LMFDB - Embedded newform 1848.2.a.t.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1848,2,Mod(1,1848)] 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("1848.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1848, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1848.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,1,0,3,0,3,0,-3,0,3] 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:	$14.7563542935$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.961.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 10x + 8 $ x^3 - x^2 - 10*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.29707$ of defining polynomial
Character	$\chi$	$=$	1848.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.29707 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +4.87067 q^{13} +3.29707 q^{15} +0.426396 q^{17} -1.29707 q^{19} +1.00000 q^{21} +5.87067 q^{25} +1.00000 q^{27} -0.870674 q^{29} -4.16774 q^{31} -1.00000 q^{33} +3.29707 q^{35} -0.870674 q^{37} +4.87067 q^{39} +6.16774 q^{41} +4.00000 q^{43} +3.29707 q^{45} -7.89121 q^{47} +1.00000 q^{49} +0.426396 q^{51} -4.59414 q^{53} -3.29707 q^{55} -1.29707 q^{57} +2.87067 q^{59} -10.3355 q^{61} +1.00000 q^{63} +16.0590 q^{65} -0.276533 q^{67} +3.14721 q^{71} +13.0384 q^{73} +5.87067 q^{75} -1.00000 q^{77} +1.00000 q^{81} -8.16774 q^{83} +1.40586 q^{85} -0.870674 q^{87} -6.00000 q^{89} +4.87067 q^{91} -4.16774 q^{93} -4.27653 q^{95} -16.9296 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + q^{15} + 4 q^{17} + 5 q^{19} + 3 q^{21} + 6 q^{25} + 3 q^{27} + 9 q^{29} + 8 q^{31} - 3 q^{33} + q^{35} + 9 q^{37} + 3 q^{39} - 2 q^{41}+ \cdots - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + q^15 + 4 * q^17 + 5 * q^19 + 3 * q^21 + 6 * q^25 + 3 * q^27 + 9 * q^29 + 8 * q^31 - 3 * q^33 + q^35 + 9 * q^37 + 3 * q^39 - 2 * q^41 + 12 * q^43 + q^45 + 3 * q^47 + 3 * q^49 + 4 * q^51 + 4 * q^53 - q^55 + 5 * q^57 - 3 * q^59 + 10 * q^61 + 3 * q^63 + q^65 - 7 * q^67 + 4 * q^71 + 7 * q^73 + 6 * q^75 - 3 * q^77 + 3 * q^81 - 4 * q^83 + 22 * q^85 + 9 * q^87 - 18 * q^89 + 3 * q^91 + 8 * q^93 - 19 * q^95 + 8 * q^97 - 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.29707	1.47449	0.737247	−	0.675623i	$-0.236125\pi$
$5$	3.29707	1.47449	0.737247	+	0.675623i	$0.236125\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.87067	1.35088	0.675441	−	0.737414i	$-0.263953\pi$
$13$	4.87067	1.35088	0.675441	+	0.737414i	$0.263953\pi$
$14$	0	0
$15$	3.29707	0.851300
$16$	0	0
$17$	0.426396	0.103416	0.0517082	−	0.998662i	$-0.483533\pi$
$17$	0.426396	0.103416	0.0517082	+	0.998662i	$0.483533\pi$
$18$	0	0
$19$	−1.29707	−0.297568	−0.148784	−	0.988870i	$-0.547536\pi$
$19$	−1.29707	−0.297568	−0.148784	+	0.988870i	$0.547536\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	5.87067	1.17413
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.870674	−0.161680	−0.0808401	−	0.996727i	$-0.525760\pi$
$29$	−0.870674	−0.161680	−0.0808401	+	0.996727i	$0.525760\pi$
$30$	0	0
$31$	−4.16774	−0.748549	−0.374275	−	0.927318i	$-0.622108\pi$
$31$	−4.16774	−0.748549	−0.374275	+	0.927318i	$0.622108\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	3.29707	0.557307
$36$	0	0
$37$	−0.870674	−0.143138	−0.0715690	−	0.997436i	$-0.522801\pi$
$37$	−0.870674	−0.143138	−0.0715690	+	0.997436i	$0.522801\pi$
$38$	0	0
$39$	4.87067	0.779932
$40$	0	0
$41$	6.16774	0.963240	0.481620	−	0.876380i	$-0.340049\pi$
$41$	6.16774	0.963240	0.481620	+	0.876380i	$0.340049\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	3.29707	0.491498
$46$	0	0
$47$	−7.89121	−1.15105	−0.575526	−	0.817784i	$-0.695203\pi$
$47$	−7.89121	−1.15105	−0.575526	+	0.817784i	$0.695203\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.426396	0.0597074
$52$	0	0
$53$	−4.59414	−0.631054	−0.315527	−	0.948917i	$-0.602181\pi$
$53$	−4.59414	−0.631054	−0.315527	+	0.948917i	$0.602181\pi$
$54$	0	0
$55$	−3.29707	−0.444577
$56$	0	0
$57$	−1.29707	−0.171801
$58$	0	0
$59$	2.87067	0.373730	0.186865	−	0.982386i	$-0.440167\pi$
$59$	2.87067	0.373730	0.186865	+	0.982386i	$0.440167\pi$
$60$	0	0
$61$	−10.3355	−1.32332	−0.661662	−	0.749802i	$-0.730149\pi$
$61$	−10.3355	−1.32332	−0.661662	+	0.749802i	$0.730149\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	16.0590	1.99187
$66$	0	0
$67$	−0.276533	−0.0337839	−0.0168919	−	0.999857i	$-0.505377\pi$
$67$	−0.276533	−0.0337839	−0.0168919	+	0.999857i	$0.505377\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.14721	0.373505	0.186752	−	0.982407i	$-0.440204\pi$
$71$	3.14721	0.373505	0.186752	+	0.982407i	$0.440204\pi$
$72$	0	0
$73$	13.0384	1.52603	0.763016	−	0.646380i	$-0.223718\pi$
$73$	13.0384	1.52603	0.763016	+	0.646380i	$0.223718\pi$
$74$	0	0
$75$	5.87067	0.677887
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.16774	−0.896526	−0.448263	−	0.893902i	$-0.647957\pi$
$83$	−8.16774	−0.896526	−0.448263	+	0.893902i	$0.647957\pi$
$84$	0	0
$85$	1.40586	0.152487
$86$	0	0
$87$	−0.870674	−0.0933461
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	4.87067	0.510585
$92$	0	0
$93$	−4.16774	−0.432175
$94$	0	0
$95$	−4.27653	−0.438763
$96$	0	0
$97$	−16.9296	−1.71894	−0.859472	−	0.511183i	$-0.829207\pi$
$97$	−16.9296	−1.71894	−0.859472	+	0.511183i	$0.829207\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−9.02054	−0.888820	−0.444410	−	0.895824i	$-0.646587\pi$
$103$	−9.02054	−0.888820	−0.444410	+	0.895824i	$0.646587\pi$
$104$	0	0
$105$	3.29707	0.321761
$106$	0	0
$107$	−10.3176	−0.997441	−0.498720	−	0.866763i	$-0.666197\pi$
$107$	−10.3176	−0.997441	−0.498720	+	0.866763i	$0.666197\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−0.870674	−0.0826407
$112$	0	0
$113$	5.14721	0.484209	0.242104	−	0.970250i	$-0.422162\pi$
$113$	5.14721	0.484209	0.242104	+	0.970250i	$0.422162\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.87067	0.450294
$118$	0	0
$119$	0.426396	0.0390877
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.16774	0.556127
$124$	0	0
$125$	2.87067	0.256761
$126$	0	0
$127$	18.9296	1.67973	0.839867	−	0.542793i	$-0.182633\pi$
$127$	18.9296	1.67973	0.839867	+	0.542793i	$0.182633\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	2.42640	0.211995	0.105998	−	0.994366i	$-0.466196\pi$
$131$	2.42640	0.211995	0.105998	+	0.994366i	$0.466196\pi$
$132$	0	0
$133$	−1.29707	−0.112470
$134$	0	0
$135$	3.29707	0.283767
$136$	0	0
$137$	−9.14721	−0.781499	−0.390749	−	0.920497i	$-0.627784\pi$
$137$	−9.14721	−0.781499	−0.390749	+	0.920497i	$0.627784\pi$
$138$	0	0
$139$	−2.42640	−0.205804	−0.102902	−	0.994691i	$-0.532813\pi$
$139$	−2.42640	−0.205804	−0.102902	+	0.994691i	$0.532813\pi$
$140$	0	0
$141$	−7.89121	−0.664560
$142$	0	0
$143$	−4.87067	−0.407306
$144$	0	0
$145$	−2.87067	−0.238397
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	15.4648	1.26693	0.633464	−	0.773773i	$-0.281633\pi$
$149$	15.4648	1.26693	0.633464	+	0.773773i	$0.281633\pi$
$150$	0	0
$151$	−0.553066	−0.0450079	−0.0225039	−	0.999747i	$-0.507164\pi$
$151$	−0.553066	−0.0450079	−0.0225039	+	0.999747i	$0.507164\pi$
$152$	0	0
$153$	0.426396	0.0344721
$154$	0	0
$155$	−13.7413	−1.10373
$156$	0	0
$157$	15.0205	1.19877	0.599385	−	0.800461i	$-0.295412\pi$
$157$	15.0205	1.19877	0.599385	+	0.800461i	$0.295412\pi$
$158$	0	0
$159$	−4.59414	−0.364339
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.276533	−0.0216597	−0.0108299	−	0.999941i	$-0.503447\pi$
$163$	−0.276533	−0.0216597	−0.0108299	+	0.999941i	$0.503447\pi$
$164$	0	0
$165$	−3.29707	−0.256677
$166$	0	0
$167$	10.0411	0.777002	0.388501	−	0.921448i	$-0.372993\pi$
$167$	10.0411	0.777002	0.388501	+	0.921448i	$0.372993\pi$
$168$	0	0
$169$	10.7235	0.824882
$170$	0	0
$171$	−1.29707	−0.0991895
$172$	0	0
$173$	−17.7824	−1.35197	−0.675986	−	0.736914i	$-0.736282\pi$
$173$	−17.7824	−1.35197	−0.675986	+	0.736914i	$0.736282\pi$
$174$	0	0
$175$	5.87067	0.443781
$176$	0	0
$177$	2.87067	0.215773
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	4.97946	0.370121	0.185060	−	0.982727i	$-0.440752\pi$
$181$	4.97946	0.370121	0.185060	+	0.982727i	$0.440752\pi$
$182$	0	0
$183$	−10.3355	−0.764021
$184$	0	0
$185$	−2.87067	−0.211056
$186$	0	0
$187$	−0.426396	−0.0311812
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	10.9296	0.790840	0.395420	−	0.918500i	$-0.370599\pi$
$191$	10.9296	0.790840	0.395420	+	0.918500i	$0.370599\pi$
$192$	0	0
$193$	20.9296	1.50655	0.753274	−	0.657707i	$-0.228473\pi$
$193$	20.9296	1.50655	0.753274	+	0.657707i	$0.228473\pi$
$194$	0	0
$195$	16.0590	1.15001
$196$	0	0
$197$	−10.3355	−0.736373	−0.368187	−	0.929752i	$-0.620021\pi$
$197$	−10.3355	−0.736373	−0.368187	+	0.929752i	$0.620021\pi$
$198$	0	0
$199$	15.0974	1.07022	0.535112	−	0.844781i	$-0.320269\pi$
$199$	15.0974	1.07022	0.535112	+	0.844781i	$0.320269\pi$
$200$	0	0
$201$	−0.276533	−0.0195051
$202$	0	0
$203$	−0.870674	−0.0611093
$204$	0	0
$205$	20.3355	1.42029
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.29707	0.0897202
$210$	0	0
$211$	3.44693	0.237297	0.118648	−	0.992936i	$-0.462144\pi$
$211$	3.44693	0.237297	0.118648	+	0.992936i	$0.462144\pi$
$212$	0	0
$213$	3.14721	0.215643
$214$	0	0
$215$	13.1883	0.899433
$216$	0	0
$217$	−4.16774	−0.282925
$218$	0	0
$219$	13.0384	0.881055
$220$	0	0
$221$	2.07684	0.139703
$222$	0	0
$223$	−4.16774	−0.279093	−0.139546	−	0.990216i	$-0.544564\pi$
$223$	−4.16774	−0.279093	−0.139546	+	0.990216i	$0.544564\pi$
$224$	0	0
$225$	5.87067	0.391378
$226$	0	0
$227$	−26.2088	−1.73954	−0.869770	−	0.493457i	$-0.835733\pi$
$227$	−26.2088	−1.73954	−0.869770	+	0.493457i	$0.835733\pi$
$228$	0	0
$229$	16.4622	1.08785	0.543925	−	0.839134i	$-0.316938\pi$
$229$	16.4622	1.08785	0.543925	+	0.839134i	$0.316938\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−26.0179	−1.69722
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.91175	−0.576453	−0.288227	−	0.957562i	$-0.593066\pi$
$239$	−8.91175	−0.576453	−0.288227	+	0.957562i	$0.593066\pi$
$240$	0	0
$241$	28.4854	1.83490	0.917451	−	0.397848i	$-0.130243\pi$
$241$	28.4854	1.83490	0.917451	+	0.397848i	$0.130243\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.29707	0.210642
$246$	0	0
$247$	−6.31761	−0.401980
$248$	0	0
$249$	−8.16774	−0.517610
$250$	0	0
$251$	−4.57626	−0.288851	−0.144425	−	0.989516i	$-0.546133\pi$
$251$	−4.57626	−0.288851	−0.144425	+	0.989516i	$0.546133\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.40586	0.0880383
$256$	0	0
$257$	−10.2765	−0.641033	−0.320516	−	0.947243i	$-0.603856\pi$
$257$	−10.2765	−0.641033	−0.320516	+	0.947243i	$0.603856\pi$
$258$	0	0
$259$	−0.870674	−0.0541011
$260$	0	0
$261$	−0.870674	−0.0538934
$262$	0	0
$263$	7.20616	0.444351	0.222176	−	0.975007i	$-0.428684\pi$
$263$	7.20616	0.444351	0.222176	+	0.975007i	$0.428684\pi$
$264$	0	0
$265$	−15.1472	−0.930486
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	12.7619	0.778106	0.389053	−	0.921215i	$-0.372802\pi$
$269$	12.7619	0.778106	0.389053	+	0.921215i	$0.372802\pi$
$270$	0	0
$271$	−28.0590	−1.70446	−0.852230	−	0.523167i	$-0.824750\pi$
$271$	−28.0590	−1.70446	−0.852230	+	0.523167i	$0.824750\pi$
$272$	0	0
$273$	4.87067	0.294787
$274$	0	0
$275$	−5.87067	−0.354015
$276$	0	0
$277$	0.258652	0.0155409	0.00777044	−	0.999970i	$-0.497527\pi$
$277$	0.258652	0.0155409	0.00777044	+	0.999970i	$0.497527\pi$
$278$	0	0
$279$	−4.16774	−0.249516
$280$	0	0
$281$	−26.3944	−1.57456	−0.787280	−	0.616595i	$-0.788512\pi$
$281$	−26.3944	−1.57456	−0.787280	+	0.616595i	$0.788512\pi$
$282$	0	0
$283$	−6.14986	−0.365571	−0.182786	−	0.983153i	$-0.558511\pi$
$283$	−6.14986	−0.365571	−0.182786	+	0.983153i	$0.558511\pi$
$284$	0	0
$285$	−4.27653	−0.253320
$286$	0	0
$287$	6.16774	0.364070
$288$	0	0
$289$	−16.8182	−0.989305
$290$	0	0
$291$	−16.9296	−0.992433
$292$	0	0
$293$	−16.0768	−0.939219	−0.469609	−	0.882874i	$-0.655605\pi$
$293$	−16.0768	−0.939219	−0.469609	+	0.882874i	$0.655605\pi$
$294$	0	0
$295$	9.46482	0.551063
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−34.0768	−1.95123
$306$	0	0
$307$	26.7619	1.52738	0.763691	−	0.645582i	$-0.223385\pi$
$307$	26.7619	1.52738	0.763691	+	0.645582i	$0.223385\pi$
$308$	0	0
$309$	−9.02054	−0.513160
$310$	0	0
$311$	−29.0563	−1.64763	−0.823816	−	0.566858i	$-0.808159\pi$
$311$	−29.0563	−1.64763	−0.823816	+	0.566858i	$0.808159\pi$
$312$	0	0
$313$	2.33549	0.132010	0.0660048	−	0.997819i	$-0.478975\pi$
$313$	2.33549	0.132010	0.0660048	+	0.997819i	$0.478975\pi$
$314$	0	0
$315$	3.29707	0.185769
$316$	0	0
$317$	−12.5941	−0.707357	−0.353679	−	0.935367i	$-0.615069\pi$
$317$	−12.5941	−0.707357	−0.353679	+	0.935367i	$0.615069\pi$
$318$	0	0
$319$	0.870674	0.0487484
$320$	0	0
$321$	−10.3176	−0.575873
$322$	0	0
$323$	−0.553066	−0.0307734
$324$	0	0
$325$	28.5941	1.58612
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	−7.89121	−0.435057
$330$	0	0
$331$	−17.1883	−0.944753	−0.472377	−	0.881397i	$-0.656604\pi$
$331$	−17.1883	−0.944753	−0.472377	+	0.881397i	$0.656604\pi$
$332$	0	0
$333$	−0.870674	−0.0477127
$334$	0	0
$335$	−0.911749	−0.0498142
$336$	0	0
$337$	17.7824	0.968670	0.484335	−	0.874883i	$-0.339062\pi$
$337$	17.7824	0.968670	0.484335	+	0.874883i	$0.339062\pi$
$338$	0	0
$339$	5.14721	0.279558
$340$	0	0
$341$	4.16774	0.225696
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.0411	−1.61269	−0.806345	−	0.591446i	$-0.798557\pi$
$347$	−30.0411	−1.61269	−0.806345	+	0.591446i	$0.798557\pi$
$348$	0	0
$349$	22.9117	1.22644	0.613219	−	0.789913i	$-0.289874\pi$
$349$	22.9117	1.22644	0.613219	+	0.789913i	$0.289874\pi$
$350$	0	0
$351$	4.87067	0.259977
$352$	0	0
$353$	−17.7235	−0.943325	−0.471662	−	0.881779i	$-0.656346\pi$
$353$	−17.7235	−0.943325	−0.471662	+	0.881779i	$0.656346\pi$
$354$	0	0
$355$	10.3766	0.550731
$356$	0	0
$357$	0.426396	0.0225673
$358$	0	0
$359$	−13.1883	−0.696051	−0.348025	−	0.937485i	$-0.613148\pi$
$359$	−13.1883	−0.696051	−0.348025	+	0.937485i	$0.613148\pi$
$360$	0	0
$361$	−17.3176	−0.911453
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	42.9886	2.25013
$366$	0	0
$367$	−25.3560	−1.32357	−0.661787	−	0.749692i	$-0.730202\pi$
$367$	−25.3560	−1.32357	−0.661787	+	0.749692i	$0.730202\pi$
$368$	0	0
$369$	6.16774	0.321080
$370$	0	0
$371$	−4.59414	−0.238516
$372$	0	0
$373$	22.3355	1.15649	0.578244	−	0.815864i	$-0.303738\pi$
$373$	22.3355	1.15649	0.578244	+	0.815864i	$0.303738\pi$
$374$	0	0
$375$	2.87067	0.148241
$376$	0	0
$377$	−4.24077	−0.218411
$378$	0	0
$379$	−6.01788	−0.309118	−0.154559	−	0.987984i	$-0.549396\pi$
$379$	−6.01788	−0.309118	−0.154559	+	0.987984i	$0.549396\pi$
$380$	0	0
$381$	18.9296	0.969794
$382$	0	0
$383$	27.6147	1.41104	0.705522	−	0.708688i	$-0.250713\pi$
$383$	27.6147	1.41104	0.705522	+	0.708688i	$0.250713\pi$
$384$	0	0
$385$	−3.29707	−0.168034
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−34.1179	−1.72985	−0.864923	−	0.501904i	$-0.832633\pi$
$389$	−34.1179	−1.72985	−0.864923	+	0.501904i	$0.832633\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	2.42640	0.122396
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−35.9091	−1.80223	−0.901113	−	0.433585i	$-0.857249\pi$
$397$	−35.9091	−1.80223	−0.901113	+	0.433585i	$0.857249\pi$
$398$	0	0
$399$	−1.29707	−0.0649347
$400$	0	0
$401$	1.78242	0.0890100	0.0445050	−	0.999009i	$-0.485829\pi$
$401$	1.78242	0.0890100	0.0445050	+	0.999009i	$0.485829\pi$
$402$	0	0
$403$	−20.2997	−1.01120
$404$	0	0
$405$	3.29707	0.163833
$406$	0	0
$407$	0.870674	0.0431577
$408$	0	0
$409$	3.23811	0.160114	0.0800572	−	0.996790i	$-0.474490\pi$
$409$	3.23811	0.160114	0.0800572	+	0.996790i	$0.474490\pi$
$410$	0	0
$411$	−9.14721	−0.451198
$412$	0	0
$413$	2.87067	0.141257
$414$	0	0
$415$	−26.9296	−1.32192
$416$	0	0
$417$	−2.42640	−0.118821
$418$	0	0
$419$	26.6531	1.30209	0.651045	−	0.759040i	$-0.274331\pi$
$419$	26.6531	1.30209	0.651045	+	0.759040i	$0.274331\pi$
$420$	0	0
$421$	37.2062	1.81332	0.906659	−	0.421865i	$-0.138624\pi$
$421$	37.2062	1.81332	0.906659	+	0.421865i	$0.138624\pi$
$422$	0	0
$423$	−7.89121	−0.383684
$424$	0	0
$425$	2.50323	0.121425
$426$	0	0
$427$	−10.3355	−0.500169
$428$	0	0
$429$	−4.87067	−0.235158
$430$	0	0
$431$	−0.793836	−0.0382378	−0.0191189	−	0.999817i	$-0.506086\pi$
$431$	−0.793836	−0.0382378	−0.0191189	+	0.999817i	$0.506086\pi$
$432$	0	0
$433$	−26.6352	−1.28001	−0.640003	−	0.768372i	$-0.721067\pi$
$433$	−26.6352	−1.28001	−0.640003	+	0.768372i	$0.721067\pi$
$434$	0	0
$435$	−2.87067	−0.137638
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−36.7299	−1.75302	−0.876512	−	0.481380i	$-0.840136\pi$
$439$	−36.7299	−1.75302	−0.876512	+	0.481380i	$0.840136\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−33.1883	−1.57682	−0.788411	−	0.615148i	$-0.789096\pi$
$443$	−33.1883	−1.57682	−0.788411	+	0.615148i	$0.789096\pi$
$444$	0	0
$445$	−19.7824	−0.937777
$446$	0	0
$447$	15.4648	0.731461
$448$	0	0
$449$	24.0768	1.13626	0.568128	−	0.822940i	$-0.307668\pi$
$449$	24.0768	1.13626	0.568128	+	0.822940i	$0.307668\pi$
$450$	0	0
$451$	−6.16774	−0.290428
$452$	0	0
$453$	−0.553066	−0.0259853
$454$	0	0
$455$	16.0590	0.752855
$456$	0	0
$457$	14.0358	0.656565	0.328283	−	0.944580i	$-0.393530\pi$
$457$	14.0358	0.656565	0.328283	+	0.944580i	$0.393530\pi$
$458$	0	0
$459$	0.426396	0.0199025
$460$	0	0
$461$	11.7413	0.546849	0.273425	−	0.961893i	$-0.411844\pi$
$461$	11.7413	0.546849	0.273425	+	0.961893i	$0.411844\pi$
$462$	0	0
$463$	−25.4648	−1.18345	−0.591725	−	0.806140i	$-0.701553\pi$
$463$	−25.4648	−1.18345	−0.591725	+	0.806140i	$0.701553\pi$
$464$	0	0
$465$	−13.7413	−0.637240
$466$	0	0
$467$	28.9117	1.33788	0.668938	−	0.743318i	$-0.266749\pi$
$467$	28.9117	1.33788	0.668938	+	0.743318i	$0.266749\pi$
$468$	0	0
$469$	−0.276533	−0.0127691
$470$	0	0
$471$	15.0205	0.692110
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−7.61468	−0.349385
$476$	0	0
$477$	−4.59414	−0.210351
$478$	0	0
$479$	13.7413	0.627858	0.313929	−	0.949446i	$-0.398355\pi$
$479$	13.7413	0.627858	0.313929	+	0.949446i	$0.398355\pi$
$480$	0	0
$481$	−4.24077	−0.193362
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−55.8182	−2.53457
$486$	0	0
$487$	−12.8528	−0.582416	−0.291208	−	0.956660i	$-0.594057\pi$
$487$	−12.8528	−0.582416	−0.291208	+	0.956660i	$0.594057\pi$
$488$	0	0
$489$	−0.276533	−0.0125053
$490$	0	0
$491$	3.20616	0.144692	0.0723461	−	0.997380i	$-0.476951\pi$
$491$	3.20616	0.144692	0.0723461	+	0.997380i	$0.476951\pi$
$492$	0	0
$493$	−0.371252	−0.0167204
$494$	0	0
$495$	−3.29707	−0.148192
$496$	0	0
$497$	3.14721	0.141172
$498$	0	0
$499$	2.65310	0.118769	0.0593845	−	0.998235i	$-0.481086\pi$
$499$	2.65310	0.118769	0.0593845	+	0.998235i	$0.481086\pi$
$500$	0	0
$501$	10.0411	0.448602
$502$	0	0
$503$	26.0411	1.16111	0.580557	−	0.814220i	$-0.302835\pi$
$503$	26.0411	1.16111	0.580557	+	0.814220i	$0.302835\pi$
$504$	0	0
$505$	19.7824	0.880306
$506$	0	0
$507$	10.7235	0.476246
$508$	0	0
$509$	−19.9091	−0.882455	−0.441228	−	0.897395i	$-0.645457\pi$
$509$	−19.9091	−0.882455	−0.441228	+	0.897395i	$0.645457\pi$
$510$	0	0
$511$	13.0384	0.576786
$512$	0	0
$513$	−1.29707	−0.0572671
$514$	0	0
$515$	−29.7413	−1.31056
$516$	0	0
$517$	7.89121	0.347055
$518$	0	0
$519$	−17.7824	−0.780562
$520$	0	0
$521$	−28.6531	−1.25531	−0.627657	−	0.778490i	$-0.715986\pi$
$521$	−28.6531	−1.25531	−0.627657	+	0.778490i	$0.715986\pi$
$522$	0	0
$523$	−4.99734	−0.218519	−0.109259	−	0.994013i	$-0.534848\pi$
$523$	−4.99734	−0.218519	−0.109259	+	0.994013i	$0.534848\pi$
$524$	0	0
$525$	5.87067	0.256217
$526$	0	0
$527$	−1.77711	−0.0774122
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	2.87067	0.124577
$532$	0	0
$533$	30.0411	1.30122
$534$	0	0
$535$	−34.0179	−1.47072
$536$	0	0
$537$	−20.0000	−0.863064
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−14.2997	−0.614793	−0.307397	−	0.951581i	$-0.599458\pi$
$541$	−14.2997	−0.614793	−0.307397	+	0.951581i	$0.599458\pi$
$542$	0	0
$543$	4.97946	0.213689
$544$	0	0
$545$	19.7824	0.847386
$546$	0	0
$547$	41.5238	1.77543	0.887714	−	0.460395i	$-0.152292\pi$
$547$	41.5238	1.77543	0.887714	+	0.460395i	$0.152292\pi$
$548$	0	0
$549$	−10.3355	−0.441108
$550$	0	0
$551$	1.12933	0.0481109
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.87067	−0.121853
$556$	0	0
$557$	−11.6824	−0.494999	−0.247499	−	0.968888i	$-0.579609\pi$
$557$	−11.6824	−0.494999	−0.247499	+	0.968888i	$0.579609\pi$
$558$	0	0
$559$	19.4827	0.824030
$560$	0	0
$561$	−0.426396	−0.0180025
$562$	0	0
$563$	46.5801	1.96312	0.981558	−	0.191165i	$-0.0612264\pi$
$563$	46.5801	1.96312	0.981558	+	0.191165i	$0.0612264\pi$
$564$	0	0
$565$	16.9707	0.713963
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−27.1883	−1.13979	−0.569896	−	0.821717i	$-0.693017\pi$
$569$	−27.1883	−1.13979	−0.569896	+	0.821717i	$0.693017\pi$
$570$	0	0
$571$	−19.2294	−0.804724	−0.402362	−	0.915481i	$-0.631811\pi$
$571$	−19.2294	−0.804724	−0.402362	+	0.915481i	$0.631811\pi$
$572$	0	0
$573$	10.9296	0.456592
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.14721	0.214281	0.107141	−	0.994244i	$-0.465831\pi$
$577$	5.14721	0.214281	0.107141	+	0.994244i	$0.465831\pi$
$578$	0	0
$579$	20.9296	0.869806
$580$	0	0
$581$	−8.16774	−0.338855
$582$	0	0
$583$	4.59414	0.190270
$584$	0	0
$585$	16.0590	0.663956
$586$	0	0
$587$	−6.35337	−0.262232	−0.131116	−	0.991367i	$-0.541856\pi$
$587$	−6.35337	−0.262232	−0.131116	+	0.991367i	$0.541856\pi$
$588$	0	0
$589$	5.40586	0.222745
$590$	0	0
$591$	−10.3355	−0.425145
$592$	0	0
$593$	−13.3150	−0.546780	−0.273390	−	0.961903i	$-0.588145\pi$
$593$	−13.3150	−0.546780	−0.273390	+	0.961903i	$0.588145\pi$
$594$	0	0
$595$	1.40586	0.0576346
$596$	0	0
$597$	15.0974	0.617894
$598$	0	0
$599$	23.2294	0.949126	0.474563	−	0.880222i	$-0.342606\pi$
$599$	23.2294	0.949126	0.474563	+	0.880222i	$0.342606\pi$
$600$	0	0
$601$	−33.5915	−1.37023	−0.685113	−	0.728437i	$-0.740247\pi$
$601$	−33.5915	−1.37023	−0.685113	+	0.728437i	$0.740247\pi$
$602$	0	0
$603$	−0.276533	−0.0112613
$604$	0	0
$605$	3.29707	0.134045
$606$	0	0
$607$	−24.9117	−1.01114	−0.505568	−	0.862787i	$-0.668717\pi$
$607$	−24.9117	−1.01114	−0.505568	+	0.862787i	$0.668717\pi$
$608$	0	0
$609$	−0.870674	−0.0352815
$610$	0	0
$611$	−38.4355	−1.55493
$612$	0	0
$613$	27.5238	1.11167	0.555837	−	0.831291i	$-0.312398\pi$
$613$	27.5238	1.11167	0.555837	+	0.831291i	$0.312398\pi$
$614$	0	0
$615$	20.3355	0.820006
$616$	0	0
$617$	−32.3766	−1.30343	−0.651716	−	0.758463i	$-0.725950\pi$
$617$	−32.3766	−1.30343	−0.651716	+	0.758463i	$0.725950\pi$
$618$	0	0
$619$	9.74135	0.391538	0.195769	−	0.980650i	$-0.437280\pi$
$619$	9.74135	0.391538	0.195769	+	0.980650i	$0.437280\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−19.8886	−0.795542
$626$	0	0
$627$	1.29707	0.0518000
$628$	0	0
$629$	−0.371252	−0.0148028
$630$	0	0
$631$	−7.66451	−0.305119	−0.152560	−	0.988294i	$-0.548752\pi$
$631$	−7.66451	−0.305119	−0.152560	+	0.988294i	$0.548752\pi$
$632$	0	0
$633$	3.44693	0.137003
$634$	0	0
$635$	62.4123	2.47676
$636$	0	0
$637$	4.87067	0.192983
$638$	0	0
$639$	3.14721	0.124502
$640$	0	0
$641$	−40.3766	−1.59478	−0.797389	−	0.603465i	$-0.793786\pi$
$641$	−40.3766	−1.59478	−0.797389	+	0.603465i	$0.793786\pi$
$642$	0	0
$643$	32.8528	1.29559	0.647794	−	0.761816i	$-0.275692\pi$
$643$	32.8528	1.29559	0.647794	+	0.761816i	$0.275692\pi$
$644$	0	0
$645$	13.1883	0.519288
$646$	0	0
$647$	37.0795	1.45775	0.728873	−	0.684649i	$-0.240045\pi$
$647$	37.0795	1.45775	0.728873	+	0.684649i	$0.240045\pi$
$648$	0	0
$649$	−2.87067	−0.112684
$650$	0	0
$651$	−4.16774	−0.163347
$652$	0	0
$653$	6.11791	0.239412	0.119706	−	0.992809i	$-0.461805\pi$
$653$	6.11791	0.239412	0.119706	+	0.992809i	$0.461805\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	0	0
$657$	13.0384	0.508677
$658$	0	0
$659$	18.8707	0.735097	0.367549	−	0.930004i	$-0.380197\pi$
$659$	18.8707	0.735097	0.367549	+	0.930004i	$0.380197\pi$
$660$	0	0
$661$	−1.31495	−0.0511457	−0.0255729	−	0.999673i	$-0.508141\pi$
$661$	−1.31495	−0.0511457	−0.0255729	+	0.999673i	$0.508141\pi$
$662$	0	0
$663$	2.07684	0.0806577
$664$	0	0
$665$	−4.27653	−0.165837
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−4.16774	−0.161134
$670$	0	0
$671$	10.3355	0.398997
$672$	0	0
$673$	−30.4534	−1.17389	−0.586946	−	0.809626i	$-0.699670\pi$
$673$	−30.4534	−1.17389	−0.586946	+	0.809626i	$0.699670\pi$
$674$	0	0
$675$	5.87067	0.225962
$676$	0	0
$677$	15.4416	0.593470	0.296735	−	0.954960i	$-0.404102\pi$
$677$	15.4416	0.593470	0.296735	+	0.954960i	$0.404102\pi$
$678$	0	0
$679$	−16.9296	−0.649700
$680$	0	0
$681$	−26.2088	−1.00432
$682$	0	0
$683$	−9.74135	−0.372742	−0.186371	−	0.982479i	$-0.559673\pi$
$683$	−9.74135	−0.372742	−0.186371	+	0.982479i	$0.559673\pi$
$684$	0	0
$685$	−30.1590	−1.15232
$686$	0	0
$687$	16.4622	0.628071
$688$	0	0
$689$	−22.3766	−0.852479
$690$	0	0
$691$	42.0768	1.60068	0.800339	−	0.599548i	$-0.204653\pi$
$691$	42.0768	1.60068	0.800339	+	0.599548i	$0.204653\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	2.62990	0.0996147
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	0.811718	0.0306582	0.0153291	−	0.999883i	$-0.495120\pi$
$701$	0.811718	0.0306582	0.0153291	+	0.999883i	$0.495120\pi$
$702$	0	0
$703$	1.12933	0.0425933
$704$	0	0
$705$	−26.0179	−0.979890
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	2.49411	0.0936683	0.0468341	−	0.998903i	$-0.485087\pi$
$709$	2.49411	0.0936683	0.0468341	+	0.998903i	$0.485087\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−16.0590	−0.600571
$716$	0	0
$717$	−8.91175	−0.332815
$718$	0	0
$719$	46.1856	1.72243	0.861217	−	0.508238i	$-0.169703\pi$
$719$	46.1856	1.72243	0.861217	+	0.508238i	$0.169703\pi$
$720$	0	0
$721$	−9.02054	−0.335942
$722$	0	0
$723$	28.4854	1.05938
$724$	0	0
$725$	−5.11144	−0.189834
$726$	0	0
$727$	43.9502	1.63002	0.815011	−	0.579446i	$-0.196731\pi$
$727$	43.9502	1.63002	0.815011	+	0.579446i	$0.196731\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.70559	0.0630834
$732$	0	0
$733$	−48.6299	−1.79619	−0.898093	−	0.439805i	$-0.855048\pi$
$733$	−48.6299	−1.79619	−0.898093	+	0.439805i	$0.855048\pi$
$734$	0	0
$735$	3.29707	0.121614
$736$	0	0
$737$	0.276533	0.0101862
$738$	0	0
$739$	−7.14721	−0.262914	−0.131457	−	0.991322i	$-0.541966\pi$
$739$	−7.14721	−0.262914	−0.131457	+	0.991322i	$0.541966\pi$
$740$	0	0
$741$	−6.31761	−0.232083
$742$	0	0
$743$	0.240770	0.00883300	0.00441650	−	0.999990i	$-0.498594\pi$
$743$	0.240770	0.00883300	0.00441650	+	0.999990i	$0.498594\pi$
$744$	0	0
$745$	50.9886	1.86808
$746$	0	0
$747$	−8.16774	−0.298842
$748$	0	0
$749$	−10.3176	−0.376997
$750$	0	0
$751$	4.05896	0.148113	0.0740567	−	0.997254i	$-0.476405\pi$
$751$	4.05896	0.148113	0.0740567	+	0.997254i	$0.476405\pi$
$752$	0	0
$753$	−4.57626	−0.166768
$754$	0	0
$755$	−1.82350	−0.0663639
$756$	0	0
$757$	15.8003	0.574272	0.287136	−	0.957890i	$-0.407297\pi$
$757$	15.8003	0.574272	0.287136	+	0.957890i	$0.407297\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−29.6504	−1.07483	−0.537414	−	0.843319i	$-0.680599\pi$
$761$	−29.6504	−1.07483	−0.537414	+	0.843319i	$0.680599\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	0	0
$765$	1.40586	0.0508289
$766$	0	0
$767$	13.9821	0.504865
$768$	0	0
$769$	−1.03842	−0.0374463	−0.0187232	−	0.999825i	$-0.505960\pi$
$769$	−1.03842	−0.0374463	−0.0187232	+	0.999825i	$0.505960\pi$
$770$	0	0
$771$	−10.2765	−0.370100
$772$	0	0
$773$	17.9270	0.644788	0.322394	−	0.946605i	$-0.395512\pi$
$773$	17.9270	0.644788	0.322394	+	0.946605i	$0.395512\pi$
$774$	0	0
$775$	−24.4675	−0.878898
$776$	0	0
$777$	−0.870674	−0.0312353
$778$	0	0
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−3.14721	−0.112616
$782$	0	0
$783$	−0.870674	−0.0311154
$784$	0	0
$785$	49.5238	1.76758
$786$	0	0
$787$	11.7733	0.419673	0.209836	−	0.977737i	$-0.432707\pi$
$787$	11.7733	0.419673	0.209836	+	0.977737i	$0.432707\pi$
$788$	0	0
$789$	7.20616	0.256546
$790$	0	0
$791$	5.14721	0.183014
$792$	0	0
$793$	−50.3408	−1.78765
$794$	0	0
$795$	−15.1472	−0.537216
$796$	0	0
$797$	43.0795	1.52595	0.762977	−	0.646426i	$-0.223737\pi$
$797$	43.0795	1.52595	0.762977	+	0.646426i	$0.223737\pi$
$798$	0	0
$799$	−3.36478	−0.119038
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−13.0384	−0.460116
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.7619	0.449240
$808$	0	0
$809$	−2.05896	−0.0723891	−0.0361945	−	0.999345i	$-0.511524\pi$
$809$	−2.05896	−0.0723891	−0.0361945	+	0.999345i	$0.511524\pi$
$810$	0	0
$811$	37.3792	1.31256	0.656281	−	0.754517i	$-0.272129\pi$
$811$	37.3792	1.31256	0.656281	+	0.754517i	$0.272129\pi$
$812$	0	0
$813$	−28.0590	−0.984071
$814$	0	0
$815$	−0.911749	−0.0319372
$816$	0	0
$817$	−5.18828	−0.181515
$818$	0	0
$819$	4.87067	0.170195
$820$	0	0
$821$	18.3944	0.641971	0.320985	−	0.947084i	$-0.395986\pi$
$821$	18.3944	0.641971	0.320985	+	0.947084i	$0.395986\pi$
$822$	0	0
$823$	−10.9064	−0.380174	−0.190087	−	0.981767i	$-0.560877\pi$
$823$	−10.9064	−0.380174	−0.190087	+	0.981767i	$0.560877\pi$
$824$	0	0
$825$	−5.87067	−0.204391
$826$	0	0
$827$	−30.1358	−1.04792	−0.523962	−	0.851742i	$-0.675547\pi$
$827$	−30.1358	−1.04792	−0.523962	+	0.851742i	$0.675547\pi$
$828$	0	0
$829$	12.9795	0.450795	0.225398	−	0.974267i	$-0.427632\pi$
$829$	12.9795	0.450795	0.225398	+	0.974267i	$0.427632\pi$
$830$	0	0
$831$	0.258652	0.00897253
$832$	0	0
$833$	0.426396	0.0147738
$834$	0	0
$835$	33.1061	1.14568
$836$	0	0
$837$	−4.16774	−0.144058
$838$	0	0
$839$	3.25600	0.112409	0.0562047	−	0.998419i	$-0.482100\pi$
$839$	3.25600	0.112409	0.0562047	+	0.998419i	$0.482100\pi$
$840$	0	0
$841$	−28.2419	−0.973860
$842$	0	0
$843$	−26.3944	−0.909073
$844$	0	0
$845$	35.3560	1.21628
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−6.14986	−0.211063
$850$	0	0
$851$	0	0
$852$	0	0
$853$	8.47623	0.290221	0.145110	−	0.989415i	$-0.453646\pi$
$853$	8.47623	0.290221	0.145110	+	0.989415i	$0.453646\pi$
$854$	0	0
$855$	−4.27653	−0.146254
$856$	0	0
$857$	−24.5085	−0.837196	−0.418598	−	0.908172i	$-0.637478\pi$
$857$	−24.5085	−0.837196	−0.418598	+	0.908172i	$0.637478\pi$
$858$	0	0
$859$	20.3355	0.693838	0.346919	−	0.937895i	$-0.387228\pi$
$859$	20.3355	0.693838	0.346919	+	0.937895i	$0.387228\pi$
$860$	0	0
$861$	6.16774	0.210196
$862$	0	0
$863$	40.0000	1.36162	0.680808	−	0.732462i	$-0.261629\pi$
$863$	40.0000	1.36162	0.680808	+	0.732462i	$0.261629\pi$
$864$	0	0
$865$	−58.6299	−1.99348
$866$	0	0
$867$	−16.8182	−0.571176
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−1.34690	−0.0456381
$872$	0	0
$873$	−16.9296	−0.572981
$874$	0	0
$875$	2.87067	0.0970465
$876$	0	0
$877$	−19.2240	−0.649150	−0.324575	−	0.945860i	$-0.605221\pi$
$877$	−19.2240	−0.649150	−0.324575	+	0.945860i	$0.605221\pi$
$878$	0	0
$879$	−16.0768	−0.542258
$880$	0	0
$881$	−29.4237	−0.991311	−0.495655	−	0.868519i	$-0.665072\pi$
$881$	−29.4237	−0.991311	−0.495655	+	0.868519i	$0.665072\pi$
$882$	0	0
$883$	24.9475	0.839551	0.419775	−	0.907628i	$-0.362109\pi$
$883$	24.9475	0.839551	0.419775	+	0.907628i	$0.362109\pi$
$884$	0	0
$885$	9.46482	0.318156
$886$	0	0
$887$	42.9296	1.44144	0.720718	−	0.693228i	$-0.243812\pi$
$887$	42.9296	1.44144	0.720718	+	0.693228i	$0.243812\pi$
$888$	0	0
$889$	18.9296	0.634879
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	10.2355	0.342517
$894$	0	0
$895$	−65.9414	−2.20418
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.62875	0.121026
$900$	0	0
$901$	−1.95893	−0.0652613
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	16.4176	0.545741
$906$	0	0
$907$	36.6710	1.21764	0.608820	−	0.793308i	$-0.291643\pi$
$907$	36.6710	1.21764	0.608820	+	0.793308i	$0.291643\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−21.7413	−0.720323	−0.360162	−	0.932890i	$-0.617278\pi$
$911$	−21.7413	−0.720323	−0.360162	+	0.932890i	$0.617278\pi$
$912$	0	0
$913$	8.16774	0.270313
$914$	0	0
$915$	−34.0768	−1.12655
$916$	0	0
$917$	2.42640	0.0801267
$918$	0	0
$919$	28.0821	0.926345	0.463172	−	0.886268i	$-0.346711\pi$
$919$	28.0821	0.926345	0.463172	+	0.886268i	$0.346711\pi$
$920$	0	0
$921$	26.7619	0.881834
$922$	0	0
$923$	15.3290	0.504561
$924$	0	0
$925$	−5.11144	−0.168063
$926$	0	0
$927$	−9.02054	−0.296273
$928$	0	0
$929$	57.3241	1.88074	0.940371	−	0.340151i	$-0.110478\pi$
$929$	57.3241	1.88074	0.940371	+	0.340151i	$0.110478\pi$
$930$	0	0
$931$	−1.29707	−0.0425098
$932$	0	0
$933$	−29.0563	−0.951261
$934$	0	0
$935$	−1.40586	−0.0459765
$936$	0	0
$937$	38.6211	1.26170	0.630849	−	0.775906i	$-0.282707\pi$
$937$	38.6211	1.26170	0.630849	+	0.775906i	$0.282707\pi$
$938$	0	0
$939$	2.33549	0.0762158
$940$	0	0
$941$	25.7003	0.837805	0.418902	−	0.908031i	$-0.362415\pi$
$941$	25.7003	0.837805	0.418902	+	0.908031i	$0.362415\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	3.29707	0.107254
$946$	0	0
$947$	−5.15252	−0.167434	−0.0837172	−	0.996490i	$-0.526679\pi$
$947$	−5.15252	−0.167434	−0.0837172	+	0.996490i	$0.526679\pi$
$948$	0	0
$949$	63.5059	2.06149
$950$	0	0
$951$	−12.5941	−0.408393
$952$	0	0
$953$	13.9410	0.451595	0.225797	−	0.974174i	$-0.427501\pi$
$953$	13.9410	0.451595	0.225797	+	0.974174i	$0.427501\pi$
$954$	0	0
$955$	36.0358	1.16609
$956$	0	0
$957$	0.870674	0.0281449
$958$	0	0
$959$	−9.14721	−0.295379
$960$	0	0
$961$	−13.6299	−0.439674
$962$	0	0
$963$	−10.3176	−0.332480
$964$	0	0
$965$	69.0065	2.22140
$966$	0	0
$967$	−4.63522	−0.149058	−0.0745292	−	0.997219i	$-0.523745\pi$
$967$	−4.63522	−0.149058	−0.0745292	+	0.997219i	$0.523745\pi$
$968$	0	0
$969$	−0.553066	−0.0177670
$970$	0	0
$971$	23.8414	0.765106	0.382553	−	0.923933i	$-0.375045\pi$
$971$	23.8414	0.765106	0.382553	+	0.923933i	$0.375045\pi$
$972$	0	0
$973$	−2.42640	−0.0777867
$974$	0	0
$975$	28.5941	0.915745
$976$	0	0
$977$	−6.55307	−0.209651	−0.104826	−	0.994491i	$-0.533428\pi$
$977$	−6.55307	−0.209651	−0.104826	+	0.994491i	$0.533428\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	55.4329	1.76803	0.884017	−	0.467455i	$-0.154829\pi$
$983$	55.4329	1.76803	0.884017	+	0.467455i	$0.154829\pi$
$984$	0	0
$985$	−34.0768	−1.08578
$986$	0	0
$987$	−7.89121	−0.251180
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−9.46482	−0.300660	−0.150330	−	0.988636i	$-0.548034\pi$
$991$	−9.46482	−0.300660	−0.150330	+	0.988636i	$0.548034\pi$
$992$	0	0
$993$	−17.1883	−0.545454
$994$	0	0
$995$	49.7771	1.57804
$996$	0	0
$997$	18.5888	0.588714	0.294357	−	0.955696i	$-0.404895\pi$
$997$	18.5888	0.588714	0.294357	+	0.955696i	$0.404895\pi$
$998$	0	0
$999$	−0.870674	−0.0275469

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1848.2.a.t.1.3	✓	3
3.2	odd	2		5544.2.a.bl.1.1		3
4.3	odd	2		3696.2.a.bn.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.a.t.1.3	✓	3	1.1	even	1	trivial
3696.2.a.bn.1.3		3	4.3	odd	2
5544.2.a.bl.1.1		3	3.2	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:	\( N \)	\(=\)	\( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1848.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.29707 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +4.87067 q^{13} +3.29707 q^{15} +0.426396 q^{17} -1.29707 q^{19} +1.00000 q^{21} +5.87067 q^{25} +1.00000 q^{27} -0.870674 q^{29} -4.16774 q^{31} -1.00000 q^{33} +3.29707 q^{35} -0.870674 q^{37} +4.87067 q^{39} +6.16774 q^{41} +4.00000 q^{43} +3.29707 q^{45} -7.89121 q^{47} +1.00000 q^{49} +0.426396 q^{51} -4.59414 q^{53} -3.29707 q^{55} -1.29707 q^{57} +2.87067 q^{59} -10.3355 q^{61} +1.00000 q^{63} +16.0590 q^{65} -0.276533 q^{67} +3.14721 q^{71} +13.0384 q^{73} +5.87067 q^{75} -1.00000 q^{77} +1.00000 q^{81} -8.16774 q^{83} +1.40586 q^{85} -0.870674 q^{87} -6.00000 q^{89} +4.87067 q^{91} -4.16774 q^{93} -4.27653 q^{95} -16.9296 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + q^{15} + 4 q^{17} + 5 q^{19} + 3 q^{21} + 6 q^{25} + 3 q^{27} + 9 q^{29} + 8 q^{31} - 3 q^{33} + q^{35} + 9 q^{37} + 3 q^{39} - 2 q^{41}+ \cdots - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + q^15 + 4 * q^17 + 5 * q^19 + 3 * q^21 + 6 * q^25 + 3 * q^27 + 9 * q^29 + 8 * q^31 - 3 * q^33 + q^35 + 9 * q^37 + 3 * q^39 - 2 * q^41 + 12 * q^43 + q^45 + 3 * q^47 + 3 * q^49 + 4 * q^51 + 4 * q^53 - q^55 + 5 * q^57 - 3 * q^59 + 10 * q^61 + 3 * q^63 + q^65 - 7 * q^67 + 4 * q^71 + 7 * q^73 + 6 * q^75 - 3 * q^77 + 3 * q^81 - 4 * q^83 + 22 * q^85 + 9 * q^87 - 18 * q^89 + 3 * q^91 + 8 * q^93 - 19 * q^95 + 8 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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