LMFDB - Embedded newform 5808.2.a.bp.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5808,2,Mod(1,5808)] 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("5808.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5808 = 2^{4} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5808.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-1,0,1,0,2,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.3771134940$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2904)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	5808.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.37228 q^{5} +3.37228 q^{7} +1.00000 q^{9} -4.37228 q^{13} -2.37228 q^{15} -1.62772 q^{17} +3.37228 q^{19} -3.37228 q^{21} -2.74456 q^{23} +0.627719 q^{25} -1.00000 q^{27} -4.37228 q^{29} +10.1168 q^{31} +8.00000 q^{35} -9.00000 q^{37} +4.37228 q^{39} +9.11684 q^{41} -4.00000 q^{43} +2.37228 q^{45} +2.74456 q^{47} +4.37228 q^{49} +1.62772 q^{51} +13.1168 q^{53} -3.37228 q^{57} +10.7446 q^{59} -6.62772 q^{61} +3.37228 q^{63} -10.3723 q^{65} +10.1168 q^{67} +2.74456 q^{69} -4.00000 q^{71} +12.1168 q^{73} -0.627719 q^{75} +8.62772 q^{79} +1.00000 q^{81} +14.7446 q^{83} -3.86141 q^{85} +4.37228 q^{87} -7.11684 q^{89} -14.7446 q^{91} -10.1168 q^{93} +8.00000 q^{95} +1.74456 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} + q^{7} + 2 q^{9} - 3 q^{13} + q^{15} - 9 q^{17} + q^{19} - q^{21} + 6 q^{23} + 7 q^{25} - 2 q^{27} - 3 q^{29} + 3 q^{31} + 16 q^{35} - 18 q^{37} + 3 q^{39} + q^{41} - 8 q^{43}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 + q^7 + 2 * q^9 - 3 * q^13 + q^15 - 9 * q^17 + q^19 - q^21 + 6 * q^23 + 7 * q^25 - 2 * q^27 - 3 * q^29 + 3 * q^31 + 16 * q^35 - 18 * q^37 + 3 * q^39 + q^41 - 8 * q^43 - q^45 - 6 * q^47 + 3 * q^49 + 9 * q^51 + 9 * q^53 - q^57 + 10 * q^59 - 19 * q^61 + q^63 - 15 * q^65 + 3 * q^67 - 6 * q^69 - 8 * q^71 + 7 * q^73 - 7 * q^75 + 23 * q^79 + 2 * q^81 + 18 * q^83 + 21 * q^85 + 3 * q^87 + 3 * q^89 - 18 * q^91 - 3 * q^93 + 16 * q^95 - 8 * q^97

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.37228	1.06092	0.530458	−	0.847711i	$-0.322020\pi$
$5$	2.37228	1.06092	0.530458	+	0.847711i	$0.322020\pi$
$6$	0	0
$7$	3.37228	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$7$	3.37228	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.37228	−1.21265	−0.606326	−	0.795216i	$-0.707357\pi$
$13$	−4.37228	−1.21265	−0.606326	+	0.795216i	$0.707357\pi$
$14$	0	0
$15$	−2.37228	−0.612520
$16$	0	0
$17$	−1.62772	−0.394780	−0.197390	−	0.980325i	$-0.563246\pi$
$17$	−1.62772	−0.394780	−0.197390	+	0.980325i	$0.563246\pi$
$18$	0	0
$19$	3.37228	0.773654	0.386827	−	0.922152i	$-0.373571\pi$
$19$	3.37228	0.773654	0.386827	+	0.922152i	$0.373571\pi$
$20$	0	0
$21$	−3.37228	−0.735892
$22$	0	0
$23$	−2.74456	−0.572281	−0.286140	−	0.958188i	$-0.592372\pi$
$23$	−2.74456	−0.572281	−0.286140	+	0.958188i	$0.592372\pi$
$24$	0	0
$25$	0.627719	0.125544
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.37228	−0.811912	−0.405956	−	0.913893i	$-0.633061\pi$
$29$	−4.37228	−0.811912	−0.405956	+	0.913893i	$0.633061\pi$
$30$	0	0
$31$	10.1168	1.81704	0.908519	−	0.417843i	$-0.137214\pi$
$31$	10.1168	1.81704	0.908519	+	0.417843i	$0.137214\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.00000	1.35225
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	0	0
$39$	4.37228	0.700125
$40$	0	0
$41$	9.11684	1.42381	0.711906	−	0.702275i	$-0.247832\pi$
$41$	9.11684	1.42381	0.711906	+	0.702275i	$0.247832\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	2.37228	0.353639
$46$	0	0
$47$	2.74456	0.400336	0.200168	−	0.979762i	$-0.435851\pi$
$47$	2.74456	0.400336	0.200168	+	0.979762i	$0.435851\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	1.62772	0.227926
$52$	0	0
$53$	13.1168	1.80174	0.900869	−	0.434092i	$-0.142931\pi$
$53$	13.1168	1.80174	0.900869	+	0.434092i	$0.142931\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.37228	−0.446670
$58$	0	0
$59$	10.7446	1.39882	0.699411	−	0.714719i	$-0.253446\pi$
$59$	10.7446	1.39882	0.699411	+	0.714719i	$0.253446\pi$
$60$	0	0
$61$	−6.62772	−0.848592	−0.424296	−	0.905523i	$-0.639478\pi$
$61$	−6.62772	−0.848592	−0.424296	+	0.905523i	$0.639478\pi$
$62$	0	0
$63$	3.37228	0.424868
$64$	0	0
$65$	−10.3723	−1.28652
$66$	0	0
$67$	10.1168	1.23597	0.617985	−	0.786190i	$-0.287949\pi$
$67$	10.1168	1.23597	0.617985	+	0.786190i	$0.287949\pi$
$68$	0	0
$69$	2.74456	0.330407
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	12.1168	1.41817	0.709085	−	0.705123i	$-0.249108\pi$
$73$	12.1168	1.41817	0.709085	+	0.705123i	$0.249108\pi$
$74$	0	0
$75$	−0.627719	−0.0724827
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.62772	0.970694	0.485347	−	0.874322i	$-0.338693\pi$
$79$	8.62772	0.970694	0.485347	+	0.874322i	$0.338693\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.7446	1.61843	0.809213	−	0.587515i	$-0.199894\pi$
$83$	14.7446	1.61843	0.809213	+	0.587515i	$0.199894\pi$
$84$	0	0
$85$	−3.86141	−0.418828
$86$	0	0
$87$	4.37228	0.468758
$88$	0	0
$89$	−7.11684	−0.754384	−0.377192	−	0.926135i	$-0.623110\pi$
$89$	−7.11684	−0.754384	−0.377192	+	0.926135i	$0.623110\pi$
$90$	0	0
$91$	−14.7446	−1.54565
$92$	0	0
$93$	−10.1168	−1.04907
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	1.74456	0.177133	0.0885667	−	0.996070i	$-0.471771\pi$
$97$	1.74456	0.177133	0.0885667	+	0.996070i	$0.471771\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	19.4891	1.93924	0.969620	−	0.244615i	$-0.0786617\pi$
$101$	19.4891	1.93924	0.969620	+	0.244615i	$0.0786617\pi$
$102$	0	0
$103$	19.3723	1.90881	0.954404	−	0.298519i	$-0.0964925\pi$
$103$	19.3723	1.90881	0.954404	+	0.298519i	$0.0964925\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	−2.74456	−0.265327	−0.132663	−	0.991161i	$-0.542353\pi$
$107$	−2.74456	−0.265327	−0.132663	+	0.991161i	$0.542353\pi$
$108$	0	0
$109$	−6.25544	−0.599162	−0.299581	−	0.954071i	$-0.596847\pi$
$109$	−6.25544	−0.599162	−0.299581	+	0.954071i	$0.596847\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	0	0
$113$	11.6277	1.09384	0.546922	−	0.837184i	$-0.315799\pi$
$113$	11.6277	1.09384	0.546922	+	0.837184i	$0.315799\pi$
$114$	0	0
$115$	−6.51087	−0.607142
$116$	0	0
$117$	−4.37228	−0.404218
$118$	0	0
$119$	−5.48913	−0.503187
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−9.11684	−0.822038
$124$	0	0
$125$	−10.3723	−0.927725
$126$	0	0
$127$	−7.37228	−0.654184	−0.327092	−	0.944992i	$-0.606069\pi$
$127$	−7.37228	−0.654184	−0.327092	+	0.944992i	$0.606069\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−20.2337	−1.76783	−0.883913	−	0.467652i	$-0.845100\pi$
$131$	−20.2337	−1.76783	−0.883913	+	0.467652i	$0.845100\pi$
$132$	0	0
$133$	11.3723	0.986102
$134$	0	0
$135$	−2.37228	−0.204173
$136$	0	0
$137$	3.25544	0.278131	0.139065	−	0.990283i	$-0.455590\pi$
$137$	3.25544	0.278131	0.139065	+	0.990283i	$0.455590\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−2.74456	−0.231134
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.3723	−0.861371
$146$	0	0
$147$	−4.37228	−0.360620
$148$	0	0
$149$	−13.6277	−1.11643	−0.558213	−	0.829698i	$-0.688513\pi$
$149$	−13.6277	−1.11643	−0.558213	+	0.829698i	$0.688513\pi$
$150$	0	0
$151$	−9.48913	−0.772214	−0.386107	−	0.922454i	$-0.626180\pi$
$151$	−9.48913	−0.772214	−0.386107	+	0.922454i	$0.626180\pi$
$152$	0	0
$153$	−1.62772	−0.131593
$154$	0	0
$155$	24.0000	1.92773
$156$	0	0
$157$	22.8614	1.82454	0.912269	−	0.409591i	$-0.134328\pi$
$157$	22.8614	1.82454	0.912269	+	0.409591i	$0.134328\pi$
$158$	0	0
$159$	−13.1168	−1.04023
$160$	0	0
$161$	−9.25544	−0.729431
$162$	0	0
$163$	5.88316	0.460804	0.230402	−	0.973095i	$-0.425996\pi$
$163$	5.88316	0.460804	0.230402	+	0.973095i	$0.425996\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.7446	1.14097	0.570484	−	0.821309i	$-0.306756\pi$
$167$	14.7446	1.14097	0.570484	+	0.821309i	$0.306756\pi$
$168$	0	0
$169$	6.11684	0.470526
$170$	0	0
$171$	3.37228	0.257885
$172$	0	0
$173$	4.74456	0.360722	0.180361	−	0.983600i	$-0.442273\pi$
$173$	4.74456	0.360722	0.180361	+	0.983600i	$0.442273\pi$
$174$	0	0
$175$	2.11684	0.160018
$176$	0	0
$177$	−10.7446	−0.807611
$178$	0	0
$179$	5.25544	0.392810	0.196405	−	0.980523i	$-0.437073\pi$
$179$	5.25544	0.392810	0.196405	+	0.980523i	$0.437073\pi$
$180$	0	0
$181$	1.74456	0.129672	0.0648362	−	0.997896i	$-0.479348\pi$
$181$	1.74456	0.129672	0.0648362	+	0.997896i	$0.479348\pi$
$182$	0	0
$183$	6.62772	0.489935
$184$	0	0
$185$	−21.3505	−1.56972
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.37228	−0.245297
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−23.9783	−1.72599	−0.862996	−	0.505211i	$-0.831415\pi$
$193$	−23.9783	−1.72599	−0.862996	+	0.505211i	$0.831415\pi$
$194$	0	0
$195$	10.3723	0.742774
$196$	0	0
$197$	−13.6277	−0.970935	−0.485467	−	0.874255i	$-0.661351\pi$
$197$	−13.6277	−0.970935	−0.485467	+	0.874255i	$0.661351\pi$
$198$	0	0
$199$	0.627719	0.0444978	0.0222489	−	0.999752i	$-0.492917\pi$
$199$	0.627719	0.0444978	0.0222489	+	0.999752i	$0.492917\pi$
$200$	0	0
$201$	−10.1168	−0.713587
$202$	0	0
$203$	−14.7446	−1.03487
$204$	0	0
$205$	21.6277	1.51055
$206$	0	0
$207$	−2.74456	−0.190760
$208$	0	0
$209$	0	0
$210$	0	0
$211$	6.11684	0.421101	0.210550	−	0.977583i	$-0.432474\pi$
$211$	6.11684	0.421101	0.210550	+	0.977583i	$0.432474\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	−9.48913	−0.647153
$216$	0	0
$217$	34.1168	2.31600
$218$	0	0
$219$	−12.1168	−0.818780
$220$	0	0
$221$	7.11684	0.478731
$222$	0	0
$223$	−9.88316	−0.661825	−0.330913	−	0.943661i	$-0.607357\pi$
$223$	−9.88316	−0.661825	−0.330913	+	0.943661i	$0.607357\pi$
$224$	0	0
$225$	0.627719	0.0418479
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−12.3723	−0.817583	−0.408792	−	0.912628i	$-0.634050\pi$
$229$	−12.3723	−0.817583	−0.408792	+	0.912628i	$0.634050\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.86141	−0.383994	−0.191997	−	0.981396i	$-0.561496\pi$
$233$	−5.86141	−0.383994	−0.191997	+	0.981396i	$0.561496\pi$
$234$	0	0
$235$	6.51087	0.424723
$236$	0	0
$237$	−8.62772	−0.560431
$238$	0	0
$239$	8.23369	0.532593	0.266296	−	0.963891i	$-0.414200\pi$
$239$	8.23369	0.532593	0.266296	+	0.963891i	$0.414200\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	10.3723	0.662661
$246$	0	0
$247$	−14.7446	−0.938174
$248$	0	0
$249$	−14.7446	−0.934399
$250$	0	0
$251$	−10.7446	−0.678191	−0.339095	−	0.940752i	$-0.610121\pi$
$251$	−10.7446	−0.678191	−0.339095	+	0.940752i	$0.610121\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	3.86141	0.241811
$256$	0	0
$257$	25.1168	1.56675	0.783373	−	0.621552i	$-0.213498\pi$
$257$	25.1168	1.56675	0.783373	+	0.621552i	$0.213498\pi$
$258$	0	0
$259$	−30.3505	−1.88589
$260$	0	0
$261$	−4.37228	−0.270637
$262$	0	0
$263$	20.2337	1.24766	0.623831	−	0.781559i	$-0.285575\pi$
$263$	20.2337	1.24766	0.623831	+	0.781559i	$0.285575\pi$
$264$	0	0
$265$	31.1168	1.91149
$266$	0	0
$267$	7.11684	0.435544
$268$	0	0
$269$	13.1168	0.799748	0.399874	−	0.916570i	$-0.369054\pi$
$269$	13.1168	0.799748	0.399874	+	0.916570i	$0.369054\pi$
$270$	0	0
$271$	9.48913	0.576423	0.288212	−	0.957567i	$-0.406939\pi$
$271$	9.48913	0.576423	0.288212	+	0.957567i	$0.406939\pi$
$272$	0	0
$273$	14.7446	0.892382
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.2337	1.39598	0.697989	−	0.716109i	$-0.254079\pi$
$277$	23.2337	1.39598	0.697989	+	0.716109i	$0.254079\pi$
$278$	0	0
$279$	10.1168	0.605680
$280$	0	0
$281$	−12.7446	−0.760277	−0.380138	−	0.924930i	$-0.624124\pi$
$281$	−12.7446	−0.760277	−0.380138	+	0.924930i	$0.624124\pi$
$282$	0	0
$283$	−15.6060	−0.927679	−0.463839	−	0.885919i	$-0.653528\pi$
$283$	−15.6060	−0.927679	−0.463839	+	0.885919i	$0.653528\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	30.7446	1.81479
$288$	0	0
$289$	−14.3505	−0.844149
$290$	0	0
$291$	−1.74456	−0.102268
$292$	0	0
$293$	−24.6060	−1.43750	−0.718748	−	0.695271i	$-0.755285\pi$
$293$	−24.6060	−1.43750	−0.718748	+	0.695271i	$0.755285\pi$
$294$	0	0
$295$	25.4891	1.48403
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	−13.4891	−0.777500
$302$	0	0
$303$	−19.4891	−1.11962
$304$	0	0
$305$	−15.7228	−0.900286
$306$	0	0
$307$	4.86141	0.277455	0.138728	−	0.990331i	$-0.455699\pi$
$307$	4.86141	0.277455	0.138728	+	0.990331i	$0.455699\pi$
$308$	0	0
$309$	−19.3723	−1.10205
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	0	0
$313$	3.62772	0.205051	0.102525	−	0.994730i	$-0.467308\pi$
$313$	3.62772	0.205051	0.102525	+	0.994730i	$0.467308\pi$
$314$	0	0
$315$	8.00000	0.450749
$316$	0	0
$317$	7.25544	0.407506	0.203753	−	0.979022i	$-0.434686\pi$
$317$	7.25544	0.407506	0.203753	+	0.979022i	$0.434686\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.74456	0.153187
$322$	0	0
$323$	−5.48913	−0.305423
$324$	0	0
$325$	−2.74456	−0.152241
$326$	0	0
$327$	6.25544	0.345926
$328$	0	0
$329$	9.25544	0.510269
$330$	0	0
$331$	−31.6060	−1.73722	−0.868611	−	0.495495i	$-0.834987\pi$
$331$	−31.6060	−1.73722	−0.868611	+	0.495495i	$0.834987\pi$
$332$	0	0
$333$	−9.00000	−0.493197
$334$	0	0
$335$	24.0000	1.31126
$336$	0	0
$337$	0.255437	0.0139146	0.00695728	−	0.999976i	$-0.497785\pi$
$337$	0.255437	0.0139146	0.00695728	+	0.999976i	$0.497785\pi$
$338$	0	0
$339$	−11.6277	−0.631531
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−8.86141	−0.478471
$344$	0	0
$345$	6.51087	0.350534
$346$	0	0
$347$	12.2337	0.656739	0.328369	−	0.944549i	$-0.393501\pi$
$347$	12.2337	0.656739	0.328369	+	0.944549i	$0.393501\pi$
$348$	0	0
$349$	3.00000	0.160586	0.0802932	−	0.996771i	$-0.474414\pi$
$349$	3.00000	0.160586	0.0802932	+	0.996771i	$0.474414\pi$
$350$	0	0
$351$	4.37228	0.233375
$352$	0	0
$353$	26.3723	1.40365	0.701827	−	0.712347i	$-0.252368\pi$
$353$	26.3723	1.40365	0.701827	+	0.712347i	$0.252368\pi$
$354$	0	0
$355$	−9.48913	−0.503630
$356$	0	0
$357$	5.48913	0.290515
$358$	0	0
$359$	−13.4891	−0.711929	−0.355964	−	0.934500i	$-0.615848\pi$
$359$	−13.4891	−0.711929	−0.355964	+	0.934500i	$0.615848\pi$
$360$	0	0
$361$	−7.62772	−0.401459
$362$	0	0
$363$	0	0
$364$	0	0
$365$	28.7446	1.50456
$366$	0	0
$367$	33.4891	1.74812	0.874059	−	0.485820i	$-0.161479\pi$
$367$	33.4891	1.74812	0.874059	+	0.485820i	$0.161479\pi$
$368$	0	0
$369$	9.11684	0.474604
$370$	0	0
$371$	44.2337	2.29650
$372$	0	0
$373$	−12.1168	−0.627386	−0.313693	−	0.949524i	$-0.601566\pi$
$373$	−12.1168	−0.627386	−0.313693	+	0.949524i	$0.601566\pi$
$374$	0	0
$375$	10.3723	0.535622
$376$	0	0
$377$	19.1168	0.984568
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	7.37228	0.377693
$382$	0	0
$383$	20.2337	1.03389	0.516947	−	0.856018i	$-0.327069\pi$
$383$	20.2337	1.03389	0.516947	+	0.856018i	$0.327069\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−13.8614	−0.702801	−0.351401	−	0.936225i	$-0.614294\pi$
$389$	−13.8614	−0.702801	−0.351401	+	0.936225i	$0.614294\pi$
$390$	0	0
$391$	4.46738	0.225925
$392$	0	0
$393$	20.2337	1.02065
$394$	0	0
$395$	20.4674	1.02983
$396$	0	0
$397$	28.4891	1.42983	0.714914	−	0.699212i	$-0.246466\pi$
$397$	28.4891	1.42983	0.714914	+	0.699212i	$0.246466\pi$
$398$	0	0
$399$	−11.3723	−0.569326
$400$	0	0
$401$	−20.6060	−1.02901	−0.514506	−	0.857486i	$-0.672025\pi$
$401$	−20.6060	−1.02901	−0.514506	+	0.857486i	$0.672025\pi$
$402$	0	0
$403$	−44.2337	−2.20344
$404$	0	0
$405$	2.37228	0.117880
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−23.7446	−1.17409	−0.587047	−	0.809553i	$-0.699709\pi$
$409$	−23.7446	−1.17409	−0.587047	+	0.809553i	$0.699709\pi$
$410$	0	0
$411$	−3.25544	−0.160579
$412$	0	0
$413$	36.2337	1.78294
$414$	0	0
$415$	34.9783	1.71701
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	32.2337	1.57472	0.787359	−	0.616494i	$-0.211448\pi$
$419$	32.2337	1.57472	0.787359	+	0.616494i	$0.211448\pi$
$420$	0	0
$421$	−29.8614	−1.45536	−0.727678	−	0.685919i	$-0.759400\pi$
$421$	−29.8614	−1.45536	−0.727678	+	0.685919i	$0.759400\pi$
$422$	0	0
$423$	2.74456	0.133445
$424$	0	0
$425$	−1.02175	−0.0495621
$426$	0	0
$427$	−22.3505	−1.08162
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.25544	−0.0604723	−0.0302361	−	0.999543i	$-0.509626\pi$
$431$	−1.25544	−0.0604723	−0.0302361	+	0.999543i	$0.509626\pi$
$432$	0	0
$433$	5.51087	0.264836	0.132418	−	0.991194i	$-0.457726\pi$
$433$	5.51087	0.264836	0.132418	+	0.991194i	$0.457726\pi$
$434$	0	0
$435$	10.3723	0.497313
$436$	0	0
$437$	−9.25544	−0.442748
$438$	0	0
$439$	−37.0951	−1.77045	−0.885226	−	0.465161i	$-0.845996\pi$
$439$	−37.0951	−1.77045	−0.885226	+	0.465161i	$0.845996\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	−16.8832	−0.800338
$446$	0	0
$447$	13.6277	0.644569
$448$	0	0
$449$	35.8614	1.69240	0.846202	−	0.532862i	$-0.178883\pi$
$449$	35.8614	1.69240	0.846202	+	0.532862i	$0.178883\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	9.48913	0.445838
$454$	0	0
$455$	−34.9783	−1.63981
$456$	0	0
$457$	28.0951	1.31423	0.657117	−	0.753789i	$-0.271776\pi$
$457$	28.0951	1.31423	0.657117	+	0.753789i	$0.271776\pi$
$458$	0	0
$459$	1.62772	0.0759754
$460$	0	0
$461$	−6.88316	−0.320581	−0.160290	−	0.987070i	$-0.551243\pi$
$461$	−6.88316	−0.320581	−0.160290	+	0.987070i	$0.551243\pi$
$462$	0	0
$463$	−6.51087	−0.302586	−0.151293	−	0.988489i	$-0.548344\pi$
$463$	−6.51087	−0.302586	−0.151293	+	0.988489i	$0.548344\pi$
$464$	0	0
$465$	−24.0000	−1.11297
$466$	0	0
$467$	5.48913	0.254006	0.127003	−	0.991902i	$-0.459464\pi$
$467$	5.48913	0.254006	0.127003	+	0.991902i	$0.459464\pi$
$468$	0	0
$469$	34.1168	1.57537
$470$	0	0
$471$	−22.8614	−1.05340
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.11684	0.0971275
$476$	0	0
$477$	13.1168	0.600579
$478$	0	0
$479$	1.25544	0.0573624	0.0286812	−	0.999589i	$-0.490869\pi$
$479$	1.25544	0.0573624	0.0286812	+	0.999589i	$0.490869\pi$
$480$	0	0
$481$	39.3505	1.79423
$482$	0	0
$483$	9.25544	0.421137
$484$	0	0
$485$	4.13859	0.187924
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	−5.88316	−0.266046
$490$	0	0
$491$	10.7446	0.484895	0.242448	−	0.970164i	$-0.422050\pi$
$491$	10.7446	0.484895	0.242448	+	0.970164i	$0.422050\pi$
$492$	0	0
$493$	7.11684	0.320527
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.4891	−0.605070
$498$	0	0
$499$	11.6060	0.519555	0.259777	−	0.965669i	$-0.416351\pi$
$499$	11.6060	0.519555	0.259777	+	0.965669i	$0.416351\pi$
$500$	0	0
$501$	−14.7446	−0.658738
$502$	0	0
$503$	−38.9783	−1.73795	−0.868977	−	0.494852i	$-0.835222\pi$
$503$	−38.9783	−1.73795	−0.868977	+	0.494852i	$0.835222\pi$
$504$	0	0
$505$	46.2337	2.05737
$506$	0	0
$507$	−6.11684	−0.271659
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	40.8614	1.80760
$512$	0	0
$513$	−3.37228	−0.148890
$514$	0	0
$515$	45.9565	2.02509
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−4.74456	−0.208263
$520$	0	0
$521$	39.4891	1.73005	0.865025	−	0.501729i	$-0.167302\pi$
$521$	39.4891	1.73005	0.865025	+	0.501729i	$0.167302\pi$
$522$	0	0
$523$	−20.6277	−0.901987	−0.450994	−	0.892527i	$-0.648930\pi$
$523$	−20.6277	−0.901987	−0.450994	+	0.892527i	$0.648930\pi$
$524$	0	0
$525$	−2.11684	−0.0923866
$526$	0	0
$527$	−16.4674	−0.717330
$528$	0	0
$529$	−15.4674	−0.672495
$530$	0	0
$531$	10.7446	0.466274
$532$	0	0
$533$	−39.8614	−1.72659
$534$	0	0
$535$	−6.51087	−0.281490
$536$	0	0
$537$	−5.25544	−0.226789
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	−1.74456	−0.0748663
$544$	0	0
$545$	−14.8397	−0.635661
$546$	0	0
$547$	30.9783	1.32453	0.662267	−	0.749268i	$-0.269594\pi$
$547$	30.9783	1.32453	0.662267	+	0.749268i	$0.269594\pi$
$548$	0	0
$549$	−6.62772	−0.282864
$550$	0	0
$551$	−14.7446	−0.628139
$552$	0	0
$553$	29.0951	1.23725
$554$	0	0
$555$	21.3505	0.906280
$556$	0	0
$557$	2.23369	0.0946444	0.0473222	−	0.998880i	$-0.484931\pi$
$557$	2.23369	0.0946444	0.0473222	+	0.998880i	$0.484931\pi$
$558$	0	0
$559$	17.4891	0.739711
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.9783	−0.462678	−0.231339	−	0.972873i	$-0.574311\pi$
$563$	−10.9783	−0.462678	−0.231339	+	0.972873i	$0.574311\pi$
$564$	0	0
$565$	27.5842	1.16048
$566$	0	0
$567$	3.37228	0.141623
$568$	0	0
$569$	−44.7446	−1.87579	−0.937895	−	0.346920i	$-0.887228\pi$
$569$	−44.7446	−1.87579	−0.937895	+	0.346920i	$0.887228\pi$
$570$	0	0
$571$	−15.3723	−0.643310	−0.321655	−	0.946857i	$-0.604239\pi$
$571$	−15.3723	−0.643310	−0.321655	+	0.946857i	$0.604239\pi$
$572$	0	0
$573$	4.00000	0.167102
$574$	0	0
$575$	−1.72281	−0.0718463
$576$	0	0
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	0	0
$579$	23.9783	0.996502
$580$	0	0
$581$	49.7228	2.06285
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−10.3723	−0.428841
$586$	0	0
$587$	−14.5109	−0.598928	−0.299464	−	0.954108i	$-0.596808\pi$
$587$	−14.5109	−0.598928	−0.299464	+	0.954108i	$0.596808\pi$
$588$	0	0
$589$	34.1168	1.40576
$590$	0	0
$591$	13.6277	0.560569
$592$	0	0
$593$	27.6277	1.13453	0.567267	−	0.823534i	$-0.308001\pi$
$593$	27.6277	1.13453	0.567267	+	0.823534i	$0.308001\pi$
$594$	0	0
$595$	−13.0217	−0.533840
$596$	0	0
$597$	−0.627719	−0.0256908
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	−10.4891	−0.427860	−0.213930	−	0.976849i	$-0.568627\pi$
$601$	−10.4891	−0.427860	−0.213930	+	0.976849i	$0.568627\pi$
$602$	0	0
$603$	10.1168	0.411990
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	14.7446	0.597480
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	8.48913	0.342873	0.171436	−	0.985195i	$-0.445159\pi$
$613$	8.48913	0.342873	0.171436	+	0.985195i	$0.445159\pi$
$614$	0	0
$615$	−21.6277	−0.872114
$616$	0	0
$617$	33.3505	1.34264	0.671321	−	0.741167i	$-0.265727\pi$
$617$	33.3505	1.34264	0.671321	+	0.741167i	$0.265727\pi$
$618$	0	0
$619$	13.4891	0.542174	0.271087	−	0.962555i	$-0.412617\pi$
$619$	13.4891	0.542174	0.271087	+	0.962555i	$0.412617\pi$
$620$	0	0
$621$	2.74456	0.110136
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	−27.7446	−1.10978
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.6495	0.584113
$630$	0	0
$631$	−25.4891	−1.01471	−0.507353	−	0.861738i	$-0.669376\pi$
$631$	−25.4891	−1.01471	−0.507353	+	0.861738i	$0.669376\pi$
$632$	0	0
$633$	−6.11684	−0.243123
$634$	0	0
$635$	−17.4891	−0.694035
$636$	0	0
$637$	−19.1168	−0.757437
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	19.8614	0.784478	0.392239	−	0.919863i	$-0.371701\pi$
$641$	19.8614	0.784478	0.392239	+	0.919863i	$0.371701\pi$
$642$	0	0
$643$	−49.3288	−1.94534	−0.972669	−	0.232198i	$-0.925408\pi$
$643$	−49.3288	−1.94534	−0.972669	+	0.232198i	$0.925408\pi$
$644$	0	0
$645$	9.48913	0.373634
$646$	0	0
$647$	−46.9783	−1.84691	−0.923453	−	0.383712i	$-0.874646\pi$
$647$	−46.9783	−1.84691	−0.923453	+	0.383712i	$0.874646\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−34.1168	−1.33714
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	−48.0000	−1.87552
$656$	0	0
$657$	12.1168	0.472723
$658$	0	0
$659$	−37.7228	−1.46947	−0.734736	−	0.678353i	$-0.762694\pi$
$659$	−37.7228	−1.46947	−0.734736	+	0.678353i	$0.762694\pi$
$660$	0	0
$661$	−45.4674	−1.76848	−0.884238	−	0.467037i	$-0.845322\pi$
$661$	−45.4674	−1.76848	−0.884238	+	0.467037i	$0.845322\pi$
$662$	0	0
$663$	−7.11684	−0.276395
$664$	0	0
$665$	26.9783	1.04617
$666$	0	0
$667$	12.0000	0.464642
$668$	0	0
$669$	9.88316	0.382105
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−18.6277	−0.718046	−0.359023	−	0.933329i	$-0.616890\pi$
$673$	−18.6277	−0.718046	−0.359023	+	0.933329i	$0.616890\pi$
$674$	0	0
$675$	−0.627719	−0.0241609
$676$	0	0
$677$	−3.11684	−0.119790	−0.0598950	−	0.998205i	$-0.519077\pi$
$677$	−3.11684	−0.119790	−0.0598950	+	0.998205i	$0.519077\pi$
$678$	0	0
$679$	5.88316	0.225775
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	−31.2119	−1.19429	−0.597146	−	0.802133i	$-0.703699\pi$
$683$	−31.2119	−1.19429	−0.597146	+	0.802133i	$0.703699\pi$
$684$	0	0
$685$	7.72281	0.295074
$686$	0	0
$687$	12.3723	0.472032
$688$	0	0
$689$	−57.3505	−2.18488
$690$	0	0
$691$	−19.3723	−0.736956	−0.368478	−	0.929636i	$-0.620121\pi$
$691$	−19.3723	−0.736956	−0.368478	+	0.929636i	$0.620121\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−18.9783	−0.719886
$696$	0	0
$697$	−14.8397	−0.562092
$698$	0	0
$699$	5.86141	0.221699
$700$	0	0
$701$	1.11684	0.0421826	0.0210913	−	0.999778i	$-0.493286\pi$
$701$	1.11684	0.0421826	0.0210913	+	0.999778i	$0.493286\pi$
$702$	0	0
$703$	−30.3505	−1.14469
$704$	0	0
$705$	−6.51087	−0.245214
$706$	0	0
$707$	65.7228	2.47176
$708$	0	0
$709$	−34.4674	−1.29445	−0.647225	−	0.762299i	$-0.724070\pi$
$709$	−34.4674	−1.29445	−0.647225	+	0.762299i	$0.724070\pi$
$710$	0	0
$711$	8.62772	0.323565
$712$	0	0
$713$	−27.7663	−1.03986
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.23369	−0.307493
$718$	0	0
$719$	−9.25544	−0.345170	−0.172585	−	0.984995i	$-0.555212\pi$
$719$	−9.25544	−0.345170	−0.172585	+	0.984995i	$0.555212\pi$
$720$	0	0
$721$	65.3288	2.43297
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	−2.74456	−0.101930
$726$	0	0
$727$	−24.4674	−0.907445	−0.453722	−	0.891143i	$-0.649904\pi$
$727$	−24.4674	−0.907445	−0.453722	+	0.891143i	$0.649904\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.51087	0.240813
$732$	0	0
$733$	39.6277	1.46368	0.731842	−	0.681475i	$-0.238661\pi$
$733$	39.6277	1.46368	0.731842	+	0.681475i	$0.238661\pi$
$734$	0	0
$735$	−10.3723	−0.382587
$736$	0	0
$737$	0	0
$738$	0	0
$739$	10.3505	0.380750	0.190375	−	0.981711i	$-0.439030\pi$
$739$	10.3505	0.380750	0.190375	+	0.981711i	$0.439030\pi$
$740$	0	0
$741$	14.7446	0.541655
$742$	0	0
$743$	6.51087	0.238861	0.119430	−	0.992843i	$-0.461893\pi$
$743$	6.51087	0.238861	0.119430	+	0.992843i	$0.461893\pi$
$744$	0	0
$745$	−32.3288	−1.18443
$746$	0	0
$747$	14.7446	0.539475
$748$	0	0
$749$	−9.25544	−0.338186
$750$	0	0
$751$	11.6060	0.423508	0.211754	−	0.977323i	$-0.432082\pi$
$751$	11.6060	0.423508	0.211754	+	0.977323i	$0.432082\pi$
$752$	0	0
$753$	10.7446	0.391554
$754$	0	0
$755$	−22.5109	−0.819255
$756$	0	0
$757$	−22.7228	−0.825875	−0.412937	−	0.910759i	$-0.635497\pi$
$757$	−22.7228	−0.825875	−0.412937	+	0.910759i	$0.635497\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	37.3505	1.35396	0.676978	−	0.736003i	$-0.263289\pi$
$761$	37.3505	1.35396	0.676978	+	0.736003i	$0.263289\pi$
$762$	0	0
$763$	−21.0951	−0.763694
$764$	0	0
$765$	−3.86141	−0.139609
$766$	0	0
$767$	−46.9783	−1.69629
$768$	0	0
$769$	−15.9783	−0.576190	−0.288095	−	0.957602i	$-0.593022\pi$
$769$	−15.9783	−0.576190	−0.288095	+	0.957602i	$0.593022\pi$
$770$	0	0
$771$	−25.1168	−0.904561
$772$	0	0
$773$	−43.7228	−1.57260	−0.786300	−	0.617845i	$-0.788006\pi$
$773$	−43.7228	−1.57260	−0.786300	+	0.617845i	$0.788006\pi$
$774$	0	0
$775$	6.35053	0.228118
$776$	0	0
$777$	30.3505	1.08882
$778$	0	0
$779$	30.7446	1.10154
$780$	0	0
$781$	0	0
$782$	0	0
$783$	4.37228	0.156253
$784$	0	0
$785$	54.2337	1.93568
$786$	0	0
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	0	0
$789$	−20.2337	−0.720338
$790$	0	0
$791$	39.2119	1.39422
$792$	0	0
$793$	28.9783	1.02905
$794$	0	0
$795$	−31.1168	−1.10360
$796$	0	0
$797$	11.4891	0.406966	0.203483	−	0.979079i	$-0.434774\pi$
$797$	11.4891	0.406966	0.203483	+	0.979079i	$0.434774\pi$
$798$	0	0
$799$	−4.46738	−0.158044
$800$	0	0
$801$	−7.11684	−0.251461
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−21.9565	−0.773865
$806$	0	0
$807$	−13.1168	−0.461735
$808$	0	0
$809$	43.7228	1.53721	0.768606	−	0.639722i	$-0.220951\pi$
$809$	43.7228	1.53721	0.768606	+	0.639722i	$0.220951\pi$
$810$	0	0
$811$	2.11684	0.0743324	0.0371662	−	0.999309i	$-0.488167\pi$
$811$	2.11684	0.0743324	0.0371662	+	0.999309i	$0.488167\pi$
$812$	0	0
$813$	−9.48913	−0.332798
$814$	0	0
$815$	13.9565	0.488875
$816$	0	0
$817$	−13.4891	−0.471925
$818$	0	0
$819$	−14.7446	−0.515217
$820$	0	0
$821$	8.51087	0.297032	0.148516	−	0.988910i	$-0.452550\pi$
$821$	8.51087	0.297032	0.148516	+	0.988910i	$0.452550\pi$
$822$	0	0
$823$	−0.627719	−0.0218809	−0.0109404	−	0.999940i	$-0.503483\pi$
$823$	−0.627719	−0.0218809	−0.0109404	+	0.999940i	$0.503483\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.25544	0.321843	0.160922	−	0.986967i	$-0.448553\pi$
$827$	9.25544	0.321843	0.160922	+	0.986967i	$0.448553\pi$
$828$	0	0
$829$	−45.2337	−1.57103	−0.785515	−	0.618842i	$-0.787602\pi$
$829$	−45.2337	−1.57103	−0.785515	+	0.618842i	$0.787602\pi$
$830$	0	0
$831$	−23.2337	−0.805968
$832$	0	0
$833$	−7.11684	−0.246584
$834$	0	0
$835$	34.9783	1.21047
$836$	0	0
$837$	−10.1168	−0.349689
$838$	0	0
$839$	−2.51087	−0.0866850	−0.0433425	−	0.999060i	$-0.513801\pi$
$839$	−2.51087	−0.0866850	−0.0433425	+	0.999060i	$0.513801\pi$
$840$	0	0
$841$	−9.88316	−0.340798
$842$	0	0
$843$	12.7446	0.438946
$844$	0	0
$845$	14.5109	0.499189
$846$	0	0
$847$	0	0
$848$	0	0
$849$	15.6060	0.535595
$850$	0	0
$851$	24.7011	0.846742
$852$	0	0
$853$	−19.9783	−0.684042	−0.342021	−	0.939692i	$-0.611111\pi$
$853$	−19.9783	−0.684042	−0.342021	+	0.939692i	$0.611111\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	−12.7446	−0.435346	−0.217673	−	0.976022i	$-0.569847\pi$
$857$	−12.7446	−0.435346	−0.217673	+	0.976022i	$0.569847\pi$
$858$	0	0
$859$	−17.8832	−0.610165	−0.305083	−	0.952326i	$-0.598684\pi$
$859$	−17.8832	−0.610165	−0.305083	+	0.952326i	$0.598684\pi$
$860$	0	0
$861$	−30.7446	−1.04777
$862$	0	0
$863$	9.25544	0.315059	0.157529	−	0.987514i	$-0.449647\pi$
$863$	9.25544	0.315059	0.157529	+	0.987514i	$0.449647\pi$
$864$	0	0
$865$	11.2554	0.382696
$866$	0	0
$867$	14.3505	0.487370
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−44.2337	−1.49880
$872$	0	0
$873$	1.74456	0.0590445
$874$	0	0
$875$	−34.9783	−1.18248
$876$	0	0
$877$	−33.0000	−1.11433	−0.557165	−	0.830402i	$-0.688111\pi$
$877$	−33.0000	−1.11433	−0.557165	+	0.830402i	$0.688111\pi$
$878$	0	0
$879$	24.6060	0.829939
$880$	0	0
$881$	−25.8614	−0.871293	−0.435646	−	0.900118i	$-0.643480\pi$
$881$	−25.8614	−0.871293	−0.435646	+	0.900118i	$0.643480\pi$
$882$	0	0
$883$	2.35053	0.0791017	0.0395508	−	0.999218i	$-0.487407\pi$
$883$	2.35053	0.0791017	0.0395508	+	0.999218i	$0.487407\pi$
$884$	0	0
$885$	−25.4891	−0.856808
$886$	0	0
$887$	−9.25544	−0.310767	−0.155384	−	0.987854i	$-0.549661\pi$
$887$	−9.25544	−0.310767	−0.155384	+	0.987854i	$0.549661\pi$
$888$	0	0
$889$	−24.8614	−0.833825
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.25544	0.309721
$894$	0	0
$895$	12.4674	0.416738
$896$	0	0
$897$	−12.0000	−0.400668
$898$	0	0
$899$	−44.2337	−1.47528
$900$	0	0
$901$	−21.3505	−0.711289
$902$	0	0
$903$	13.4891	0.448890
$904$	0	0
$905$	4.13859	0.137571
$906$	0	0
$907$	4.39403	0.145901	0.0729507	−	0.997336i	$-0.476758\pi$
$907$	4.39403	0.145901	0.0729507	+	0.997336i	$0.476758\pi$
$908$	0	0
$909$	19.4891	0.646413
$910$	0	0
$911$	−16.4674	−0.545589	−0.272794	−	0.962072i	$-0.587948\pi$
$911$	−16.4674	−0.545589	−0.272794	+	0.962072i	$0.587948\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	15.7228	0.519780
$916$	0	0
$917$	−68.2337	−2.25327
$918$	0	0
$919$	53.0951	1.75145	0.875723	−	0.482814i	$-0.160385\pi$
$919$	53.0951	1.75145	0.875723	+	0.482814i	$0.160385\pi$
$920$	0	0
$921$	−4.86141	−0.160189
$922$	0	0
$923$	17.4891	0.575662
$924$	0	0
$925$	−5.64947	−0.185753
$926$	0	0
$927$	19.3723	0.636269
$928$	0	0
$929$	42.6060	1.39786	0.698928	−	0.715192i	$-0.253661\pi$
$929$	42.6060	1.39786	0.698928	+	0.715192i	$0.253661\pi$
$930$	0	0
$931$	14.7446	0.483234
$932$	0	0
$933$	−28.0000	−0.916679
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.4674	1.15867	0.579334	−	0.815090i	$-0.303313\pi$
$937$	35.4674	1.15867	0.579334	+	0.815090i	$0.303313\pi$
$938$	0	0
$939$	−3.62772	−0.118386
$940$	0	0
$941$	14.1386	0.460905	0.230452	−	0.973084i	$-0.425979\pi$
$941$	14.1386	0.460905	0.230452	+	0.973084i	$0.425979\pi$
$942$	0	0
$943$	−25.0217	−0.814820
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	38.7446	1.25903	0.629515	−	0.776989i	$-0.283254\pi$
$947$	38.7446	1.25903	0.629515	+	0.776989i	$0.283254\pi$
$948$	0	0
$949$	−52.9783	−1.71975
$950$	0	0
$951$	−7.25544	−0.235274
$952$	0	0
$953$	−17.8614	−0.578588	−0.289294	−	0.957240i	$-0.593420\pi$
$953$	−17.8614	−0.578588	−0.289294	+	0.957240i	$0.593420\pi$
$954$	0	0
$955$	−9.48913	−0.307061
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.9783	0.354506
$960$	0	0
$961$	71.3505	2.30163
$962$	0	0
$963$	−2.74456	−0.0884423
$964$	0	0
$965$	−56.8832	−1.83113
$966$	0	0
$967$	−8.62772	−0.277449	−0.138724	−	0.990331i	$-0.544300\pi$
$967$	−8.62772	−0.277449	−0.138724	+	0.990331i	$0.544300\pi$
$968$	0	0
$969$	5.48913	0.176336
$970$	0	0
$971$	44.2337	1.41953	0.709763	−	0.704440i	$-0.248802\pi$
$971$	44.2337	1.41953	0.709763	+	0.704440i	$0.248802\pi$
$972$	0	0
$973$	−26.9783	−0.864883
$974$	0	0
$975$	2.74456	0.0878963
$976$	0	0
$977$	7.62772	0.244032	0.122016	−	0.992528i	$-0.461064\pi$
$977$	7.62772	0.244032	0.122016	+	0.992528i	$0.461064\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−6.25544	−0.199721
$982$	0	0
$983$	−15.7663	−0.502867	−0.251434	−	0.967875i	$-0.580902\pi$
$983$	−15.7663	−0.502867	−0.251434	+	0.967875i	$0.580902\pi$
$984$	0	0
$985$	−32.3288	−1.03008
$986$	0	0
$987$	−9.25544	−0.294604
$988$	0	0
$989$	10.9783	0.349088
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	31.6060	1.00299
$994$	0	0
$995$	1.48913	0.0472084
$996$	0	0
$997$	−22.4891	−0.712238	−0.356119	−	0.934441i	$-0.615900\pi$
$997$	−22.4891	−0.712238	−0.356119	+	0.934441i	$0.615900\pi$
$998$	0	0
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5808.2.a.bp.1.2		2
4.3	odd	2		2904.2.a.t.1.2	✓	2
11.10	odd	2		5808.2.a.bo.1.2		2
12.11	even	2		8712.2.a.bl.1.1		2
44.43	even	2		2904.2.a.u.1.2	yes	2
132.131	odd	2		8712.2.a.bq.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2904.2.a.t.1.2	✓	2	4.3	odd	2
2904.2.a.u.1.2	yes	2	44.43	even	2
5808.2.a.bo.1.2		2	11.10	odd	2
5808.2.a.bp.1.2		2	1.1	even	1	trivial
8712.2.a.bl.1.1		2	12.11	even	2
8712.2.a.bq.1.1		2	132.131	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:	\( N \)	\(=\)	\( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5808.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.37228 q^{5} +3.37228 q^{7} +1.00000 q^{9} -4.37228 q^{13} -2.37228 q^{15} -1.62772 q^{17} +3.37228 q^{19} -3.37228 q^{21} -2.74456 q^{23} +0.627719 q^{25} -1.00000 q^{27} -4.37228 q^{29} +10.1168 q^{31} +8.00000 q^{35} -9.00000 q^{37} +4.37228 q^{39} +9.11684 q^{41} -4.00000 q^{43} +2.37228 q^{45} +2.74456 q^{47} +4.37228 q^{49} +1.62772 q^{51} +13.1168 q^{53} -3.37228 q^{57} +10.7446 q^{59} -6.62772 q^{61} +3.37228 q^{63} -10.3723 q^{65} +10.1168 q^{67} +2.74456 q^{69} -4.00000 q^{71} +12.1168 q^{73} -0.627719 q^{75} +8.62772 q^{79} +1.00000 q^{81} +14.7446 q^{83} -3.86141 q^{85} +4.37228 q^{87} -7.11684 q^{89} -14.7446 q^{91} -10.1168 q^{93} +8.00000 q^{95} +1.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} + q^{7} + 2 q^{9} - 3 q^{13} + q^{15} - 9 q^{17} + q^{19} - q^{21} + 6 q^{23} + 7 q^{25} - 2 q^{27} - 3 q^{29} + 3 q^{31} + 16 q^{35} - 18 q^{37} + 3 q^{39} + q^{41} - 8 q^{43}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 + q^7 + 2 * q^9 - 3 * q^13 + q^15 - 9 * q^17 + q^19 - q^21 + 6 * q^23 + 7 * q^25 - 2 * q^27 - 3 * q^29 + 3 * q^31 + 16 * q^35 - 18 * q^37 + 3 * q^39 + q^41 - 8 * q^43 - q^45 - 6 * q^47 + 3 * q^49 + 9 * q^51 + 9 * q^53 - q^57 + 10 * q^59 - 19 * q^61 + q^63 - 15 * q^65 + 3 * q^67 - 6 * q^69 - 8 * q^71 + 7 * q^73 - 7 * q^75 + 23 * q^79 + 2 * q^81 + 18 * q^83 + 21 * q^85 + 3 * q^87 + 3 * q^89 - 18 * q^91 - 3 * q^93 + 16 * q^95 - 8 * q^97

Introduction

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