LMFDB - Embedded newform 3808.2.a.e.1.1

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)

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")

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);

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 = [5,0,0,0,-2,0,-5] 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:	$0$
Dimension:	$5$
Coefficient field:	5.5.804272.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - 6x^{2} + 3x + 2 $ x^5 - 9x^3 - 6x^2 + 3*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.49947$ 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)

$f(q)$	$=$	$q-2.49947 q^{3} -2.69931 q^{5} -1.00000 q^{7} +3.24737 q^{9} +3.96654 q^{11} -0.387119 q^{13} +6.74685 q^{15} -1.00000 q^{17} -4.75333 q^{19} +2.49947 q^{21} -0.353662 q^{23} +2.28626 q^{25} -0.618307 q^{27} -9.19231 q^{29} +3.98921 q^{31} -9.91427 q^{33} +2.69931 q^{35} +5.51461 q^{37} +0.967593 q^{39} -8.86568 q^{41} -10.8134 q^{43} -8.76566 q^{45} +11.1275 q^{47} +1.00000 q^{49} +2.49947 q^{51} -5.13829 q^{53} -10.7069 q^{55} +11.8808 q^{57} -9.89336 q^{59} -9.79982 q^{61} -3.24737 q^{63} +1.04495 q^{65} -2.62263 q^{67} +0.883969 q^{69} +12.5784 q^{71} -8.73606 q^{73} -5.71444 q^{75} -3.96654 q^{77} +7.03241 q^{79} -8.19668 q^{81} -7.21112 q^{83} +2.69931 q^{85} +22.9759 q^{87} -14.7638 q^{89} +0.387119 q^{91} -9.97092 q^{93} +12.8307 q^{95} +7.34182 q^{97} +12.8809 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{5} - 5 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 5 q^{17} - 6 q^{19} + 18 q^{23} + 5 q^{25} + 18 q^{27} - 14 q^{29} + 16 q^{31} - 26 q^{33} + 2 q^{35} + 2 q^{37} + 6 q^{39} - 10 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q - 2 * q^5 - 5 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 8 * q^15 - 5 * q^17 - 6 * q^19 + 18 * q^23 + 5 * q^25 + 18 * q^27 - 14 * q^29 + 16 * q^31 - 26 * q^33 + 2 * q^35 + 2 * q^37 + 6 * q^39 - 10 * q^41 - 20 * q^43 + 6 * q^45 + 26 * q^47 + 5 * q^49 + 4 * q^55 + 12 * q^57 + 8 * q^61 - 3 * q^63 + 2 * q^65 + 12 * q^67 + 32 * q^69 + 10 * q^71 - 14 * q^73 - 4 * q^75 + 4 * q^77 + 34 * q^79 + 33 * q^81 + 20 * q^83 + 2 * q^85 + 14 * q^87 - 4 * q^89 + 6 * q^91 + 16 * q^93 + 42 * q^95 - 12 * q^97 - 8 * 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
$3$	−2.49947	−1.44307	−0.721536	−	0.692377i	$-0.756564\pi$
$3$	−2.49947	−1.44307	−0.721536	+	0.692377i	$0.756564\pi$
$4$	0	0
$5$	−2.69931	−1.20717	−0.603583	−	0.797300i	$-0.706261\pi$
$5$	−2.69931	−1.20717	−0.603583	+	0.797300i	$0.706261\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.24737	1.08246
$10$	0	0
$11$	3.96654	1.19596	0.597979	−	0.801512i	$-0.295971\pi$
$11$	3.96654	1.19596	0.597979	+	0.801512i	$0.295971\pi$
$12$	0	0
$13$	−0.387119	−0.107367	−0.0536837	−	0.998558i	$-0.517096\pi$
$13$	−0.387119	−0.107367	−0.0536837	+	0.998558i	$0.517096\pi$
$14$	0	0
$15$	6.74685	1.74203
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.75333	−1.09049	−0.545244	−	0.838277i	$-0.683563\pi$
$19$	−4.75333	−1.09049	−0.545244	+	0.838277i	$0.683563\pi$
$20$	0	0
$21$	2.49947	0.545430
$22$	0	0
$23$	−0.353662	−0.0737436	−0.0368718	−	0.999320i	$-0.511739\pi$
$23$	−0.353662	−0.0737436	−0.0368718	+	0.999320i	$0.511739\pi$
$24$	0	0
$25$	2.28626	0.457251
$26$	0	0
$27$	−0.618307	−0.118993
$28$	0	0
$29$	−9.19231	−1.70697	−0.853484	−	0.521119i	$-0.825515\pi$
$29$	−9.19231	−1.70697	−0.853484	+	0.521119i	$0.825515\pi$
$30$	0	0
$31$	3.98921	0.716482	0.358241	−	0.933629i	$-0.383377\pi$
$31$	3.98921	0.716482	0.358241	+	0.933629i	$0.383377\pi$
$32$	0	0
$33$	−9.91427	−1.72585
$34$	0	0
$35$	2.69931	0.456266
$36$	0	0
$37$	5.51461	0.906596	0.453298	−	0.891359i	$-0.350247\pi$
$37$	5.51461	0.906596	0.453298	+	0.891359i	$0.350247\pi$
$38$	0	0
$39$	0.967593	0.154939
$40$	0	0
$41$	−8.86568	−1.38459	−0.692293	−	0.721616i	$-0.743400\pi$
$41$	−8.86568	−1.38459	−0.692293	+	0.721616i	$0.743400\pi$
$42$	0	0
$43$	−10.8134	−1.64903	−0.824515	−	0.565840i	$-0.808552\pi$
$43$	−10.8134	−1.64903	−0.824515	+	0.565840i	$0.808552\pi$
$44$	0	0
$45$	−8.76566	−1.30671
$46$	0	0
$47$	11.1275	1.62311	0.811556	−	0.584275i	$-0.198621\pi$
$47$	11.1275	1.62311	0.811556	+	0.584275i	$0.198621\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.49947	0.349996
$52$	0	0
$53$	−5.13829	−0.705798	−0.352899	−	0.935661i	$-0.614804\pi$
$53$	−5.13829	−0.705798	−0.352899	+	0.935661i	$0.614804\pi$
$54$	0	0
$55$	−10.7069	−1.44372
$56$	0	0
$57$	11.8808	1.57365
$58$	0	0
$59$	−9.89336	−1.28801	−0.644003	−	0.765023i	$-0.722728\pi$
$59$	−9.89336	−1.28801	−0.644003	+	0.765023i	$0.722728\pi$
$60$	0	0
$61$	−9.79982	−1.25474	−0.627369	−	0.778722i	$-0.715868\pi$
$61$	−9.79982	−1.25474	−0.627369	+	0.778722i	$0.715868\pi$
$62$	0	0
$63$	−3.24737	−0.409131
$64$	0	0
$65$	1.04495	0.129610
$66$	0	0
$67$	−2.62263	−0.320405	−0.160202	−	0.987084i	$-0.551215\pi$
$67$	−2.62263	−0.320405	−0.160202	+	0.987084i	$0.551215\pi$
$68$	0	0
$69$	0.883969	0.106417
$70$	0	0
$71$	12.5784	1.49278	0.746389	−	0.665510i	$-0.231786\pi$
$71$	12.5784	1.49278	0.746389	+	0.665510i	$0.231786\pi$
$72$	0	0
$73$	−8.73606	−1.02248	−0.511239	−	0.859439i	$-0.670813\pi$
$73$	−8.73606	−1.02248	−0.511239	+	0.859439i	$0.670813\pi$
$74$	0	0
$75$	−5.71444	−0.659847
$76$	0	0
$77$	−3.96654	−0.452030
$78$	0	0
$79$	7.03241	0.791208	0.395604	−	0.918421i	$-0.370535\pi$
$79$	7.03241	0.791208	0.395604	+	0.918421i	$0.370535\pi$
$80$	0	0
$81$	−8.19668	−0.910742
$82$	0	0
$83$	−7.21112	−0.791523	−0.395761	−	0.918353i	$-0.629519\pi$
$83$	−7.21112	−0.791523	−0.395761	+	0.918353i	$0.629519\pi$
$84$	0	0
$85$	2.69931	0.292781
$86$	0	0
$87$	22.9759	2.46328
$88$	0	0
$89$	−14.7638	−1.56496	−0.782478	−	0.622678i	$-0.786045\pi$
$89$	−14.7638	−1.56496	−0.782478	+	0.622678i	$0.786045\pi$
$90$	0	0
$91$	0.387119	0.0405811
$92$	0	0
$93$	−9.97092	−1.03394
$94$	0	0
$95$	12.8307	1.31640
$96$	0	0
$97$	7.34182	0.745449	0.372724	−	0.927942i	$-0.378424\pi$
$97$	7.34182	0.745449	0.372724	+	0.927942i	$0.378424\pi$
$98$	0	0
$99$	12.8809	1.29457
$100$	0	0
$101$	7.86895	0.782990	0.391495	−	0.920180i	$-0.371958\pi$
$101$	7.86895	0.782990	0.391495	+	0.920180i	$0.371958\pi$
$102$	0	0
$103$	−7.10658	−0.700232	−0.350116	−	0.936706i	$-0.613858\pi$
$103$	−7.10658	−0.700232	−0.350116	+	0.936706i	$0.613858\pi$
$104$	0	0
$105$	−6.74685	−0.658425
$106$	0	0
$107$	9.38607	0.907385	0.453693	−	0.891158i	$-0.350106\pi$
$107$	9.38607	0.907385	0.453693	+	0.891158i	$0.350106\pi$
$108$	0	0
$109$	−6.13831	−0.587943	−0.293972	−	0.955814i	$-0.594977\pi$
$109$	−6.13831	−0.587943	−0.293972	+	0.955814i	$0.594977\pi$
$110$	0	0
$111$	−13.7836	−1.30828
$112$	0	0
$113$	−4.47424	−0.420902	−0.210451	−	0.977604i	$-0.567493\pi$
$113$	−4.47424	−0.420902	−0.210451	+	0.977604i	$0.567493\pi$
$114$	0	0
$115$	0.954641	0.0890208
$116$	0	0
$117$	−1.25712	−0.116221
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	4.73346	0.430315
$122$	0	0
$123$	22.1595	1.99806
$124$	0	0
$125$	7.32522	0.655188
$126$	0	0
$127$	2.77249	0.246018	0.123009	−	0.992406i	$-0.460746\pi$
$127$	2.77249	0.246018	0.123009	+	0.992406i	$0.460746\pi$
$128$	0	0
$129$	27.0279	2.37967
$130$	0	0
$131$	6.89441	0.602368	0.301184	−	0.953566i	$-0.402618\pi$
$131$	6.89441	0.602368	0.301184	+	0.953566i	$0.402618\pi$
$132$	0	0
$133$	4.75333	0.412166
$134$	0	0
$135$	1.66900	0.143645
$136$	0	0
$137$	21.9353	1.87406	0.937030	−	0.349249i	$-0.113563\pi$
$137$	21.9353	1.87406	0.937030	+	0.349249i	$0.113563\pi$
$138$	0	0
$139$	−8.16206	−0.692297	−0.346148	−	0.938180i	$-0.612511\pi$
$139$	−8.16206	−0.692297	−0.346148	+	0.938180i	$0.612511\pi$
$140$	0	0
$141$	−27.8129	−2.34227
$142$	0	0
$143$	−1.53552	−0.128407
$144$	0	0
$145$	24.8129	2.06060
$146$	0	0
$147$	−2.49947	−0.206153
$148$	0	0
$149$	9.78101	0.801291	0.400646	−	0.916233i	$-0.368786\pi$
$149$	9.78101	0.801291	0.400646	+	0.916233i	$0.368786\pi$
$150$	0	0
$151$	16.5952	1.35050	0.675251	−	0.737588i	$-0.264035\pi$
$151$	16.5952	1.35050	0.675251	+	0.737588i	$0.264035\pi$
$152$	0	0
$153$	−3.24737	−0.262535
$154$	0	0
$155$	−10.7681	−0.864914
$156$	0	0
$157$	−6.45261	−0.514974	−0.257487	−	0.966282i	$-0.582895\pi$
$157$	−6.45261	−0.514974	−0.257487	+	0.966282i	$0.582895\pi$
$158$	0	0
$159$	12.8430	1.01852
$160$	0	0
$161$	0.353662	0.0278724
$162$	0	0
$163$	22.0386	1.72620	0.863099	−	0.505035i	$-0.168520\pi$
$163$	22.0386	1.72620	0.863099	+	0.505035i	$0.168520\pi$
$164$	0	0
$165$	26.7617	2.08339
$166$	0	0
$167$	5.69497	0.440690	0.220345	−	0.975422i	$-0.429282\pi$
$167$	5.69497	0.440690	0.220345	+	0.975422i	$0.429282\pi$
$168$	0	0
$169$	−12.8501	−0.988472
$170$	0	0
$171$	−15.4358	−1.18041
$172$	0	0
$173$	−1.49216	−0.113447	−0.0567234	−	0.998390i	$-0.518065\pi$
$173$	−1.49216	−0.113447	−0.0567234	+	0.998390i	$0.518065\pi$
$174$	0	0
$175$	−2.28626	−0.172825
$176$	0	0
$177$	24.7282	1.85869
$178$	0	0
$179$	−0.790424	−0.0590790	−0.0295395	−	0.999564i	$-0.509404\pi$
$179$	−0.790424	−0.0590790	−0.0295395	+	0.999564i	$0.509404\pi$
$180$	0	0
$181$	−11.4418	−0.850465	−0.425233	−	0.905084i	$-0.639808\pi$
$181$	−11.4418	−0.850465	−0.425233	+	0.905084i	$0.639808\pi$
$182$	0	0
$183$	24.4944	1.81068
$184$	0	0
$185$	−14.8856	−1.09441
$186$	0	0
$187$	−3.96654	−0.290062
$188$	0	0
$189$	0.618307	0.0449752
$190$	0	0
$191$	3.90456	0.282524	0.141262	−	0.989972i	$-0.454884\pi$
$191$	3.90456	0.282524	0.141262	+	0.989972i	$0.454884\pi$
$192$	0	0
$193$	21.4272	1.54236	0.771182	−	0.636615i	$-0.219666\pi$
$193$	21.4272	1.54236	0.771182	+	0.636615i	$0.219666\pi$
$194$	0	0
$195$	−2.61183	−0.187037
$196$	0	0
$197$	19.8139	1.41168	0.705841	−	0.708370i	$-0.250569\pi$
$197$	19.8139	1.41168	0.705841	+	0.708370i	$0.250569\pi$
$198$	0	0
$199$	8.38280	0.594241	0.297120	−	0.954840i	$-0.403974\pi$
$199$	8.38280	0.594241	0.297120	+	0.954840i	$0.403974\pi$
$200$	0	0
$201$	6.55519	0.462367
$202$	0	0
$203$	9.19231	0.645173
$204$	0	0
$205$	23.9312	1.67143
$206$	0	0
$207$	−1.14847	−0.0798243
$208$	0	0
$209$	−18.8543	−1.30418
$210$	0	0
$211$	17.2251	1.18582	0.592911	−	0.805268i	$-0.297979\pi$
$211$	17.2251	1.18582	0.592911	+	0.805268i	$0.297979\pi$
$212$	0	0
$213$	−31.4393	−2.15419
$214$	0	0
$215$	29.1887	1.99065
$216$	0	0
$217$	−3.98921	−0.270805
$218$	0	0
$219$	21.8356	1.47551
$220$	0	0
$221$	0.387119	0.0260404
$222$	0	0
$223$	−15.2676	−1.02239	−0.511196	−	0.859464i	$-0.670797\pi$
$223$	−15.2676	−1.02239	−0.511196	+	0.859464i	$0.670797\pi$
$224$	0	0
$225$	7.42433	0.494956
$226$	0	0
$227$	4.20523	0.279111	0.139556	−	0.990214i	$-0.455433\pi$
$227$	4.20523	0.279111	0.139556	+	0.990214i	$0.455433\pi$
$228$	0	0
$229$	9.12399	0.602930	0.301465	−	0.953477i	$-0.402524\pi$
$229$	9.12399	0.602930	0.301465	+	0.953477i	$0.402524\pi$
$230$	0	0
$231$	9.91427	0.652311
$232$	0	0
$233$	22.6986	1.48703	0.743516	−	0.668718i	$-0.233157\pi$
$233$	22.6986	1.48703	0.743516	+	0.668718i	$0.233157\pi$
$234$	0	0
$235$	−30.0365	−1.95937
$236$	0	0
$237$	−17.5773	−1.14177
$238$	0	0
$239$	2.94682	0.190614	0.0953070	−	0.995448i	$-0.469617\pi$
$239$	2.94682	0.190614	0.0953070	+	0.995448i	$0.469617\pi$
$240$	0	0
$241$	8.41632	0.542143	0.271071	−	0.962559i	$-0.412622\pi$
$241$	8.41632	0.542143	0.271071	+	0.962559i	$0.412622\pi$
$242$	0	0
$243$	22.3423	1.43326
$244$	0	0
$245$	−2.69931	−0.172452
$246$	0	0
$247$	1.84010	0.117083
$248$	0	0
$249$	18.0240	1.14222
$250$	0	0
$251$	−8.44525	−0.533059	−0.266530	−	0.963827i	$-0.585877\pi$
$251$	−8.44525	−0.533059	−0.266530	+	0.963827i	$0.585877\pi$
$252$	0	0
$253$	−1.40281	−0.0881942
$254$	0	0
$255$	−6.74685	−0.422504
$256$	0	0
$257$	−7.91427	−0.493679	−0.246839	−	0.969056i	$-0.579392\pi$
$257$	−7.91427	−0.493679	−0.246839	+	0.969056i	$0.579392\pi$
$258$	0	0
$259$	−5.51461	−0.342661
$260$	0	0
$261$	−29.8509	−1.84772
$262$	0	0
$263$	−26.1843	−1.61459	−0.807296	−	0.590146i	$-0.799070\pi$
$263$	−26.1843	−1.61459	−0.807296	+	0.590146i	$0.799070\pi$
$264$	0	0
$265$	13.8698	0.852016
$266$	0	0
$267$	36.9017	2.25835
$268$	0	0
$269$	2.15990	0.131691	0.0658457	−	0.997830i	$-0.479025\pi$
$269$	2.15990	0.131691	0.0658457	+	0.997830i	$0.479025\pi$
$270$	0	0
$271$	7.44630	0.452330	0.226165	−	0.974089i	$-0.427381\pi$
$271$	7.44630	0.452330	0.226165	+	0.974089i	$0.427381\pi$
$272$	0	0
$273$	−0.967593	−0.0585614
$274$	0	0
$275$	9.06854	0.546853
$276$	0	0
$277$	−18.2125	−1.09428	−0.547141	−	0.837040i	$-0.684284\pi$
$277$	−18.2125	−1.09428	−0.547141	+	0.837040i	$0.684284\pi$
$278$	0	0
$279$	12.9544	0.775562
$280$	0	0
$281$	−27.4473	−1.63737	−0.818686	−	0.574242i	$-0.805297\pi$
$281$	−27.4473	−1.63737	−0.818686	+	0.574242i	$0.805297\pi$
$282$	0	0
$283$	19.7003	1.17106	0.585532	−	0.810650i	$-0.300886\pi$
$283$	19.7003	1.17106	0.585532	+	0.810650i	$0.300886\pi$
$284$	0	0
$285$	−32.0700	−1.89966
$286$	0	0
$287$	8.86568	0.523325
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−18.3507	−1.07574
$292$	0	0
$293$	22.0000	1.28525	0.642626	−	0.766180i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642626	+	0.766180i	$0.277845\pi$
$294$	0	0
$295$	26.7052	1.55484
$296$	0	0
$297$	−2.45254	−0.142311
$298$	0	0
$299$	0.136909	0.00791765
$300$	0	0
$301$	10.8134	0.623275
$302$	0	0
$303$	−19.6682	−1.12991
$304$	0	0
$305$	26.4527	1.51468
$306$	0	0
$307$	−3.87424	−0.221114	−0.110557	−	0.993870i	$-0.535264\pi$
$307$	−3.87424	−0.221114	−0.110557	+	0.993870i	$0.535264\pi$
$308$	0	0
$309$	17.7627	1.01049
$310$	0	0
$311$	−24.3558	−1.38109	−0.690545	−	0.723289i	$-0.742629\pi$
$311$	−24.3558	−1.38109	−0.690545	+	0.723289i	$0.742629\pi$
$312$	0	0
$313$	8.39473	0.474498	0.237249	−	0.971449i	$-0.423754\pi$
$313$	8.39473	0.474498	0.237249	+	0.971449i	$0.423754\pi$
$314$	0	0
$315$	8.76566	0.493889
$316$	0	0
$317$	5.36556	0.301360	0.150680	−	0.988583i	$-0.451854\pi$
$317$	5.36556	0.301360	0.150680	+	0.988583i	$0.451854\pi$
$318$	0	0
$319$	−36.4617	−2.04146
$320$	0	0
$321$	−23.4602	−1.30942
$322$	0	0
$323$	4.75333	0.264482
$324$	0	0
$325$	−0.885053	−0.0490939
$326$	0	0
$327$	15.3426	0.848445
$328$	0	0
$329$	−11.1275	−0.613479
$330$	0	0
$331$	−1.02789	−0.0564980	−0.0282490	−	0.999601i	$-0.508993\pi$
$331$	−1.02789	−0.0564980	−0.0282490	+	0.999601i	$0.508993\pi$
$332$	0	0
$333$	17.9080	0.981353
$334$	0	0
$335$	7.07927	0.386782
$336$	0	0
$337$	−28.3557	−1.54464	−0.772318	−	0.635236i	$-0.780903\pi$
$337$	−28.3557	−1.54464	−0.772318	+	0.635236i	$0.780903\pi$
$338$	0	0
$339$	11.1833	0.607391
$340$	0	0
$341$	15.8234	0.856883
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.38610	−0.128463
$346$	0	0
$347$	−24.9874	−1.34139	−0.670697	−	0.741732i	$-0.734005\pi$
$347$	−24.9874	−1.34139	−0.670697	+	0.741732i	$0.734005\pi$
$348$	0	0
$349$	9.15191	0.489890	0.244945	−	0.969537i	$-0.421230\pi$
$349$	9.15191	0.489890	0.244945	+	0.969537i	$0.421230\pi$
$350$	0	0
$351$	0.239358	0.0127760
$352$	0	0
$353$	0.166585	0.00886642	0.00443321	−	0.999990i	$-0.498589\pi$
$353$	0.166585	0.00886642	0.00443321	+	0.999990i	$0.498589\pi$
$354$	0	0
$355$	−33.9529	−1.80203
$356$	0	0
$357$	−2.49947	−0.132286
$358$	0	0
$359$	−15.0289	−0.793194	−0.396597	−	0.917993i	$-0.629809\pi$
$359$	−15.0289	−0.793194	−0.396597	+	0.917993i	$0.629809\pi$
$360$	0	0
$361$	3.59410	0.189163
$362$	0	0
$363$	−11.8312	−0.620976
$364$	0	0
$365$	23.5813	1.23430
$366$	0	0
$367$	−14.7601	−0.770472	−0.385236	−	0.922818i	$-0.625880\pi$
$367$	−14.7601	−0.770472	−0.385236	+	0.922818i	$0.625880\pi$
$368$	0	0
$369$	−28.7902	−1.49876
$370$	0	0
$371$	5.13829	0.266767
$372$	0	0
$373$	36.8479	1.90791	0.953955	−	0.299951i	$-0.0969702\pi$
$373$	36.8479	1.90791	0.953955	+	0.299951i	$0.0969702\pi$
$374$	0	0
$375$	−18.3092	−0.945484
$376$	0	0
$377$	3.55851	0.183273
$378$	0	0
$379$	−28.8891	−1.48393	−0.741967	−	0.670436i	$-0.766107\pi$
$379$	−28.8891	−1.48393	−0.741967	+	0.670436i	$0.766107\pi$
$380$	0	0
$381$	−6.92976	−0.355022
$382$	0	0
$383$	30.4571	1.55628	0.778142	−	0.628088i	$-0.216162\pi$
$383$	30.4571	1.55628	0.778142	+	0.628088i	$0.216162\pi$
$384$	0	0
$385$	10.7069	0.545675
$386$	0	0
$387$	−35.1152	−1.78501
$388$	0	0
$389$	15.9773	0.810083	0.405042	−	0.914298i	$-0.367257\pi$
$389$	15.9773	0.810083	0.405042	+	0.914298i	$0.367257\pi$
$390$	0	0
$391$	0.353662	0.0178854
$392$	0	0
$393$	−17.2324	−0.869260
$394$	0	0
$395$	−18.9826	−0.955119
$396$	0	0
$397$	21.7662	1.09241	0.546206	−	0.837651i	$-0.316072\pi$
$397$	21.7662	1.09241	0.546206	+	0.837651i	$0.316072\pi$
$398$	0	0
$399$	−11.8808	−0.594785
$400$	0	0
$401$	4.80600	0.240000	0.120000	−	0.992774i	$-0.461711\pi$
$401$	4.80600	0.240000	0.120000	+	0.992774i	$0.461711\pi$
$402$	0	0
$403$	−1.54430	−0.0769268
$404$	0	0
$405$	22.1254	1.09942
$406$	0	0
$407$	21.8739	1.08425
$408$	0	0
$409$	11.1234	0.550019	0.275009	−	0.961442i	$-0.411319\pi$
$409$	11.1234	0.550019	0.275009	+	0.961442i	$0.411319\pi$
$410$	0	0
$411$	−54.8267	−2.70440
$412$	0	0
$413$	9.89336	0.486821
$414$	0	0
$415$	19.4650	0.955500
$416$	0	0
$417$	20.4009	0.999034
$418$	0	0
$419$	14.2304	0.695199	0.347600	−	0.937643i	$-0.386997\pi$
$419$	14.2304	0.695199	0.347600	+	0.937643i	$0.386997\pi$
$420$	0	0
$421$	14.0092	0.682768	0.341384	−	0.939924i	$-0.389104\pi$
$421$	14.0092	0.682768	0.341384	+	0.939924i	$0.389104\pi$
$422$	0	0
$423$	36.1351	1.75695
$424$	0	0
$425$	−2.28626	−0.110900
$426$	0	0
$427$	9.79982	0.474246
$428$	0	0
$429$	3.83800	0.185300
$430$	0	0
$431$	2.96523	0.142830	0.0714152	−	0.997447i	$-0.477248\pi$
$431$	2.96523	0.142830	0.0714152	+	0.997447i	$0.477248\pi$
$432$	0	0
$433$	−25.1851	−1.21032	−0.605159	−	0.796105i	$-0.706890\pi$
$433$	−25.1851	−1.21032	−0.605159	+	0.796105i	$0.706890\pi$
$434$	0	0
$435$	−62.0191	−2.97359
$436$	0	0
$437$	1.68107	0.0804165
$438$	0	0
$439$	2.59316	0.123765	0.0618824	−	0.998083i	$-0.480290\pi$
$439$	2.59316	0.123765	0.0618824	+	0.998083i	$0.480290\pi$
$440$	0	0
$441$	3.24737	0.154637
$442$	0	0
$443$	18.0389	0.857054	0.428527	−	0.903529i	$-0.359033\pi$
$443$	18.0389	0.857054	0.428527	+	0.903529i	$0.359033\pi$
$444$	0	0
$445$	39.8519	1.88916
$446$	0	0
$447$	−24.4474	−1.15632
$448$	0	0
$449$	2.35993	0.111372	0.0556859	−	0.998448i	$-0.482265\pi$
$449$	2.35993	0.111372	0.0556859	+	0.998448i	$0.482265\pi$
$450$	0	0
$451$	−35.1661	−1.65591
$452$	0	0
$453$	−41.4794	−1.94887
$454$	0	0
$455$	−1.04495	−0.0489881
$456$	0	0
$457$	−14.7983	−0.692235	−0.346118	−	0.938191i	$-0.612500\pi$
$457$	−14.7983	−0.692235	−0.346118	+	0.938191i	$0.612500\pi$
$458$	0	0
$459$	0.618307	0.0288601
$460$	0	0
$461$	−21.6568	−1.00866	−0.504328	−	0.863512i	$-0.668260\pi$
$461$	−21.6568	−1.00866	−0.504328	+	0.863512i	$0.668260\pi$
$462$	0	0
$463$	−24.8480	−1.15478	−0.577392	−	0.816467i	$-0.695930\pi$
$463$	−24.8480	−1.15478	−0.577392	+	0.816467i	$0.695930\pi$
$464$	0	0
$465$	26.9146	1.24813
$466$	0	0
$467$	15.9147	0.736444	0.368222	−	0.929738i	$-0.379967\pi$
$467$	15.9147	0.736444	0.368222	+	0.929738i	$0.379967\pi$
$468$	0	0
$469$	2.62263	0.121102
$470$	0	0
$471$	16.1281	0.743145
$472$	0	0
$473$	−42.8919	−1.97217
$474$	0	0
$475$	−10.8673	−0.498627
$476$	0	0
$477$	−16.6859	−0.763997
$478$	0	0
$479$	21.9729	1.00397	0.501985	−	0.864877i	$-0.332603\pi$
$479$	21.9729	1.00397	0.501985	+	0.864877i	$0.332603\pi$
$480$	0	0
$481$	−2.13481	−0.0973389
$482$	0	0
$483$	−0.883969	−0.0402220
$484$	0	0
$485$	−19.8178	−0.899881
$486$	0	0
$487$	−20.0473	−0.908428	−0.454214	−	0.890893i	$-0.650080\pi$
$487$	−20.0473	−0.908428	−0.454214	+	0.890893i	$0.650080\pi$
$488$	0	0
$489$	−55.0850	−2.49103
$490$	0	0
$491$	22.9554	1.03596	0.517981	−	0.855392i	$-0.326684\pi$
$491$	22.9554	1.03596	0.517981	+	0.855392i	$0.326684\pi$
$492$	0	0
$493$	9.19231	0.414001
$494$	0	0
$495$	−34.7694	−1.56277
$496$	0	0
$497$	−12.5784	−0.564217
$498$	0	0
$499$	11.9586	0.535340	0.267670	−	0.963511i	$-0.413746\pi$
$499$	11.9586	0.535340	0.267670	+	0.963511i	$0.413746\pi$
$500$	0	0
$501$	−14.2344	−0.635947
$502$	0	0
$503$	33.2164	1.48104	0.740522	−	0.672032i	$-0.234578\pi$
$503$	33.2164	1.48104	0.740522	+	0.672032i	$0.234578\pi$
$504$	0	0
$505$	−21.2407	−0.945199
$506$	0	0
$507$	32.1186	1.42644
$508$	0	0
$509$	5.92264	0.262516	0.131258	−	0.991348i	$-0.458098\pi$
$509$	5.92264	0.262516	0.131258	+	0.991348i	$0.458098\pi$
$510$	0	0
$511$	8.73606	0.386460
$512$	0	0
$513$	2.93901	0.129761
$514$	0	0
$515$	19.1828	0.845297
$516$	0	0
$517$	44.1377	1.94117
$518$	0	0
$519$	3.72961	0.163712
$520$	0	0
$521$	43.1459	1.89025	0.945127	−	0.326702i	$-0.105937\pi$
$521$	43.1459	1.89025	0.945127	+	0.326702i	$0.105937\pi$
$522$	0	0
$523$	39.0197	1.70621	0.853107	−	0.521736i	$-0.174715\pi$
$523$	39.0197	1.70621	0.853107	+	0.521736i	$0.174715\pi$
$524$	0	0
$525$	5.71444	0.249399
$526$	0	0
$527$	−3.98921	−0.173773
$528$	0	0
$529$	−22.8749	−0.994562
$530$	0	0
$531$	−32.1275	−1.39421
$532$	0	0
$533$	3.43207	0.148659
$534$	0	0
$535$	−25.3359	−1.09537
$536$	0	0
$537$	1.97564	0.0852553
$538$	0	0
$539$	3.96654	0.170851
$540$	0	0
$541$	−23.5658	−1.01317	−0.506586	−	0.862190i	$-0.669093\pi$
$541$	−23.5658	−1.01317	−0.506586	+	0.862190i	$0.669093\pi$
$542$	0	0
$543$	28.5986	1.22728
$544$	0	0
$545$	16.5692	0.709746
$546$	0	0
$547$	−27.0101	−1.15487	−0.577434	−	0.816438i	$-0.695946\pi$
$547$	−27.0101	−1.15487	−0.577434	+	0.816438i	$0.695946\pi$
$548$	0	0
$549$	−31.8237	−1.35820
$550$	0	0
$551$	43.6940	1.86143
$552$	0	0
$553$	−7.03241	−0.299048
$554$	0	0
$555$	37.2063	1.57932
$556$	0	0
$557$	44.1385	1.87021	0.935105	−	0.354370i	$-0.115305\pi$
$557$	44.1385	1.87021	0.935105	+	0.354370i	$0.115305\pi$
$558$	0	0
$559$	4.18607	0.177052
$560$	0	0
$561$	9.91427	0.418581
$562$	0	0
$563$	−2.31430	−0.0975361	−0.0487681	−	0.998810i	$-0.515530\pi$
$563$	−2.31430	−0.0975361	−0.0487681	+	0.998810i	$0.515530\pi$
$564$	0	0
$565$	12.0774	0.508098
$566$	0	0
$567$	8.19668	0.344228
$568$	0	0
$569$	−7.49662	−0.314275	−0.157137	−	0.987577i	$-0.550227\pi$
$569$	−7.49662	−0.314275	−0.157137	+	0.987577i	$0.550227\pi$
$570$	0	0
$571$	−38.8032	−1.62386	−0.811932	−	0.583752i	$-0.801584\pi$
$571$	−38.8032	−1.62386	−0.811932	+	0.583752i	$0.801584\pi$
$572$	0	0
$573$	−9.75936	−0.407703
$574$	0	0
$575$	−0.808562	−0.0337194
$576$	0	0
$577$	−5.38257	−0.224079	−0.112040	−	0.993704i	$-0.535738\pi$
$577$	−5.38257	−0.224079	−0.112040	+	0.993704i	$0.535738\pi$
$578$	0	0
$579$	−53.5567	−2.22574
$580$	0	0
$581$	7.21112	0.299168
$582$	0	0
$583$	−20.3812	−0.844105
$584$	0	0
$585$	3.39335	0.140298
$586$	0	0
$587$	8.92647	0.368435	0.184217	−	0.982886i	$-0.441025\pi$
$587$	8.92647	0.368435	0.184217	+	0.982886i	$0.441025\pi$
$588$	0	0
$589$	−18.9620	−0.781315
$590$	0	0
$591$	−49.5244	−2.03716
$592$	0	0
$593$	36.1707	1.48535	0.742675	−	0.669651i	$-0.233557\pi$
$593$	36.1707	1.48535	0.742675	+	0.669651i	$0.233557\pi$
$594$	0	0
$595$	−2.69931	−0.110661
$596$	0	0
$597$	−20.9526	−0.857533
$598$	0	0
$599$	−19.7708	−0.807814	−0.403907	−	0.914800i	$-0.632348\pi$
$599$	−19.7708	−0.807814	−0.403907	+	0.914800i	$0.632348\pi$
$600$	0	0
$601$	−37.1676	−1.51610	−0.758049	−	0.652198i	$-0.773847\pi$
$601$	−37.1676	−1.51610	−0.758049	+	0.652198i	$0.773847\pi$
$602$	0	0
$603$	−8.51665	−0.346825
$604$	0	0
$605$	−12.7771	−0.519462
$606$	0	0
$607$	21.5923	0.876404	0.438202	−	0.898876i	$-0.355615\pi$
$607$	21.5923	0.876404	0.438202	+	0.898876i	$0.355615\pi$
$608$	0	0
$609$	−22.9759	−0.931032
$610$	0	0
$611$	−4.30766	−0.174269
$612$	0	0
$613$	5.06652	0.204635	0.102317	−	0.994752i	$-0.467374\pi$
$613$	5.06652	0.204635	0.102317	+	0.994752i	$0.467374\pi$
$614$	0	0
$615$	−59.8154	−2.41199
$616$	0	0
$617$	45.0672	1.81434	0.907168	−	0.420769i	$-0.138240\pi$
$617$	45.0672	1.81434	0.907168	+	0.420769i	$0.138240\pi$
$618$	0	0
$619$	−11.5442	−0.463999	−0.231999	−	0.972716i	$-0.574527\pi$
$619$	−11.5442	−0.463999	−0.231999	+	0.972716i	$0.574527\pi$
$620$	0	0
$621$	0.218672	0.00877499
$622$	0	0
$623$	14.7638	0.591498
$624$	0	0
$625$	−31.2043	−1.24817
$626$	0	0
$627$	47.1258	1.88202
$628$	0	0
$629$	−5.51461	−0.219882
$630$	0	0
$631$	6.73416	0.268083	0.134041	−	0.990976i	$-0.457204\pi$
$631$	6.73416	0.268083	0.134041	+	0.990976i	$0.457204\pi$
$632$	0	0
$633$	−43.0536	−1.71123
$634$	0	0
$635$	−7.48379	−0.296985
$636$	0	0
$637$	−0.387119	−0.0153382
$638$	0	0
$639$	40.8467	1.61587
$640$	0	0
$641$	−11.6749	−0.461131	−0.230565	−	0.973057i	$-0.574058\pi$
$641$	−11.6749	−0.461131	−0.230565	+	0.973057i	$0.574058\pi$
$642$	0	0
$643$	−5.78351	−0.228080	−0.114040	−	0.993476i	$-0.536379\pi$
$643$	−5.78351	−0.228080	−0.114040	+	0.993476i	$0.536379\pi$
$644$	0	0
$645$	−72.9565	−2.87266
$646$	0	0
$647$	0.247318	0.00972308	0.00486154	−	0.999988i	$-0.498453\pi$
$647$	0.247318	0.00972308	0.00486154	+	0.999988i	$0.498453\pi$
$648$	0	0
$649$	−39.2425	−1.54040
$650$	0	0
$651$	9.97092	0.390791
$652$	0	0
$653$	21.1488	0.827617	0.413808	−	0.910364i	$-0.364198\pi$
$653$	21.1488	0.827617	0.413808	+	0.910364i	$0.364198\pi$
$654$	0	0
$655$	−18.6101	−0.727158
$656$	0	0
$657$	−28.3692	−1.10679
$658$	0	0
$659$	10.8669	0.423313	0.211657	−	0.977344i	$-0.432114\pi$
$659$	10.8669	0.423313	0.211657	+	0.977344i	$0.432114\pi$
$660$	0	0
$661$	7.55573	0.293884	0.146942	−	0.989145i	$-0.453057\pi$
$661$	7.55573	0.293884	0.146942	+	0.989145i	$0.453057\pi$
$662$	0	0
$663$	−0.967593	−0.0375782
$664$	0	0
$665$	−12.8307	−0.497553
$666$	0	0
$667$	3.25097	0.125878
$668$	0	0
$669$	38.1609	1.47539
$670$	0	0
$671$	−38.8714	−1.50061
$672$	0	0
$673$	3.21471	0.123918	0.0619589	−	0.998079i	$-0.480265\pi$
$673$	3.21471	0.123918	0.0619589	+	0.998079i	$0.480265\pi$
$674$	0	0
$675$	−1.41361	−0.0544098
$676$	0	0
$677$	48.2707	1.85519	0.927597	−	0.373583i	$-0.121871\pi$
$677$	48.2707	1.85519	0.927597	+	0.373583i	$0.121871\pi$
$678$	0	0
$679$	−7.34182	−0.281753
$680$	0	0
$681$	−10.5109	−0.402778
$682$	0	0
$683$	29.4018	1.12503	0.562515	−	0.826787i	$-0.309834\pi$
$683$	29.4018	1.12503	0.562515	+	0.826787i	$0.309834\pi$
$684$	0	0
$685$	−59.2101	−2.26230
$686$	0	0
$687$	−22.8052	−0.870072
$688$	0	0
$689$	1.98913	0.0757797
$690$	0	0
$691$	24.8169	0.944079	0.472040	−	0.881577i	$-0.343518\pi$
$691$	24.8169	0.944079	0.472040	+	0.881577i	$0.343518\pi$
$692$	0	0
$693$	−12.8809	−0.489303
$694$	0	0
$695$	22.0319	0.835717
$696$	0	0
$697$	8.86568	0.335812
$698$	0	0
$699$	−56.7345	−2.14589
$700$	0	0
$701$	−47.0769	−1.77807	−0.889035	−	0.457840i	$-0.848623\pi$
$701$	−47.0769	−1.77807	−0.889035	+	0.457840i	$0.848623\pi$
$702$	0	0
$703$	−26.2127	−0.988632
$704$	0	0
$705$	75.0755	2.82751
$706$	0	0
$707$	−7.86895	−0.295942
$708$	0	0
$709$	−10.4576	−0.392745	−0.196373	−	0.980529i	$-0.562916\pi$
$709$	−10.4576	−0.392745	−0.196373	+	0.980529i	$0.562916\pi$
$710$	0	0
$711$	22.8369	0.856449
$712$	0	0
$713$	−1.41083	−0.0528360
$714$	0	0
$715$	4.14485	0.155008
$716$	0	0
$717$	−7.36551	−0.275070
$718$	0	0
$719$	28.3406	1.05692	0.528462	−	0.848957i	$-0.322769\pi$
$719$	28.3406	1.05692	0.528462	+	0.848957i	$0.322769\pi$
$720$	0	0
$721$	7.10658	0.264663
$722$	0	0
$723$	−21.0364	−0.782351
$724$	0	0
$725$	−21.0160	−0.780514
$726$	0	0
$727$	50.3347	1.86681	0.933406	−	0.358823i	$-0.116822\pi$
$727$	50.3347	1.86681	0.933406	+	0.358823i	$0.116822\pi$
$728$	0	0
$729$	−31.2540	−1.15756
$730$	0	0
$731$	10.8134	0.399949
$732$	0	0
$733$	−20.9397	−0.773426	−0.386713	−	0.922200i	$-0.626390\pi$
$733$	−20.9397	−0.773426	−0.386713	+	0.922200i	$0.626390\pi$
$734$	0	0
$735$	6.74685	0.248861
$736$	0	0
$737$	−10.4028	−0.383191
$738$	0	0
$739$	15.8002	0.581221	0.290610	−	0.956841i	$-0.406142\pi$
$739$	15.8002	0.581221	0.290610	+	0.956841i	$0.406142\pi$
$740$	0	0
$741$	−4.59929	−0.168959
$742$	0	0
$743$	12.8376	0.470968	0.235484	−	0.971878i	$-0.424333\pi$
$743$	12.8376	0.470968	0.235484	+	0.971878i	$0.424333\pi$
$744$	0	0
$745$	−26.4019	−0.967292
$746$	0	0
$747$	−23.4172	−0.856791
$748$	0	0
$749$	−9.38607	−0.342959
$750$	0	0
$751$	23.2906	0.849886	0.424943	−	0.905220i	$-0.360294\pi$
$751$	23.2906	0.849886	0.424943	+	0.905220i	$0.360294\pi$
$752$	0	0
$753$	21.1087	0.769243
$754$	0	0
$755$	−44.7956	−1.63028
$756$	0	0
$757$	1.14207	0.0415091	0.0207546	−	0.999785i	$-0.493393\pi$
$757$	1.14207	0.0415091	0.0207546	+	0.999785i	$0.493393\pi$
$758$	0	0
$759$	3.50630	0.127271
$760$	0	0
$761$	−20.4442	−0.741103	−0.370552	−	0.928812i	$-0.620831\pi$
$761$	−20.4442	−0.741103	−0.370552	+	0.928812i	$0.620831\pi$
$762$	0	0
$763$	6.13831	0.222222
$764$	0	0
$765$	8.76566	0.316923
$766$	0	0
$767$	3.82991	0.138290
$768$	0	0
$769$	19.4240	0.700448	0.350224	−	0.936666i	$-0.386105\pi$
$769$	19.4240	0.700448	0.350224	+	0.936666i	$0.386105\pi$
$770$	0	0
$771$	19.7815	0.712414
$772$	0	0
$773$	35.6453	1.28207	0.641037	−	0.767510i	$-0.278505\pi$
$773$	35.6453	1.28207	0.641037	+	0.767510i	$0.278505\pi$
$774$	0	0
$775$	9.12035	0.327613
$776$	0	0
$777$	13.7836	0.494485
$778$	0	0
$779$	42.1415	1.50987
$780$	0	0
$781$	49.8927	1.78530
$782$	0	0
$783$	5.68367	0.203118
$784$	0	0
$785$	17.4176	0.621660
$786$	0	0
$787$	−18.3271	−0.653290	−0.326645	−	0.945147i	$-0.605918\pi$
$787$	−18.3271	−0.653290	−0.326645	+	0.945147i	$0.605918\pi$
$788$	0	0
$789$	65.4470	2.32997
$790$	0	0
$791$	4.47424	0.159086
$792$	0	0
$793$	3.79369	0.134718
$794$	0	0
$795$	−34.6673	−1.22952
$796$	0	0
$797$	−29.4813	−1.04428	−0.522140	−	0.852860i	$-0.674866\pi$
$797$	−29.4813	−1.04428	−0.522140	+	0.852860i	$0.674866\pi$
$798$	0	0
$799$	−11.1275	−0.393662
$800$	0	0
$801$	−47.9435	−1.69400
$802$	0	0
$803$	−34.6519	−1.22284
$804$	0	0
$805$	−0.954641	−0.0336467
$806$	0	0
$807$	−5.39861	−0.190040
$808$	0	0
$809$	38.8713	1.36664	0.683320	−	0.730119i	$-0.260535\pi$
$809$	38.8713	1.36664	0.683320	+	0.730119i	$0.260535\pi$
$810$	0	0
$811$	−6.94096	−0.243730	−0.121865	−	0.992547i	$-0.538888\pi$
$811$	−6.94096	−0.243730	−0.121865	+	0.992547i	$0.538888\pi$
$812$	0	0
$813$	−18.6118	−0.652745
$814$	0	0
$815$	−59.4890	−2.08381
$816$	0	0
$817$	51.3997	1.79825
$818$	0	0
$819$	1.25712	0.0439273
$820$	0	0
$821$	−3.68396	−0.128571	−0.0642856	−	0.997932i	$-0.520477\pi$
$821$	−3.68396	−0.128571	−0.0642856	+	0.997932i	$0.520477\pi$
$822$	0	0
$823$	−27.3693	−0.954032	−0.477016	−	0.878895i	$-0.658282\pi$
$823$	−27.3693	−0.954032	−0.477016	+	0.878895i	$0.658282\pi$
$824$	0	0
$825$	−22.6666	−0.789149
$826$	0	0
$827$	−13.4756	−0.468592	−0.234296	−	0.972165i	$-0.575279\pi$
$827$	−13.4756	−0.468592	−0.234296	+	0.972165i	$0.575279\pi$
$828$	0	0
$829$	−50.0184	−1.73721	−0.868605	−	0.495505i	$-0.834983\pi$
$829$	−50.0184	−1.73721	−0.868605	+	0.495505i	$0.834983\pi$
$830$	0	0
$831$	45.5216	1.57913
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−15.3725	−0.531986
$836$	0	0
$837$	−2.46655	−0.0852566
$838$	0	0
$839$	48.4292	1.67196	0.835981	−	0.548759i	$-0.184899\pi$
$839$	48.4292	1.67196	0.835981	+	0.548759i	$0.184899\pi$
$840$	0	0
$841$	55.4985	1.91374
$842$	0	0
$843$	68.6040	2.36285
$844$	0	0
$845$	34.6865	1.19325
$846$	0	0
$847$	−4.73346	−0.162644
$848$	0	0
$849$	−49.2405	−1.68993
$850$	0	0
$851$	−1.95031	−0.0668557
$852$	0	0
$853$	3.53342	0.120982	0.0604910	−	0.998169i	$-0.480733\pi$
$853$	3.53342	0.120982	0.0604910	+	0.998169i	$0.480733\pi$
$854$	0	0
$855$	41.6660	1.42495
$856$	0	0
$857$	41.3779	1.41344	0.706721	−	0.707493i	$-0.250174\pi$
$857$	41.3779	1.41344	0.706721	+	0.707493i	$0.250174\pi$
$858$	0	0
$859$	−7.62294	−0.260091	−0.130046	−	0.991508i	$-0.541512\pi$
$859$	−7.62294	−0.260091	−0.130046	+	0.991508i	$0.541512\pi$
$860$	0	0
$861$	−22.1595	−0.755195
$862$	0	0
$863$	−41.2894	−1.40551	−0.702754	−	0.711433i	$-0.748047\pi$
$863$	−41.2894	−1.40551	−0.702754	+	0.711433i	$0.748047\pi$
$864$	0	0
$865$	4.02779	0.136949
$866$	0	0
$867$	−2.49947	−0.0848866
$868$	0	0
$869$	27.8943	0.946251
$870$	0	0
$871$	1.01527	0.0344010
$872$	0	0
$873$	23.8416	0.806917
$874$	0	0
$875$	−7.32522	−0.247638
$876$	0	0
$877$	−50.2264	−1.69603	−0.848013	−	0.529976i	$-0.822201\pi$
$877$	−50.2264	−1.69603	−0.848013	+	0.529976i	$0.822201\pi$
$878$	0	0
$879$	−54.9884	−1.85471
$880$	0	0
$881$	−45.4974	−1.53285	−0.766423	−	0.642336i	$-0.777965\pi$
$881$	−45.4974	−1.53285	−0.766423	+	0.642336i	$0.777965\pi$
$882$	0	0
$883$	−12.2921	−0.413662	−0.206831	−	0.978377i	$-0.566315\pi$
$883$	−12.2921	−0.413662	−0.206831	+	0.978377i	$0.566315\pi$
$884$	0	0
$885$	−66.7490	−2.24374
$886$	0	0
$887$	−33.7665	−1.13377	−0.566885	−	0.823797i	$-0.691852\pi$
$887$	−33.7665	−1.13377	−0.566885	+	0.823797i	$0.691852\pi$
$888$	0	0
$889$	−2.77249	−0.0929862
$890$	0	0
$891$	−32.5125	−1.08921
$892$	0	0
$893$	−52.8926	−1.76998
$894$	0	0
$895$	2.13360	0.0713182
$896$	0	0
$897$	−0.342201	−0.0114257
$898$	0	0
$899$	−36.6700	−1.22301
$900$	0	0
$901$	5.13829	0.171181
$902$	0	0
$903$	−27.0279	−0.899431
$904$	0	0
$905$	30.8850	1.02665
$906$	0	0
$907$	6.72961	0.223453	0.111727	−	0.993739i	$-0.464362\pi$
$907$	6.72961	0.223453	0.111727	+	0.993739i	$0.464362\pi$
$908$	0	0
$909$	25.5534	0.847554
$910$	0	0
$911$	−9.94245	−0.329408	−0.164704	−	0.986343i	$-0.552667\pi$
$911$	−9.94245	−0.329408	−0.164704	+	0.986343i	$0.552667\pi$
$912$	0	0
$913$	−28.6032	−0.946628
$914$	0	0
$915$	−66.1179	−2.18579
$916$	0	0
$917$	−6.89441	−0.227674
$918$	0	0
$919$	−26.1101	−0.861293	−0.430646	−	0.902521i	$-0.641714\pi$
$919$	−26.1101	−0.861293	−0.430646	+	0.902521i	$0.641714\pi$
$920$	0	0
$921$	9.68355	0.319084
$922$	0	0
$923$	−4.86932	−0.160276
$924$	0	0
$925$	12.6078	0.414543
$926$	0	0
$927$	−23.0777	−0.757972
$928$	0	0
$929$	−32.7024	−1.07293	−0.536465	−	0.843922i	$-0.680241\pi$
$929$	−32.7024	−1.07293	−0.536465	+	0.843922i	$0.680241\pi$
$930$	0	0
$931$	−4.75333	−0.155784
$932$	0	0
$933$	60.8767	1.99301
$934$	0	0
$935$	10.7069	0.350154
$936$	0	0
$937$	13.2825	0.433921	0.216961	−	0.976180i	$-0.430386\pi$
$937$	13.2825	0.433921	0.216961	+	0.976180i	$0.430386\pi$
$938$	0	0
$939$	−20.9824	−0.684735
$940$	0	0
$941$	−25.1141	−0.818696	−0.409348	−	0.912378i	$-0.634244\pi$
$941$	−25.1141	−0.818696	−0.409348	+	0.912378i	$0.634244\pi$
$942$	0	0
$943$	3.13545	0.102104
$944$	0	0
$945$	−1.66900	−0.0542926
$946$	0	0
$947$	−43.7909	−1.42301	−0.711506	−	0.702680i	$-0.751987\pi$
$947$	−43.7909	−1.42301	−0.711506	+	0.702680i	$0.751987\pi$
$948$	0	0
$949$	3.38189	0.109781
$950$	0	0
$951$	−13.4111	−0.434884
$952$	0	0
$953$	−33.5962	−1.08829	−0.544144	−	0.838992i	$-0.683145\pi$
$953$	−33.5962	−1.08829	−0.544144	+	0.838992i	$0.683145\pi$
$954$	0	0
$955$	−10.5396	−0.341054
$956$	0	0
$957$	91.1350	2.94598
$958$	0	0
$959$	−21.9353	−0.708328
$960$	0	0
$961$	−15.0862	−0.486653
$962$	0	0
$963$	30.4801	0.982207
$964$	0	0
$965$	−57.8386	−1.86189
$966$	0	0
$967$	−28.9747	−0.931763	−0.465881	−	0.884847i	$-0.654263\pi$
$967$	−28.9747	−0.931763	−0.465881	+	0.884847i	$0.654263\pi$
$968$	0	0
$969$	−11.8808	−0.381667
$970$	0	0
$971$	31.2076	1.00150	0.500750	−	0.865592i	$-0.333058\pi$
$971$	31.2076	1.00150	0.500750	+	0.865592i	$0.333058\pi$
$972$	0	0
$973$	8.16206	0.261664
$974$	0	0
$975$	2.21217	0.0708461
$976$	0	0
$977$	23.9617	0.766603	0.383301	−	0.923623i	$-0.374787\pi$
$977$	23.9617	0.766603	0.383301	+	0.923623i	$0.374787\pi$
$978$	0	0
$979$	−58.5611	−1.87162
$980$	0	0
$981$	−19.9334	−0.636424
$982$	0	0
$983$	37.4195	1.19350	0.596748	−	0.802429i	$-0.296459\pi$
$983$	37.4195	1.19350	0.596748	+	0.802429i	$0.296459\pi$
$984$	0	0
$985$	−53.4838	−1.70414
$986$	0	0
$987$	27.8129	0.885294
$988$	0	0
$989$	3.82429	0.121605
$990$	0	0
$991$	24.2658	0.770830	0.385415	−	0.922743i	$-0.374058\pi$
$991$	24.2658	0.770830	0.385415	+	0.922743i	$0.374058\pi$
$992$	0	0
$993$	2.56919	0.0815307
$994$	0	0
$995$	−22.6277	−0.717348
$996$	0	0
$997$	5.71337	0.180944	0.0904721	−	0.995899i	$-0.471162\pi$
$997$	5.71337	0.180944	0.0904721	+	0.995899i	$0.471162\pi$
$998$	0	0
$999$	−3.40972	−0.107879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.e.1.1	✓	5
4.3	odd	2		3808.2.a.f.1.5	yes	5
8.3	odd	2		7616.2.a.bs.1.1		5
8.5	even	2		7616.2.a.br.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.e.1.1	✓	5	1.1	even	1	trivial
3808.2.a.f.1.5	yes	5	4.3	odd	2
7616.2.a.br.1.5		5	8.5	even	2
7616.2.a.bs.1.1		5	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)

\(f(q)\)	\(=\)	\(q-2.49947 q^{3} -2.69931 q^{5} -1.00000 q^{7} +3.24737 q^{9} +3.96654 q^{11} -0.387119 q^{13} +6.74685 q^{15} -1.00000 q^{17} -4.75333 q^{19} +2.49947 q^{21} -0.353662 q^{23} +2.28626 q^{25} -0.618307 q^{27} -9.19231 q^{29} +3.98921 q^{31} -9.91427 q^{33} +2.69931 q^{35} +5.51461 q^{37} +0.967593 q^{39} -8.86568 q^{41} -10.8134 q^{43} -8.76566 q^{45} +11.1275 q^{47} +1.00000 q^{49} +2.49947 q^{51} -5.13829 q^{53} -10.7069 q^{55} +11.8808 q^{57} -9.89336 q^{59} -9.79982 q^{61} -3.24737 q^{63} +1.04495 q^{65} -2.62263 q^{67} +0.883969 q^{69} +12.5784 q^{71} -8.73606 q^{73} -5.71444 q^{75} -3.96654 q^{77} +7.03241 q^{79} -8.19668 q^{81} -7.21112 q^{83} +2.69931 q^{85} +22.9759 q^{87} -14.7638 q^{89} +0.387119 q^{91} -9.97092 q^{93} +12.8307 q^{95} +7.34182 q^{97} +12.8809 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{5} - 5 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 5 q^{17} - 6 q^{19} + 18 q^{23} + 5 q^{25} + 18 q^{27} - 14 q^{29} + 16 q^{31} - 26 q^{33} + 2 q^{35} + 2 q^{37} + 6 q^{39} - 10 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q - 2 * q^5 - 5 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 8 * q^15 - 5 * q^17 - 6 * q^19 + 18 * q^23 + 5 * q^25 + 18 * q^27 - 14 * q^29 + 16 * q^31 - 26 * q^33 + 2 * q^35 + 2 * q^37 + 6 * q^39 - 10 * q^41 - 20 * q^43 + 6 * q^45 + 26 * q^47 + 5 * q^49 + 4 * q^55 + 12 * q^57 + 8 * q^61 - 3 * q^63 + 2 * q^65 + 12 * q^67 + 32 * q^69 + 10 * q^71 - 14 * q^73 - 4 * q^75 + 4 * q^77 + 34 * q^79 + 33 * q^81 + 20 * q^83 + 2 * q^85 + 14 * q^87 - 4 * q^89 + 6 * q^91 + 16 * q^93 + 42 * q^95 - 12 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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