	0	0
$303$	−0.911368	−0.0523567
$304$	0	0
$305$	−5.79092	−0.331587
$306$	0	0
$307$	−3.14124	−0.179280	−0.0896402	−	0.995974i	$-0.528572\pi$
$307$	−3.14124	−0.179280	−0.0896402	+	0.995974i	$0.528572\pi$
$308$	0	0
$309$	−11.8329	−0.673150
$310$	0	0
$311$	11.8061	0.669465	0.334732	−	0.942313i	$-0.391354\pi$
$311$	11.8061	0.669465	0.334732	+	0.942313i	$0.391354\pi$
$312$	0	0
$313$	5.16859	0.292146	0.146073	−	0.989274i	$-0.453337\pi$
$313$	5.16859	0.292146	0.146073	+	0.989274i	$0.453337\pi$
$314$	0	0
$315$	2.64034	0.148766
$316$	0	0
$317$	9.45956	0.531302	0.265651	−	0.964069i	$-0.414413\pi$
$317$	9.45956	0.531302	0.265651	+	0.964069i	$0.414413\pi$
$318$	0	0
$319$	−20.3276	−1.13813
$320$	0	0
$321$	15.9635	0.890994
$322$	0	0
$323$	7.98338	0.444207
$324$	0	0
$325$	1.35929	0.0753998
$326$	0	0
$327$	−7.29831	−0.403598
$328$	0	0
$329$	−3.77537	−0.208143
$330$	0	0
$331$	27.2197	1.49613	0.748065	−	0.663625i	$-0.230983\pi$
$331$	27.2197	1.49613	0.748065	+	0.663625i	$0.230983\pi$
$332$	0	0
$333$	3.93417	0.215591
$334$	0	0
$335$	0.536788	0.0293278
$336$	0	0
$337$	−23.7410	−1.29325	−0.646626	−	0.762807i	$-0.723821\pi$
$337$	−23.7410	−1.29325	−0.646626	+	0.762807i	$0.723821\pi$
$338$	0	0
$339$	−3.39983	−0.184653
$340$	0	0
$341$	−31.7469	−1.71919
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.64907	0.0887830
$346$	0	0
$347$	6.39856	0.343493	0.171746	−	0.985141i	$-0.445059\pi$
$347$	6.39856	0.343493	0.171746	+	0.985141i	$0.445059\pi$
$348$	0	0
$349$	−15.2053	−0.813921	−0.406960	−	0.913446i	$-0.633411\pi$
$349$	−15.2053	−0.813921	−0.406960	+	0.913446i	$0.633411\pi$
$350$	0	0
$351$	2.21093	0.118011
$352$	0	0
$353$	31.9967	1.70301	0.851505	−	0.524346i	$-0.175690\pi$
$353$	31.9967	1.70301	0.851505	+	0.524346i	$0.175690\pi$
$354$	0	0
$355$	8.84128	0.469247
$356$	0	0
$357$	1.03321	0.0546834
$358$	0	0
$359$	−11.5311	−0.608586	−0.304293	−	0.952578i	$-0.598420\pi$
$359$	−11.5311	−0.608586	−0.304293	+	0.952578i	$0.598420\pi$
$360$	0	0
$361$	44.7344	2.35444
$362$	0	0
$363$	2.16073	0.113409
$364$	0	0
$365$	−7.02139	−0.367516
$366$	0	0
$367$	17.0671	0.890896	0.445448	−	0.895308i	$-0.353044\pi$
$367$	17.0671	0.890896	0.445448	+	0.895308i	$0.353044\pi$
$368$	0	0
$369$	−4.05316	−0.210999
$370$	0	0
$371$	12.3925	0.643384
$372$	0	0
$373$	−9.13050	−0.472759	−0.236380	−	0.971661i	$-0.575961\pi$
$373$	−9.13050	−0.472759	−0.236380	+	0.971661i	$0.575961\pi$
$374$	0	0
$375$	−11.4815	−0.592903
$376$	0	0
$377$	2.43734	0.125530
$378$	0	0
$379$	−0.961890	−0.0494090	−0.0247045	−	0.999695i	$-0.507864\pi$
$379$	−0.961890	−0.0494090	−0.0247045	+	0.999695i	$0.507864\pi$
$380$	0	0
$381$	2.44721	0.125374
$382$	0	0
$383$	−9.18606	−0.469386	−0.234693	−	0.972070i	$-0.575408\pi$
$383$	−9.18606	−0.469386	−0.234693	+	0.972070i	$0.575408\pi$
$384$	0	0
$385$	−4.94353	−0.251946
$386$	0	0
$387$	8.27869	0.420830
$388$	0	0
$389$	−30.1569	−1.52902	−0.764509	−	0.644614i	$-0.777018\pi$
$389$	−30.1569	−1.52902	−0.764509	+	0.644614i	$0.777018\pi$
$390$	0	0
$391$	−1.16816	−0.0590765
$392$	0	0
$393$	0.823009	0.0415153
$394$	0	0
$395$	2.47708	0.124635
$396$	0	0
$397$	7.97859	0.400434	0.200217	−	0.979752i	$-0.435835\pi$
$397$	7.97859	0.400434	0.200217	+	0.979752i	$0.435835\pi$
$398$	0	0
$399$	8.24854	0.412943
$400$	0	0
$401$	−30.1922	−1.50772	−0.753862	−	0.657032i	$-0.771811\pi$
$401$	−30.1922	−1.50772	−0.753862	+	0.657032i	$0.771811\pi$
$402$	0	0
$403$	3.80655	0.189618
$404$	0	0
$405$	0.726671	0.0361086
$406$	0	0
$407$	−7.36599	−0.365119
$408$	0	0
$409$	37.8398	1.87106	0.935528	−	0.353253i	$-0.114924\pi$
$409$	37.8398	1.87106	0.935528	+	0.353253i	$0.114924\pi$
$410$	0	0
$411$	−18.3399	−0.904642
$412$	0	0
$413$	−13.6386	−0.671111
$414$	0	0
$415$	4.13765	0.203109
$416$	0	0
$417$	6.22560	0.304869
$418$	0	0
$419$	−5.97012	−0.291659	−0.145830	−	0.989310i	$-0.546585\pi$
$419$	−5.97012	−0.291659	−0.145830	+	0.989310i	$0.546585\pi$
$420$	0	0
$421$	6.38246	0.311062	0.155531	−	0.987831i	$-0.450291\pi$
$421$	6.38246	0.311062	0.155531	+	0.987831i	$0.450291\pi$
$422$	0	0
$423$	−7.29579	−0.354734
$424$	0	0
$425$	3.13322	0.151984
$426$	0	0
$427$	4.23839	0.205110
$428$	0	0
$429$	−1.62182	−0.0783020
$430$	0	0
$431$	0.0788856	0.00379978	0.00189989	−	0.999998i	$-0.499395\pi$
$431$	0.0788856	0.00379978	0.00189989	+	0.999998i	$0.499395\pi$
$432$	0	0
$433$	−29.6701	−1.42585	−0.712926	−	0.701239i	$-0.752630\pi$
$433$	−29.6701	−1.42585	−0.712926	+	0.701239i	$0.752630\pi$
$434$	0	0
$435$	−7.93109	−0.380266
$436$	0	0
$437$	−9.32588	−0.446117
$438$	0	0
$439$	−14.1016	−0.673033	−0.336516	−	0.941678i	$-0.609249\pi$
$439$	−14.1016	−0.673033	−0.336516	+	0.941678i	$0.609249\pi$
$440$	0	0
$441$	−1.93247	−0.0920224
$442$	0	0
$443$	7.36923	0.350123	0.175061	−	0.984558i	$-0.443988\pi$
$443$	7.36923	0.350123	0.175061	+	0.984558i	$0.443988\pi$
$444$	0	0
$445$	21.0047	0.995720
$446$	0	0
$447$	−9.24867	−0.437447
$448$	0	0
$449$	8.75753	0.413293	0.206647	−	0.978416i	$-0.433745\pi$
$449$	8.75753	0.413293	0.206647	+	0.978416i	$0.433745\pi$
$450$	0	0
$451$	7.58877	0.357341
$452$	0	0
$453$	−12.4609	−0.585466
$454$	0	0
$455$	0.592744	0.0277883
$456$	0	0
$457$	−20.9184	−0.978519	−0.489260	−	0.872138i	$-0.662733\pi$
$457$	−20.9184	−0.978519	−0.489260	+	0.872138i	$0.662733\pi$
$458$	0	0
$459$	5.09629	0.237875
$460$	0	0
$461$	25.3568	1.18099	0.590493	−	0.807043i	$-0.298933\pi$
$461$	25.3568	1.18099	0.590493	+	0.807043i	$0.298933\pi$
$462$	0	0
$463$	−29.4000	−1.36633	−0.683167	−	0.730262i	$-0.739398\pi$
$463$	−29.4000	−1.36633	−0.683167	+	0.730262i	$0.739398\pi$
$464$	0	0
$465$	−12.3865	−0.574409
$466$	0	0
$467$	−34.7409	−1.60762	−0.803809	−	0.594888i	$-0.797197\pi$
$467$	−34.7409	−1.60762	−0.803809	+	0.594888i	$0.797197\pi$
$468$	0	0
$469$	−0.392876	−0.0181413
$470$	0	0
$471$	−11.5918	−0.534121
$472$	0	0
$473$	−15.5003	−0.712704
$474$	0	0
$475$	25.0137	1.14771
$476$	0	0
$477$	23.9481	1.09651
$478$	0	0
$479$	−21.3771	−0.976743	−0.488371	−	0.872636i	$-0.662409\pi$
$479$	−21.3771	−0.976743	−0.488371	+	0.872636i	$0.662409\pi$
$480$	0	0
$481$	0.883203	0.0402706
$482$	0	0
$483$	−1.20696	−0.0549186
$484$	0	0
$485$	1.34747	0.0611854
$486$	0	0
$487$	−31.1361	−1.41091	−0.705456	−	0.708754i	$-0.749258\pi$
$487$	−31.1361	−1.41091	−0.705456	+	0.708754i	$0.749258\pi$
$488$	0	0
$489$	−2.98866	−0.135152
$490$	0	0
$491$	−23.8418	−1.07596	−0.537982	−	0.842957i	$-0.680813\pi$
$491$	−23.8418	−1.07596	−0.537982	+	0.842957i	$0.680813\pi$
$492$	0	0
$493$	5.61819	0.253030
$494$	0	0
$495$	−9.55323	−0.429386
$496$	0	0
$497$	−6.47096	−0.290262
$498$	0	0
$499$	−31.5541	−1.41256	−0.706278	−	0.707934i	$-0.749627\pi$
$499$	−31.5541	−1.41256	−0.706278	+	0.707934i	$0.749627\pi$
$500$	0	0
$501$	10.7563	0.480558
$502$	0	0
$503$	9.12790	0.406993	0.203496	−	0.979076i	$-0.434769\pi$
$503$	9.12790	0.406993	0.203496	+	0.979076i	$0.434769\pi$
$504$	0	0
$505$	−1.20517	−0.0536296
$506$	0	0
$507$	−13.2373	−0.587890
$508$	0	0
$509$	28.7629	1.27489	0.637447	−	0.770494i	$-0.279991\pi$
$509$	28.7629	1.27489	0.637447	+	0.770494i	$0.279991\pi$
$510$	0	0
$511$	5.13898	0.227335
$512$	0	0
$513$	40.6857	1.79632
$514$	0	0
$515$	−15.6476	−0.689516
$516$	0	0
$517$	13.6600	0.600766
$518$	0	0
$519$	8.67214	0.380665
$520$	0	0
$521$	18.9095	0.828439	0.414220	−	0.910177i	$-0.364055\pi$
$521$	18.9095	0.828439	0.414220	+	0.910177i	$0.364055\pi$
$522$	0	0
$523$	0.124841	0.00545890	0.00272945	−	0.999996i	$-0.499131\pi$
$523$	0.124841	0.00545890	0.00272945	+	0.999996i	$0.499131\pi$
$524$	0	0
$525$	3.23729	0.141287
$526$	0	0
$527$	8.77427	0.382213
$528$	0	0
$529$	−21.6354	−0.940670
$530$	0	0
$531$	−26.3562	−1.14376
$532$	0	0
$533$	−0.909916	−0.0394128
$534$	0	0
$535$	21.1098	0.912656
$536$	0	0
$537$	−15.1394	−0.653312
$538$	0	0
$539$	3.61819	0.155846
$540$	0	0
$541$	−6.38332	−0.274440	−0.137220	−	0.990541i	$-0.543817\pi$
$541$	−6.38332	−0.274440	−0.137220	+	0.990541i	$0.543817\pi$
$542$	0	0
$543$	0.300986	0.0129166
$544$	0	0
$545$	−9.65115	−0.413410
$546$	0	0
$547$	34.4049	1.47105	0.735523	−	0.677500i	$-0.236937\pi$
$547$	34.4049	1.47105	0.735523	+	0.677500i	$0.236937\pi$
$548$	0	0
$549$	8.19056	0.349565
$550$	0	0
$551$	44.8521	1.91077
$552$	0	0
$553$	−1.81298	−0.0770958
$554$	0	0
$555$	−2.87393	−0.121992
$556$	0	0
$557$	10.7281	0.454565	0.227282	−	0.973829i	$-0.427016\pi$
$557$	10.7281	0.454565	0.227282	+	0.973829i	$0.427016\pi$
$558$	0	0
$559$	1.85853	0.0786075
$560$	0	0
$561$	−3.73836	−0.157834
$562$	0	0
$563$	−6.18082	−0.260491	−0.130245	−	0.991482i	$-0.541576\pi$
$563$	−6.18082	−0.260491	−0.130245	+	0.991482i	$0.541576\pi$
$564$	0	0
$565$	−4.49587	−0.189143
$566$	0	0
$567$	−0.531853	−0.0223357
$568$	0	0
$569$	2.11127	0.0885090	0.0442545	−	0.999020i	$-0.485909\pi$
$569$	2.11127	0.0885090	0.0442545	+	0.999020i	$0.485909\pi$
$570$	0	0
$571$	18.3029	0.765952	0.382976	−	0.923758i	$-0.374899\pi$
$571$	18.3029	0.765952	0.382976	+	0.923758i	$0.374899\pi$
$572$	0	0
$573$	11.8037	0.493109
$574$	0	0
$575$	−3.66011	−0.152637
$576$	0	0
$577$	−14.6759	−0.610966	−0.305483	−	0.952197i	$-0.598818\pi$
$577$	−14.6759	−0.610966	−0.305483	+	0.952197i	$0.598818\pi$
$578$	0	0
$579$	−6.19081	−0.257281
$580$	0	0
$581$	−3.02836	−0.125638
$582$	0	0
$583$	−44.8382	−1.85701
$584$	0	0
$585$	1.14546	0.0473589
$586$	0	0
$587$	21.3369	0.880669	0.440334	−	0.897834i	$-0.354860\pi$
$587$	21.3369	0.880669	0.440334	+	0.897834i	$0.354860\pi$
$588$	0	0
$589$	70.0484	2.88629
$590$	0	0
$591$	21.2115	0.872526
$592$	0	0
$593$	20.7197	0.850857	0.425428	−	0.904992i	$-0.360123\pi$
$593$	20.7197	0.850857	0.425428	+	0.904992i	$0.360123\pi$
$594$	0	0
$595$	1.36630	0.0560129
$596$	0	0
$597$	−11.5521	−0.472798
$598$	0	0
$599$	−33.9551	−1.38737	−0.693684	−	0.720279i	$-0.744014\pi$
$599$	−33.9551	−1.38737	−0.693684	+	0.720279i	$0.744014\pi$
$600$	0	0
$601$	2.47422	0.100925	0.0504627	−	0.998726i	$-0.483930\pi$
$601$	2.47422	0.100925	0.0504627	+	0.998726i	$0.483930\pi$
$602$	0	0
$603$	−0.759222	−0.0309179
$604$	0	0
$605$	2.85731	0.116166
$606$	0	0
$607$	10.6420	0.431947	0.215974	−	0.976399i	$-0.430707\pi$
$607$	10.6420	0.431947	0.215974	+	0.976399i	$0.430707\pi$
$608$	0	0
$609$	5.80478	0.235222
$610$	0	0
$611$	−1.63787	−0.0662613
$612$	0	0
$613$	−32.6990	−1.32070	−0.660349	−	0.750959i	$-0.729592\pi$
$613$	−32.6990	−1.32070	−0.660349	+	0.750959i	$0.729592\pi$
$614$	0	0
$615$	2.96085	0.119393
$616$	0	0
$617$	−26.4645	−1.06542	−0.532711	−	0.846297i	$-0.678827\pi$
$617$	−26.4645	−1.06542	−0.532711	+	0.846297i	$0.678827\pi$
$618$	0	0
$619$	19.7169	0.792490	0.396245	−	0.918145i	$-0.370313\pi$
$619$	19.7169	0.792490	0.396245	+	0.918145i	$0.370313\pi$
$620$	0	0
$621$	−5.95329	−0.238897
$622$	0	0
$623$	−15.3734	−0.615923
$624$	0	0
$625$	0.483174	0.0193269
$626$	0	0
$627$	−29.8448	−1.19188
$628$	0	0
$629$	2.03582	0.0811736
$630$	0	0
$631$	33.1790	1.32083	0.660417	−	0.750899i	$-0.270379\pi$
$631$	33.1790	1.32083	0.660417	+	0.750899i	$0.270379\pi$
$632$	0	0
$633$	27.4560	1.09128
$634$	0	0
$635$	3.23615	0.128423
$636$	0	0
$637$	−0.433831	−0.0171890
$638$	0	0
$639$	−12.5049	−0.494688
$640$	0	0
$641$	−28.5498	−1.12765	−0.563825	−	0.825894i	$-0.690671\pi$
$641$	−28.5498	−1.12765	−0.563825	+	0.825894i	$0.690671\pi$
$642$	0	0
$643$	−0.629729	−0.0248341	−0.0124170	−	0.999923i	$-0.503953\pi$
$643$	−0.629729	−0.0248341	−0.0124170	+	0.999923i	$0.503953\pi$
$644$	0	0
$645$	−6.04763	−0.238125
$646$	0	0
$647$	26.9956	1.06131	0.530654	−	0.847588i	$-0.321946\pi$
$647$	26.9956	1.06131	0.530654	+	0.847588i	$0.321946\pi$
$648$	0	0
$649$	49.3469	1.93704
$650$	0	0
$651$	9.06569	0.355312
$652$	0	0
$653$	41.7136	1.63238	0.816189	−	0.577785i	$-0.196083\pi$
$653$	41.7136	1.63238	0.816189	+	0.577785i	$0.196083\pi$
$654$	0	0
$655$	1.08833	0.0425246
$656$	0	0
$657$	9.93092	0.387442
$658$	0	0
$659$	36.2324	1.41141	0.705707	−	0.708503i	$-0.250629\pi$
$659$	36.2324	1.41141	0.705707	+	0.708503i	$0.250629\pi$
$660$	0	0
$661$	31.9036	1.24091	0.620453	−	0.784244i	$-0.286949\pi$
$661$	31.9036	1.24091	0.620453	+	0.784244i	$0.286949\pi$
$662$	0	0
$663$	0.448240	0.0174082
$664$	0	0
$665$	10.9077	0.422983
$666$	0	0
$667$	−6.56295	−0.254118
$668$	0	0
$669$	−7.99537	−0.309119
$670$	0	0
$671$	−15.3353	−0.592012
$672$	0	0
$673$	41.4779	1.59885	0.799427	−	0.600763i	$-0.205136\pi$
$673$	41.4779	1.59885	0.799427	+	0.600763i	$0.205136\pi$
$674$	0	0
$675$	15.9678	0.614601
$676$	0	0
$677$	36.9997	1.42201	0.711007	−	0.703185i	$-0.248239\pi$
$677$	36.9997	1.42201	0.711007	+	0.703185i	$0.248239\pi$
$678$	0	0
$679$	−0.986216	−0.0378475
$680$	0	0
$681$	3.21469	0.123187
$682$	0	0
$683$	−24.2529	−0.928012	−0.464006	−	0.885832i	$-0.653588\pi$
$683$	−24.2529	−0.928012	−0.464006	+	0.885832i	$0.653588\pi$
$684$	0	0
$685$	−24.2524	−0.926636
$686$	0	0
$687$	−20.7309	−0.790933
$688$	0	0
$689$	5.37623	0.204818
$690$	0	0
$691$	8.62679	0.328178	0.164089	−	0.986446i	$-0.447532\pi$
$691$	8.62679	0.328178	0.164089	+	0.986446i	$0.447532\pi$
$692$	0	0
$693$	6.99204	0.265606
$694$	0	0
$695$	8.23261	0.312281
$696$	0	0
$697$	−2.09740	−0.0794446
$698$	0	0
$699$	21.2103	0.802248
$700$	0	0
$701$	46.9485	1.77322	0.886610	−	0.462517i	$-0.153054\pi$
$701$	46.9485	1.77322	0.886610	+	0.462517i	$0.153054\pi$
$702$	0	0
$703$	16.2528	0.612985
$704$	0	0
$705$	5.32962	0.200725
$706$	0	0
$707$	0.882071	0.0331737
$708$	0	0
$709$	−7.55462	−0.283720	−0.141860	−	0.989887i	$-0.545308\pi$
$709$	−7.55462	−0.283720	−0.141860	+	0.989887i	$0.545308\pi$
$710$	0	0
$711$	−3.50353	−0.131393
$712$	0	0
$713$	−10.2498	−0.383857
$714$	0	0
$715$	−2.14466	−0.0802057
$716$	0	0
$717$	−16.7547	−0.625715
$718$	0	0
$719$	−27.7902	−1.03640	−0.518201	−	0.855259i	$-0.673398\pi$
$719$	−27.7902	−1.03640	−0.518201	+	0.855259i	$0.673398\pi$
$720$	0	0
$721$	11.4525	0.426514
$722$	0	0
$723$	−19.9358	−0.741419
$724$	0	0
$725$	17.6030	0.653760
$726$	0	0
$727$	−33.3418	−1.23658	−0.618289	−	0.785951i	$-0.712174\pi$
$727$	−33.3418	−1.23658	−0.618289	+	0.785951i	$0.712174\pi$
$728$	0	0
$729$	14.7689	0.546996
$730$	0	0
$731$	4.28400	0.158449
$732$	0	0
$733$	21.8265	0.806179	0.403089	−	0.915161i	$-0.367936\pi$
$733$	21.8265	0.806179	0.403089	+	0.915161i	$0.367936\pi$
$734$	0	0
$735$	1.41168	0.0520706
$736$	0	0
$737$	1.42150	0.0523616
$738$	0	0
$739$	7.99175	0.293981	0.146991	−	0.989138i	$-0.453041\pi$
$739$	7.99175	0.293981	0.146991	+	0.989138i	$0.453041\pi$
$740$	0	0
$741$	3.57847	0.131458
$742$	0	0
$743$	21.5451	0.790412	0.395206	−	0.918592i	$-0.370673\pi$
$743$	21.5451	0.790412	0.395206	+	0.918592i	$0.370673\pi$
$744$	0	0
$745$	−12.2303	−0.448082
$746$	0	0
$747$	−5.85222	−0.214121
$748$	0	0
$749$	−15.4503	−0.564542
$750$	0	0
$751$	−31.1273	−1.13585	−0.567926	−	0.823079i	$-0.692254\pi$
$751$	−31.1273	−1.13585	−0.567926	+	0.823079i	$0.692254\pi$
$752$	0	0
$753$	18.1699	0.662149
$754$	0	0
$755$	−16.4781	−0.599700
$756$	0	0
$757$	51.6759	1.87819	0.939096	−	0.343655i	$-0.111665\pi$
$757$	51.6759	1.87819	0.939096	+	0.343655i	$0.111665\pi$
$758$	0	0
$759$	4.36701	0.158512
$760$	0	0
$761$	13.0392	0.472671	0.236335	−	0.971672i	$-0.424054\pi$
$761$	13.0392	0.472671	0.236335	+	0.971672i	$0.424054\pi$
$762$	0	0
$763$	7.06371	0.255723
$764$	0	0
$765$	2.64034	0.0954616
$766$	0	0
$767$	−5.91684	−0.213645
$768$	0	0
$769$	12.8190	0.462264	0.231132	−	0.972922i	$-0.425757\pi$
$769$	12.8190	0.462264	0.231132	+	0.972922i	$0.425757\pi$
$770$	0	0
$771$	−7.98348	−0.287518
$772$	0	0
$773$	−30.1833	−1.08562	−0.542808	−	0.839857i	$-0.682639\pi$
$773$	−30.1833	−1.08562	−0.542808	+	0.839857i	$0.682639\pi$
$774$	0	0
$775$	27.4917	0.987532
$776$	0	0
$777$	2.10344	0.0754605
$778$	0	0
$779$	−16.7443	−0.599928
$780$	0	0
$781$	23.4131	0.837788
$782$	0	0
$783$	28.6319	1.02322
$784$	0	0
$785$	−15.3288	−0.547107
$786$	0	0
$787$	−44.2604	−1.57771	−0.788857	−	0.614577i	$-0.789327\pi$
$787$	−44.2604	−1.57771	−0.788857	+	0.614577i	$0.789327\pi$
$788$	0	0
$789$	10.0212	0.356765
$790$	0	0
$791$	3.29054	0.116998
$792$	0	0
$793$	1.83874	0.0652957
$794$	0	0
$795$	−17.4942	−0.620455
$796$	0	0
$797$	−53.6382	−1.89996	−0.949982	−	0.312306i	$-0.898899\pi$
$797$	−53.6382	−1.89996	−0.949982	+	0.312306i	$0.898899\pi$
$798$	0	0
$799$	−3.77537	−0.133563
$800$	0	0
$801$	−29.7087	−1.04971
$802$	0	0
$803$	−18.5938	−0.656160
$804$	0	0
$805$	−1.59606	−0.0562537
$806$	0	0
$807$	−11.9303	−0.419967
$808$	0	0
$809$	−19.0440	−0.669551	−0.334776	−	0.942298i	$-0.608661\pi$
$809$	−19.0440	−0.669551	−0.334776	+	0.942298i	$0.608661\pi$
$810$	0	0
$811$	51.4731	1.80747	0.903733	−	0.428096i	$-0.140816\pi$
$811$	51.4731	1.80747	0.903733	+	0.428096i	$0.140816\pi$
$812$	0	0
$813$	−12.1922	−0.427600
$814$	0	0
$815$	−3.95215	−0.138438
$816$	0	0
$817$	34.2008	1.19653
$818$	0	0
$819$	−0.838366	−0.0292949
$820$	0	0
$821$	4.50392	0.157188	0.0785940	−	0.996907i	$-0.474957\pi$
$821$	4.50392	0.157188	0.0785940	+	0.996907i	$0.474957\pi$
$822$	0	0
$823$	−37.6342	−1.31185	−0.655923	−	0.754828i	$-0.727720\pi$
$823$	−37.6342	−1.31185	−0.655923	+	0.754828i	$0.727720\pi$
$824$	0	0
$825$	−11.7131	−0.407798
$826$	0	0
$827$	3.09731	0.107704	0.0538520	−	0.998549i	$-0.482850\pi$
$827$	3.09731	0.107704	0.0538520	+	0.998549i	$0.482850\pi$
$828$	0	0
$829$	36.0877	1.25338	0.626688	−	0.779270i	$-0.284410\pi$
$829$	36.0877	1.25338	0.626688	+	0.779270i	$0.284410\pi$
$830$	0	0
$831$	11.1770	0.387726
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	14.2240	0.492241
$836$	0	0
$837$	44.7163	1.54562
$838$	0	0
$839$	41.0497	1.41719	0.708597	−	0.705614i	$-0.249329\pi$
$839$	41.0497	1.41719	0.708597	+	0.705614i	$0.249329\pi$
$840$	0	0
$841$	2.56402	0.0884145
$842$	0	0
$843$	−0.796400	−0.0274295
$844$	0	0
$845$	−17.5048	−0.602182
$846$	0	0
$847$	−2.09127	−0.0718570
$848$	0	0
$849$	25.3150	0.868809
$850$	0	0
$851$	−2.37817	−0.0815226
$852$	0	0
$853$	−2.23419	−0.0764972	−0.0382486	−	0.999268i	$-0.512178\pi$
$853$	−2.23419	−0.0764972	−0.0382486	+	0.999268i	$0.512178\pi$
$854$	0	0
$855$	21.0788	0.720881
$856$	0	0
$857$	−43.1912	−1.47538	−0.737692	−	0.675137i	$-0.764084\pi$
$857$	−43.1912	−1.47538	−0.737692	+	0.675137i	$0.764084\pi$
$858$	0	0
$859$	−44.1166	−1.50524	−0.752619	−	0.658456i	$-0.771210\pi$
$859$	−44.1166	−1.50524	−0.752619	+	0.658456i	$0.771210\pi$
$860$	0	0
$861$	−2.16706	−0.0738532
$862$	0	0
$863$	−25.1589	−0.856419	−0.428209	−	0.903680i	$-0.640855\pi$
$863$	−25.1589	−0.856419	−0.428209	+	0.903680i	$0.640855\pi$
$864$	0	0
$865$	11.4679	0.389919
$866$	0	0
$867$	1.03321	0.0350898
$868$	0	0
$869$	6.55970	0.222523
$870$	0	0
$871$	−0.170442	−0.00577521
$872$	0	0
$873$	−1.90583	−0.0645027
$874$	0	0
$875$	11.1124	0.375669
$876$	0	0
$877$	−50.4841	−1.70473	−0.852363	−	0.522950i	$-0.824831\pi$
$877$	−50.4841	−1.70473	−0.852363	+	0.522950i	$0.824831\pi$
$878$	0	0
$879$	−27.7718	−0.936718
$880$	0	0
$881$	45.5434	1.53440	0.767198	−	0.641410i	$-0.221650\pi$
$881$	45.5434	1.53440	0.767198	+	0.641410i	$0.221650\pi$
$882$	0	0
$883$	4.57926	0.154104	0.0770522	−	0.997027i	$-0.475449\pi$
$883$	4.57926	0.154104	0.0770522	+	0.997027i	$0.475449\pi$
$884$	0	0
$885$	19.2533	0.647194
$886$	0	0
$887$	9.75709	0.327611	0.163806	−	0.986493i	$-0.447623\pi$
$887$	9.75709	0.327611	0.163806	+	0.986493i	$0.447623\pi$
$888$	0	0
$889$	−2.36854	−0.0794384
$890$	0	0
$891$	1.92434	0.0644679
$892$	0	0
$893$	−30.1402	−1.00860
$894$	0	0
$895$	−20.0200	−0.669195
$896$	0	0
$897$	−0.523617	−0.0174831
$898$	0	0
$899$	49.2955	1.64410
$900$	0	0
$901$	12.3925	0.412853
$902$	0	0
$903$	4.42628	0.147297
$904$	0	0
$905$	0.398019	0.0132306
$906$	0	0
$907$	−3.29156	−0.109294	−0.0546472	−	0.998506i	$-0.517403\pi$
$907$	−3.29156	−0.109294	−0.0546472	+	0.998506i	$0.517403\pi$
$908$	0	0
$909$	1.70458	0.0565372
$910$	0	0
$911$	2.19783	0.0728173	0.0364086	−	0.999337i	$-0.488408\pi$
$911$	2.19783	0.0728173	0.0364086	+	0.999337i	$0.488408\pi$
$912$	0	0
$913$	10.9572	0.362629
$914$	0	0
$915$	−5.98325	−0.197800
$916$	0	0
$917$	−0.796553	−0.0263045
$918$	0	0
$919$	23.4899	0.774861	0.387430	−	0.921899i	$-0.373363\pi$
$919$	23.4899	0.774861	0.387430	+	0.921899i	$0.373363\pi$
$920$	0	0
$921$	−3.24558	−0.106945
$922$	0	0
$923$	−2.80730	−0.0924035
$924$	0	0
$925$	6.37868	0.209730
$926$	0	0
$927$	22.1317	0.726899
$928$	0	0
$929$	−47.6752	−1.56417	−0.782087	−	0.623170i	$-0.785845\pi$
$929$	−47.6752	−1.56417	−0.782087	+	0.623170i	$0.785845\pi$
$930$	0	0
$931$	−7.98338	−0.261645
$932$	0	0
$933$	12.1983	0.399353
$934$	0	0
$935$	−4.94353	−0.161671
$936$	0	0
$937$	52.8484	1.72648	0.863241	−	0.504792i	$-0.168431\pi$
$937$	52.8484	1.72648	0.863241	+	0.504792i	$0.168431\pi$
$938$	0	0
$939$	5.34026	0.174273
$940$	0	0
$941$	19.9202	0.649379	0.324689	−	0.945821i	$-0.394740\pi$
$941$	19.9202	0.649379	0.324689	+	0.945821i	$0.394740\pi$
$942$	0	0
$943$	2.45010	0.0797862
$944$	0	0
$945$	6.96307	0.226509
$946$	0	0
$947$	22.3510	0.726311	0.363156	−	0.931729i	$-0.381699\pi$
$947$	22.3510	0.726311	0.363156	+	0.931729i	$0.381699\pi$
$948$	0	0
$949$	2.22945	0.0723709
$950$	0	0
$951$	9.77375	0.316935
$952$	0	0
$953$	−0.395664	−0.0128168	−0.00640841	−	0.999979i	$-0.502040\pi$
$953$	−0.395664	−0.0128168	−0.00640841	+	0.999979i	$0.502040\pi$
$954$	0	0
$955$	15.6090	0.505097
$956$	0	0
$957$	−21.0028	−0.678924
$958$	0	0
$959$	17.7504	0.573190
$960$	0	0
$961$	45.9878	1.48348
$962$	0	0
$963$	−29.8573	−0.962137
$964$	0	0
$965$	−8.18661	−0.263536
$966$	0	0
$967$	57.7244	1.85629	0.928145	−	0.372218i	$-0.121403\pi$
$967$	57.7244	1.85629	0.928145	+	0.372218i	$0.121403\pi$
$968$	0	0
$969$	8.24854	0.264981
$970$	0	0
$971$	−47.8636	−1.53602	−0.768008	−	0.640440i	$-0.778752\pi$
$971$	−47.8636	−1.53602	−0.768008	+	0.640440i	$0.778752\pi$
$972$	0	0
$973$	−6.02547	−0.193168
$974$	0	0
$975$	1.40443	0.0449779
$976$	0	0
$977$	−3.15545	−0.100952	−0.0504759	−	0.998725i	$-0.516074\pi$
$977$	−3.15545	−0.100952	−0.0504759	+	0.998725i	$0.516074\pi$
$978$	0	0
$979$	55.6239	1.77775
$980$	0	0
$981$	13.6504	0.435824
$982$	0	0
$983$	18.4200	0.587507	0.293754	−	0.955881i	$-0.405096\pi$
$983$	18.4200	0.587507	0.293754	+	0.955881i	$0.405096\pi$
$984$	0	0
$985$	28.0497	0.893739
$986$	0	0
$987$	−3.90076	−0.124163
$988$	0	0
$989$	−5.00440	−0.159131
$990$	0	0
$991$	−53.7196	−1.70646	−0.853229	−	0.521536i	$-0.825359\pi$
$991$	−53.7196	−1.70646	−0.853229	+	0.521536i	$0.825359\pi$
$992$	0	0
$993$	28.1238	0.892481
$994$	0	0
$995$	−15.2763	−0.484292
$996$	0	0
$997$	−52.3921	−1.65928	−0.829638	−	0.558302i	$-0.811453\pi$
$997$	−52.3921	−1.65928	−0.829638	+	0.558302i	$0.811453\pi$
$998$	0	0
$999$	10.3752	0.328255

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.m.1.4	yes	6
4.3	odd	2		3808.2.a.i.1.3	✓	6
8.3	odd	2		7616.2.a.cb.1.4		6
8.5	even	2		7616.2.a.bx.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.i.1.3	✓	6	4.3	odd	2
3808.2.a.m.1.4	yes	6	1.1	even	1	trivial
7616.2.a.bx.1.3		6	8.5	even	2
7616.2.a.cb.1.4		6	8.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q+1.03321 q^{3} +1.36630 q^{5} -1.00000 q^{7} -1.93247 q^{9} +3.61819 q^{11} -0.433831 q^{13} +1.41168 q^{15} -1.00000 q^{17} -7.98338 q^{19} -1.03321 q^{21} +1.16816 q^{23} -3.13322 q^{25} -5.09629 q^{27} -5.61819 q^{29} -8.77427 q^{31} +3.73836 q^{33} -1.36630 q^{35} -2.03582 q^{37} -0.448240 q^{39} +2.09740 q^{41} -4.28400 q^{43} -2.64034 q^{45} +3.77537 q^{47} +1.00000 q^{49} -1.03321 q^{51} -12.3925 q^{53} +4.94353 q^{55} -8.24854 q^{57} +13.6386 q^{59} -4.23839 q^{61} +1.93247 q^{63} -0.592744 q^{65} +0.392876 q^{67} +1.20696 q^{69} +6.47096 q^{71} -5.13898 q^{73} -3.23729 q^{75} -3.61819 q^{77} +1.81298 q^{79} +0.531853 q^{81} +3.02836 q^{83} -1.36630 q^{85} -5.80478 q^{87} +15.3734 q^{89} +0.433831 q^{91} -9.06569 q^{93} -10.9077 q^{95} +0.986216 q^{97} -6.99204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 6 q^{5} - 6 q^{7} + 4 q^{9} + 2 q^{11} - 4 q^{13} - 2 q^{15} - 6 q^{17} + 10 q^{19} - 2 q^{21} - 4 q^{23} + 4 q^{25} + 8 q^{27} - 14 q^{29} - 8 q^{31} - 8 q^{33} + 6 q^{35} - 4 q^{37} - 14 q^{39}+ \cdots + 4 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 6 * q^5 - 6 * q^7 + 4 * q^9 + 2 * q^11 - 4 * q^13 - 2 * q^15 - 6 * q^17 + 10 * q^19 - 2 * q^21 - 4 * q^23 + 4 * q^25 + 8 * q^27 - 14 * q^29 - 8 * q^31 - 8 * q^33 + 6 * q^35 - 4 * q^37 - 14 * q^39 - 2 * q^41 + 8 * q^43 - 8 * q^45 + 2 * q^47 + 6 * q^49 - 2 * q^51 - 10 * q^53 - 2 * q^55 - 18 * q^57 + 14 * q^59 - 20 * q^61 - 4 * q^63 - 20 * q^65 + 6 * q^67 - 14 * q^69 - 20 * q^71 - 6 * q^73 + 10 * q^75 - 2 * q^77 - 38 * q^79 - 18 * q^81 + 16 * q^83 + 6 * q^85 + 4 * q^87 - 2 * q^89 + 4 * q^91 - 16 * q^93 - 42 * q^95 - 12 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more