LMFDB - Embedded newform 4864.2.a.bt.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4864,2,Mod(1,4864)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4864, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("4864.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0,4,0,0,0,0,0,14,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 23x^{8} + 44x^{7} + 167x^{6} - 266x^{5} - 491x^{4} + 460x^{3} + 546x^{2} + 56x - 8 $ x^10 - 2x^9 - 23x^8 + 44x^7 + 167x^6 - 266x^5 - 491x^4 + 460x^3 + 546x^2 + 56*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$3.22490$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.30364 q^{3} -2.55295 q^{5} -1.32515 q^{7} +7.91406 q^{9} +2.51756 q^{11} +1.22781 q^{13} -8.43404 q^{15} +0.210784 q^{17} +1.00000 q^{19} -4.37781 q^{21} +8.11631 q^{23} +1.51756 q^{25} +16.2343 q^{27} -5.97458 q^{29} -6.01598 q^{31} +8.31712 q^{33} +3.38303 q^{35} +11.2625 q^{37} +4.05623 q^{39} +0.996870 q^{41} -7.83781 q^{43} -20.2042 q^{45} +0.910938 q^{47} -5.24399 q^{49} +0.696356 q^{51} +3.32429 q^{53} -6.42720 q^{55} +3.30364 q^{57} -2.04859 q^{59} +10.0769 q^{61} -10.4873 q^{63} -3.13453 q^{65} +11.7584 q^{67} +26.8134 q^{69} +11.9944 q^{71} +13.0038 q^{73} +5.01347 q^{75} -3.33613 q^{77} -12.0761 q^{79} +29.8902 q^{81} -4.50788 q^{83} -0.538122 q^{85} -19.7379 q^{87} -6.64553 q^{89} -1.62702 q^{91} -19.8746 q^{93} -2.55295 q^{95} +9.92753 q^{97} +19.9241 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{3} + 14 q^{9} + 20 q^{11} + 4 q^{17} + 10 q^{19} + 10 q^{25} + 28 q^{27} - 8 q^{33} + 36 q^{35} - 12 q^{41} - 4 q^{43} + 26 q^{49} + 36 q^{51} + 4 q^{57} + 52 q^{59} - 24 q^{65} + 12 q^{67}+ \cdots + 60 q^{99}+O(q^{100}) $ 10 * q + 4 * q^3 + 14 * q^9 + 20 * q^11 + 4 * q^17 + 10 * q^19 + 10 * q^25 + 28 * q^27 - 8 * q^33 + 36 * q^35 - 12 * q^41 - 4 * q^43 + 26 * q^49 + 36 * q^51 + 4 * q^57 + 52 * q^59 - 24 * q^65 + 12 * q^67 + 12 * q^73 - 12 * q^75 + 34 * q^81 + 16 * q^83 - 20 * q^89 + 60 * q^91 - 28 * q^97 + 60 * 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
$3$	3.30364	1.90736	0.953680	−	0.300824i	$-0.0972616\pi$
$3$	3.30364	1.90736	0.953680	+	0.300824i	$0.0972616\pi$
$4$	0	0
$5$	−2.55295	−1.14171	−0.570857	−	0.821049i	$-0.693389\pi$
$5$	−2.55295	−1.14171	−0.570857	+	0.821049i	$0.693389\pi$
$6$	0	0
$7$	−1.32515	−0.500858	−0.250429	−	0.968135i	$-0.580572\pi$
$7$	−1.32515	−0.500858	−0.250429	+	0.968135i	$0.580572\pi$
$8$	0	0
$9$	7.91406	2.63802
$10$	0	0
$11$	2.51756	0.759072	0.379536	−	0.925177i	$-0.376084\pi$
$11$	2.51756	0.759072	0.379536	+	0.925177i	$0.376084\pi$
$12$	0	0
$13$	1.22781	0.340532	0.170266	−	0.985398i	$-0.445537\pi$
$13$	1.22781	0.340532	0.170266	+	0.985398i	$0.445537\pi$
$14$	0	0
$15$	−8.43404	−2.17766
$16$	0	0
$17$	0.210784	0.0511227	0.0255614	−	0.999673i	$-0.491863\pi$
$17$	0.210784	0.0511227	0.0255614	+	0.999673i	$0.491863\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.37781	−0.955316
$22$	0	0
$23$	8.11631	1.69237	0.846184	−	0.532891i	$-0.178895\pi$
$23$	8.11631	1.69237	0.846184	+	0.532891i	$0.178895\pi$
$24$	0	0
$25$	1.51756	0.303512
$26$	0	0
$27$	16.2343	3.12429
$28$	0	0
$29$	−5.97458	−1.10945	−0.554726	−	0.832033i	$-0.687177\pi$
$29$	−5.97458	−1.10945	−0.554726	+	0.832033i	$0.687177\pi$
$30$	0	0
$31$	−6.01598	−1.08050	−0.540251	−	0.841504i	$-0.681671\pi$
$31$	−6.01598	−1.08050	−0.540251	+	0.841504i	$0.681671\pi$
$32$	0	0
$33$	8.31712	1.44782
$34$	0	0
$35$	3.38303	0.571837
$36$	0	0
$37$	11.2625	1.85154	0.925769	−	0.378089i	$-0.123419\pi$
$37$	11.2625	1.85154	0.925769	+	0.378089i	$0.123419\pi$
$38$	0	0
$39$	4.05623	0.649517
$40$	0	0
$41$	0.996870	0.155685	0.0778424	−	0.996966i	$-0.475197\pi$
$41$	0.996870	0.155685	0.0778424	+	0.996966i	$0.475197\pi$
$42$	0	0
$43$	−7.83781	−1.19525	−0.597627	−	0.801774i	$-0.703890\pi$
$43$	−7.83781	−1.19525	−0.597627	+	0.801774i	$0.703890\pi$
$44$	0	0
$45$	−20.2042	−3.01187
$46$	0	0
$47$	0.910938	0.132874	0.0664369	−	0.997791i	$-0.478837\pi$
$47$	0.910938	0.132874	0.0664369	+	0.997791i	$0.478837\pi$
$48$	0	0
$49$	−5.24399	−0.749141
$50$	0	0
$51$	0.696356	0.0975094
$52$	0	0
$53$	3.32429	0.456627	0.228313	−	0.973588i	$-0.426679\pi$
$53$	3.32429	0.456627	0.228313	+	0.973588i	$0.426679\pi$
$54$	0	0
$55$	−6.42720	−0.866644
$56$	0	0
$57$	3.30364	0.437578
$58$	0	0
$59$	−2.04859	−0.266704	−0.133352	−	0.991069i	$-0.542574\pi$
$59$	−2.04859	−0.266704	−0.133352	+	0.991069i	$0.542574\pi$
$60$	0	0
$61$	10.0769	1.29022	0.645108	−	0.764091i	$-0.276812\pi$
$61$	10.0769	1.29022	0.645108	+	0.764091i	$0.276812\pi$
$62$	0	0
$63$	−10.4873	−1.32127
$64$	0	0
$65$	−3.13453	−0.388790
$66$	0	0
$67$	11.7584	1.43652	0.718260	−	0.695775i	$-0.244939\pi$
$67$	11.7584	1.43652	0.718260	+	0.695775i	$0.244939\pi$
$68$	0	0
$69$	26.8134	3.22795
$70$	0	0
$71$	11.9944	1.42347	0.711737	−	0.702446i	$-0.247909\pi$
$71$	11.9944	1.42347	0.711737	+	0.702446i	$0.247909\pi$
$72$	0	0
$73$	13.0038	1.52198	0.760989	−	0.648764i	$-0.224714\pi$
$73$	13.0038	1.52198	0.760989	+	0.648764i	$0.224714\pi$
$74$	0	0
$75$	5.01347	0.578906
$76$	0	0
$77$	−3.33613	−0.380187
$78$	0	0
$79$	−12.0761	−1.35866	−0.679332	−	0.733831i	$-0.737730\pi$
$79$	−12.0761	−1.35866	−0.679332	+	0.733831i	$0.737730\pi$
$80$	0	0
$81$	29.8902	3.32113
$82$	0	0
$83$	−4.50788	−0.494804	−0.247402	−	0.968913i	$-0.579577\pi$
$83$	−4.50788	−0.494804	−0.247402	+	0.968913i	$0.579577\pi$
$84$	0	0
$85$	−0.538122	−0.0583675
$86$	0	0
$87$	−19.7379	−2.11612
$88$	0	0
$89$	−6.64553	−0.704425	−0.352213	−	0.935920i	$-0.614571\pi$
$89$	−6.64553	−0.704425	−0.352213	+	0.935920i	$0.614571\pi$
$90$	0	0
$91$	−1.62702	−0.170558
$92$	0	0
$93$	−19.8746	−2.06090
$94$	0	0
$95$	−2.55295	−0.261927
$96$	0	0
$97$	9.92753	1.00799	0.503994	−	0.863707i	$-0.331863\pi$
$97$	9.92753	1.00799	0.503994	+	0.863707i	$0.331863\pi$
$98$	0	0
$99$	19.9241	2.00245
$100$	0	0
$101$	−1.26920	−0.126290	−0.0631450	−	0.998004i	$-0.520113\pi$
$101$	−1.26920	−0.126290	−0.0631450	+	0.998004i	$0.520113\pi$
$102$	0	0
$103$	14.4500	1.42380	0.711901	−	0.702280i	$-0.247834\pi$
$103$	14.4500	1.42380	0.711901	+	0.702280i	$0.247834\pi$
$104$	0	0
$105$	11.1763	1.09070
$106$	0	0
$107$	−10.3868	−1.00413	−0.502066	−	0.864829i	$-0.667427\pi$
$107$	−10.3868	−1.00413	−0.502066	+	0.864829i	$0.667427\pi$
$108$	0	0
$109$	15.3639	1.47160	0.735799	−	0.677200i	$-0.236807\pi$
$109$	15.3639	1.47160	0.735799	+	0.677200i	$0.236807\pi$
$110$	0	0
$111$	37.2072	3.53155
$112$	0	0
$113$	16.2509	1.52876	0.764379	−	0.644768i	$-0.223046\pi$
$113$	16.2509	1.52876	0.764379	+	0.644768i	$0.223046\pi$
$114$	0	0
$115$	−20.7205	−1.93220
$116$	0	0
$117$	9.71693	0.898331
$118$	0	0
$119$	−0.279320	−0.0256052
$120$	0	0
$121$	−4.66190	−0.423809
$122$	0	0
$123$	3.29330	0.296947
$124$	0	0
$125$	8.89050	0.795191
$126$	0	0
$127$	6.56693	0.582721	0.291360	−	0.956613i	$-0.405892\pi$
$127$	6.56693	0.582721	0.291360	+	0.956613i	$0.405892\pi$
$128$	0	0
$129$	−25.8933	−2.27978
$130$	0	0
$131$	−10.0235	−0.875759	−0.437880	−	0.899034i	$-0.644270\pi$
$131$	−10.0235	−0.875759	−0.437880	+	0.899034i	$0.644270\pi$
$132$	0	0
$133$	−1.32515	−0.114905
$134$	0	0
$135$	−41.4454	−3.56705
$136$	0	0
$137$	−7.01005	−0.598909	−0.299455	−	0.954111i	$-0.596805\pi$
$137$	−7.01005	−0.598909	−0.299455	+	0.954111i	$0.596805\pi$
$138$	0	0
$139$	−3.13111	−0.265577	−0.132789	−	0.991144i	$-0.542393\pi$
$139$	−3.13111	−0.265577	−0.132789	+	0.991144i	$0.542393\pi$
$140$	0	0
$141$	3.00941	0.253438
$142$	0	0
$143$	3.09107	0.258489
$144$	0	0
$145$	15.2528	1.26668
$146$	0	0
$147$	−17.3243	−1.42888
$148$	0	0
$149$	−7.47186	−0.612119	−0.306059	−	0.952012i	$-0.599011\pi$
$149$	−7.47186	−0.612119	−0.306059	+	0.952012i	$0.599011\pi$
$150$	0	0
$151$	−18.1373	−1.47599	−0.737995	−	0.674806i	$-0.764227\pi$
$151$	−18.1373	−1.47599	−0.737995	+	0.674806i	$0.764227\pi$
$152$	0	0
$153$	1.66816	0.134863
$154$	0	0
$155$	15.3585	1.23362
$156$	0	0
$157$	−6.69383	−0.534225	−0.267113	−	0.963665i	$-0.586070\pi$
$157$	−6.69383	−0.534225	−0.267113	+	0.963665i	$0.586070\pi$
$158$	0	0
$159$	10.9823	0.870952
$160$	0	0
$161$	−10.7553	−0.847635
$162$	0	0
$163$	15.6424	1.22521	0.612604	−	0.790390i	$-0.290122\pi$
$163$	15.6424	1.22521	0.612604	+	0.790390i	$0.290122\pi$
$164$	0	0
$165$	−21.2332	−1.65300
$166$	0	0
$167$	4.23338	0.327588	0.163794	−	0.986495i	$-0.447627\pi$
$167$	4.23338	0.327588	0.163794	+	0.986495i	$0.447627\pi$
$168$	0	0
$169$	−11.4925	−0.884038
$170$	0	0
$171$	7.91406	0.605203
$172$	0	0
$173$	−4.25476	−0.323483	−0.161742	−	0.986833i	$-0.551711\pi$
$173$	−4.25476	−0.323483	−0.161742	+	0.986833i	$0.551711\pi$
$174$	0	0
$175$	−2.01099	−0.152016
$176$	0	0
$177$	−6.76781	−0.508700
$178$	0	0
$179$	20.3171	1.51857	0.759286	−	0.650757i	$-0.225548\pi$
$179$	20.3171	1.51857	0.759286	+	0.650757i	$0.225548\pi$
$180$	0	0
$181$	−16.6430	−1.23706	−0.618532	−	0.785760i	$-0.712272\pi$
$181$	−16.6430	−1.23706	−0.618532	+	0.785760i	$0.712272\pi$
$182$	0	0
$183$	33.2905	2.46091
$184$	0	0
$185$	−28.7525	−2.11393
$186$	0	0
$187$	0.530662	0.0388058
$188$	0	0
$189$	−21.5128	−1.56483
$190$	0	0
$191$	−15.9246	−1.15226	−0.576132	−	0.817357i	$-0.695439\pi$
$191$	−15.9246	−1.15226	−0.576132	+	0.817357i	$0.695439\pi$
$192$	0	0
$193$	−25.8762	−1.86261	−0.931304	−	0.364242i	$-0.881328\pi$
$193$	−25.8762	−1.86261	−0.931304	+	0.364242i	$0.881328\pi$
$194$	0	0
$195$	−10.3554	−0.741563
$196$	0	0
$197$	18.2850	1.30275	0.651376	−	0.758755i	$-0.274192\pi$
$197$	18.2850	1.30275	0.651376	+	0.758755i	$0.274192\pi$
$198$	0	0
$199$	10.0808	0.714606	0.357303	−	0.933989i	$-0.383696\pi$
$199$	10.0808	0.714606	0.357303	+	0.933989i	$0.383696\pi$
$200$	0	0
$201$	38.8456	2.73996
$202$	0	0
$203$	7.91719	0.555678
$204$	0	0
$205$	−2.54496	−0.177748
$206$	0	0
$207$	64.2330	4.46450
$208$	0	0
$209$	2.51756	0.174143
$210$	0	0
$211$	15.8127	1.08859	0.544297	−	0.838893i	$-0.316796\pi$
$211$	15.8127	1.08859	0.544297	+	0.838893i	$0.316796\pi$
$212$	0	0
$213$	39.6252	2.71508
$214$	0	0
$215$	20.0095	1.36464
$216$	0	0
$217$	7.97204	0.541178
$218$	0	0
$219$	42.9599	2.90296
$220$	0	0
$221$	0.258802	0.0174089
$222$	0	0
$223$	4.15828	0.278459	0.139229	−	0.990260i	$-0.455537\pi$
$223$	4.15828	0.278459	0.139229	+	0.990260i	$0.455537\pi$
$224$	0	0
$225$	12.0101	0.800670
$226$	0	0
$227$	16.7714	1.11315	0.556577	−	0.830796i	$-0.312114\pi$
$227$	16.7714	1.11315	0.556577	+	0.830796i	$0.312114\pi$
$228$	0	0
$229$	1.32899	0.0878220	0.0439110	−	0.999035i	$-0.486018\pi$
$229$	1.32899	0.0878220	0.0439110	+	0.999035i	$0.486018\pi$
$230$	0	0
$231$	−11.0214	−0.725154
$232$	0	0
$233$	15.4631	1.01302	0.506510	−	0.862234i	$-0.330935\pi$
$233$	15.4631	1.01302	0.506510	+	0.862234i	$0.330935\pi$
$234$	0	0
$235$	−2.32558	−0.151704
$236$	0	0
$237$	−39.8950	−2.59146
$238$	0	0
$239$	15.4564	0.999794	0.499897	−	0.866085i	$-0.333371\pi$
$239$	15.4564	0.999794	0.499897	+	0.866085i	$0.333371\pi$
$240$	0	0
$241$	−14.0126	−0.902632	−0.451316	−	0.892364i	$-0.649045\pi$
$241$	−14.0126	−0.902632	−0.451316	+	0.892364i	$0.649045\pi$
$242$	0	0
$243$	50.0436	3.21030
$244$	0	0
$245$	13.3877	0.855306
$246$	0	0
$247$	1.22781	0.0781234
$248$	0	0
$249$	−14.8924	−0.943769
$250$	0	0
$251$	4.36102	0.275265	0.137632	−	0.990483i	$-0.456051\pi$
$251$	4.36102	0.275265	0.137632	+	0.990483i	$0.456051\pi$
$252$	0	0
$253$	20.4333	1.28463
$254$	0	0
$255$	−1.77776	−0.111328
$256$	0	0
$257$	21.4237	1.33637	0.668186	−	0.743994i	$-0.267071\pi$
$257$	21.4237	1.33637	0.668186	+	0.743994i	$0.267071\pi$
$258$	0	0
$259$	−14.9244	−0.927357
$260$	0	0
$261$	−47.2832	−2.92676
$262$	0	0
$263$	−5.37369	−0.331356	−0.165678	−	0.986180i	$-0.552981\pi$
$263$	−5.37369	−0.331356	−0.165678	+	0.986180i	$0.552981\pi$
$264$	0	0
$265$	−8.48676	−0.521338
$266$	0	0
$267$	−21.9545	−1.34359
$268$	0	0
$269$	−10.0684	−0.613879	−0.306940	−	0.951729i	$-0.599305\pi$
$269$	−10.0684	−0.613879	−0.306940	+	0.951729i	$0.599305\pi$
$270$	0	0
$271$	−27.6347	−1.67869	−0.839344	−	0.543601i	$-0.817060\pi$
$271$	−27.6347	−1.67869	−0.839344	+	0.543601i	$0.817060\pi$
$272$	0	0
$273$	−5.37510	−0.325316
$274$	0	0
$275$	3.82054	0.230387
$276$	0	0
$277$	9.52596	0.572360	0.286180	−	0.958176i	$-0.407615\pi$
$277$	9.52596	0.572360	0.286180	+	0.958176i	$0.407615\pi$
$278$	0	0
$279$	−47.6108	−2.85038
$280$	0	0
$281$	−27.6168	−1.64748	−0.823739	−	0.566969i	$-0.808116\pi$
$281$	−27.6168	−1.64748	−0.823739	+	0.566969i	$0.808116\pi$
$282$	0	0
$283$	17.0192	1.01168	0.505842	−	0.862626i	$-0.331182\pi$
$283$	17.0192	1.01168	0.505842	+	0.862626i	$0.331182\pi$
$284$	0	0
$285$	−8.43404	−0.499589
$286$	0	0
$287$	−1.32100	−0.0779760
$288$	0	0
$289$	−16.9556	−0.997386
$290$	0	0
$291$	32.7970	1.92260
$292$	0	0
$293$	13.8952	0.811767	0.405884	−	0.913925i	$-0.366964\pi$
$293$	13.8952	0.811767	0.405884	+	0.913925i	$0.366964\pi$
$294$	0	0
$295$	5.22995	0.304499
$296$	0	0
$297$	40.8708	2.37157
$298$	0	0
$299$	9.96525	0.576305
$300$	0	0
$301$	10.3862	0.598652
$302$	0	0
$303$	−4.19298	−0.240880
$304$	0	0
$305$	−25.7259	−1.47306
$306$	0	0
$307$	−8.57307	−0.489291	−0.244645	−	0.969613i	$-0.578672\pi$
$307$	−8.57307	−0.489291	−0.244645	+	0.969613i	$0.578672\pi$
$308$	0	0
$309$	47.7377	2.71570
$310$	0	0
$311$	−18.7442	−1.06288	−0.531442	−	0.847094i	$-0.678350\pi$
$311$	−18.7442	−1.06288	−0.531442	+	0.847094i	$0.678350\pi$
$312$	0	0
$313$	−0.267307	−0.0151091	−0.00755454	−	0.999971i	$-0.502405\pi$
$313$	−0.267307	−0.0151091	−0.00755454	+	0.999971i	$0.502405\pi$
$314$	0	0
$315$	26.7735	1.50852
$316$	0	0
$317$	24.3865	1.36968	0.684841	−	0.728693i	$-0.259872\pi$
$317$	24.3865	1.36968	0.684841	+	0.728693i	$0.259872\pi$
$318$	0	0
$319$	−15.0414	−0.842155
$320$	0	0
$321$	−34.3144	−1.91524
$322$	0	0
$323$	0.210784	0.0117284
$324$	0	0
$325$	1.86327	0.103355
$326$	0	0
$327$	50.7570	2.80687
$328$	0	0
$329$	−1.20712	−0.0665509
$330$	0	0
$331$	15.8978	0.873824	0.436912	−	0.899504i	$-0.356072\pi$
$331$	15.8978	0.873824	0.436912	+	0.899504i	$0.356072\pi$
$332$	0	0
$333$	89.1319	4.88440
$334$	0	0
$335$	−30.0187	−1.64009
$336$	0	0
$337$	−23.4174	−1.27563	−0.637814	−	0.770190i	$-0.720161\pi$
$337$	−23.4174	−1.27563	−0.637814	+	0.770190i	$0.720161\pi$
$338$	0	0
$339$	53.6872	2.91589
$340$	0	0
$341$	−15.1456	−0.820179
$342$	0	0
$343$	16.2251	0.876071
$344$	0	0
$345$	−68.4533	−3.68540
$346$	0	0
$347$	−0.166224	−0.00892336	−0.00446168	−	0.999990i	$-0.501420\pi$
$347$	−0.166224	−0.00892336	−0.00446168	+	0.999990i	$0.501420\pi$
$348$	0	0
$349$	3.92636	0.210173	0.105087	−	0.994463i	$-0.466488\pi$
$349$	3.92636	0.210173	0.105087	+	0.994463i	$0.466488\pi$
$350$	0	0
$351$	19.9326	1.06392
$352$	0	0
$353$	9.45111	0.503032	0.251516	−	0.967853i	$-0.419071\pi$
$353$	9.45111	0.503032	0.251516	+	0.967853i	$0.419071\pi$
$354$	0	0
$355$	−30.6211	−1.62520
$356$	0	0
$357$	−0.922773	−0.0488383
$358$	0	0
$359$	−7.81041	−0.412218	−0.206109	−	0.978529i	$-0.566080\pi$
$359$	−7.81041	−0.412218	−0.206109	+	0.978529i	$0.566080\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−15.4013	−0.808356
$364$	0	0
$365$	−33.1980	−1.73766
$366$	0	0
$367$	−34.3412	−1.79260	−0.896298	−	0.443452i	$-0.853754\pi$
$367$	−34.3412	−1.79260	−0.896298	+	0.443452i	$0.853754\pi$
$368$	0	0
$369$	7.88929	0.410700
$370$	0	0
$371$	−4.40517	−0.228705
$372$	0	0
$373$	−33.9919	−1.76004	−0.880018	−	0.474941i	$-0.842469\pi$
$373$	−33.9919	−1.76004	−0.880018	+	0.474941i	$0.842469\pi$
$374$	0	0
$375$	29.3711	1.51671
$376$	0	0
$377$	−7.33563	−0.377804
$378$	0	0
$379$	−4.11882	−0.211570	−0.105785	−	0.994389i	$-0.533735\pi$
$379$	−4.11882	−0.211570	−0.105785	+	0.994389i	$0.533735\pi$
$380$	0	0
$381$	21.6948	1.11146
$382$	0	0
$383$	−17.5442	−0.896467	−0.448233	−	0.893917i	$-0.647947\pi$
$383$	−17.5442	−0.896467	−0.448233	+	0.893917i	$0.647947\pi$
$384$	0	0
$385$	8.51698	0.434065
$386$	0	0
$387$	−62.0289	−3.15310
$388$	0	0
$389$	−2.23423	−0.113280	−0.0566399	−	0.998395i	$-0.518039\pi$
$389$	−2.23423	−0.113280	−0.0566399	+	0.998395i	$0.518039\pi$
$390$	0	0
$391$	1.71079	0.0865184
$392$	0	0
$393$	−33.1142	−1.67039
$394$	0	0
$395$	30.8296	1.55121
$396$	0	0
$397$	−15.4349	−0.774655	−0.387327	−	0.921942i	$-0.626602\pi$
$397$	−15.4349	−0.774655	−0.387327	+	0.921942i	$0.626602\pi$
$398$	0	0
$399$	−4.37781	−0.219164
$400$	0	0
$401$	−22.3503	−1.11612	−0.558061	−	0.829800i	$-0.688454\pi$
$401$	−22.3503	−1.11612	−0.558061	+	0.829800i	$0.688454\pi$
$402$	0	0
$403$	−7.38645	−0.367945
$404$	0	0
$405$	−76.3082	−3.79178
$406$	0	0
$407$	28.3539	1.40545
$408$	0	0
$409$	−8.58617	−0.424559	−0.212279	−	0.977209i	$-0.568089\pi$
$409$	−8.58617	−0.424559	−0.212279	+	0.977209i	$0.568089\pi$
$410$	0	0
$411$	−23.1587	−1.14234
$412$	0	0
$413$	2.71468	0.133581
$414$	0	0
$415$	11.5084	0.564925
$416$	0	0
$417$	−10.3441	−0.506551
$418$	0	0
$419$	21.9670	1.07316	0.536579	−	0.843850i	$-0.319716\pi$
$419$	21.9670	1.07316	0.536579	+	0.843850i	$0.319716\pi$
$420$	0	0
$421$	−22.3942	−1.09143	−0.545713	−	0.837972i	$-0.683741\pi$
$421$	−22.3942	−1.09143	−0.545713	+	0.837972i	$0.683741\pi$
$422$	0	0
$423$	7.20922	0.350524
$424$	0	0
$425$	0.319878	0.0155163
$426$	0	0
$427$	−13.3534	−0.646215
$428$	0	0
$429$	10.2118	0.493031
$430$	0	0
$431$	7.39422	0.356167	0.178084	−	0.984015i	$-0.443010\pi$
$431$	7.39422	0.356167	0.178084	+	0.984015i	$0.443010\pi$
$432$	0	0
$433$	−30.0766	−1.44539	−0.722694	−	0.691168i	$-0.757097\pi$
$433$	−30.0766	−1.44539	−0.722694	+	0.691168i	$0.757097\pi$
$434$	0	0
$435$	50.3899	2.41601
$436$	0	0
$437$	8.11631	0.388256
$438$	0	0
$439$	−16.3397	−0.779850	−0.389925	−	0.920847i	$-0.627499\pi$
$439$	−16.3397	−0.779850	−0.389925	+	0.920847i	$0.627499\pi$
$440$	0	0
$441$	−41.5013	−1.97625
$442$	0	0
$443$	4.07589	0.193651	0.0968256	−	0.995301i	$-0.469131\pi$
$443$	4.07589	0.193651	0.0968256	+	0.995301i	$0.469131\pi$
$444$	0	0
$445$	16.9657	0.804252
$446$	0	0
$447$	−24.6844	−1.16753
$448$	0	0
$449$	4.21892	0.199103	0.0995517	−	0.995032i	$-0.468259\pi$
$449$	4.21892	0.199103	0.0995517	+	0.995032i	$0.468259\pi$
$450$	0	0
$451$	2.50968	0.118176
$452$	0	0
$453$	−59.9191	−2.81525
$454$	0	0
$455$	4.15370	0.194729
$456$	0	0
$457$	−28.8382	−1.34899	−0.674496	−	0.738278i	$-0.735639\pi$
$457$	−28.8382	−1.34899	−0.674496	+	0.738278i	$0.735639\pi$
$458$	0	0
$459$	3.42194	0.159722
$460$	0	0
$461$	−31.0261	−1.44503	−0.722514	−	0.691356i	$-0.757014\pi$
$461$	−31.0261	−1.44503	−0.722514	+	0.691356i	$0.757014\pi$
$462$	0	0
$463$	−12.5849	−0.584870	−0.292435	−	0.956285i	$-0.594465\pi$
$463$	−12.5849	−0.584870	−0.292435	+	0.956285i	$0.594465\pi$
$464$	0	0
$465$	50.7390	2.35296
$466$	0	0
$467$	−27.8849	−1.29036	−0.645179	−	0.764032i	$-0.723217\pi$
$467$	−27.8849	−1.29036	−0.645179	+	0.764032i	$0.723217\pi$
$468$	0	0
$469$	−15.5816	−0.719492
$470$	0	0
$471$	−22.1140	−1.01896
$472$	0	0
$473$	−19.7321	−0.907284
$474$	0	0
$475$	1.51756	0.0696304
$476$	0	0
$477$	26.3087	1.20459
$478$	0	0
$479$	11.7138	0.535217	0.267608	−	0.963528i	$-0.413767\pi$
$479$	11.7138	0.535217	0.267608	+	0.963528i	$0.413767\pi$
$480$	0	0
$481$	13.8281	0.630508
$482$	0	0
$483$	−35.5316	−1.61675
$484$	0	0
$485$	−25.3445	−1.15083
$486$	0	0
$487$	−8.07492	−0.365909	−0.182955	−	0.983121i	$-0.558566\pi$
$487$	−8.07492	−0.365909	−0.182955	+	0.983121i	$0.558566\pi$
$488$	0	0
$489$	51.6769	2.33691
$490$	0	0
$491$	−15.0953	−0.681240	−0.340620	−	0.940201i	$-0.610637\pi$
$491$	−15.0953	−0.681240	−0.340620	+	0.940201i	$0.610637\pi$
$492$	0	0
$493$	−1.25935	−0.0567182
$494$	0	0
$495$	−50.8653	−2.28622
$496$	0	0
$497$	−15.8943	−0.712958
$498$	0	0
$499$	31.2381	1.39841	0.699205	−	0.714922i	$-0.253538\pi$
$499$	31.2381	1.39841	0.699205	+	0.714922i	$0.253538\pi$
$500$	0	0
$501$	13.9856	0.624829
$502$	0	0
$503$	21.7755	0.970920	0.485460	−	0.874259i	$-0.338652\pi$
$503$	21.7755	0.970920	0.485460	+	0.874259i	$0.338652\pi$
$504$	0	0
$505$	3.24020	0.144187
$506$	0	0
$507$	−37.9671	−1.68618
$508$	0	0
$509$	−4.88080	−0.216338	−0.108169	−	0.994133i	$-0.534499\pi$
$509$	−4.88080	−0.216338	−0.108169	+	0.994133i	$0.534499\pi$
$510$	0	0
$511$	−17.2319	−0.762295
$512$	0	0
$513$	16.2343	0.716762
$514$	0	0
$515$	−36.8902	−1.62558
$516$	0	0
$517$	2.29334	0.100861
$518$	0	0
$519$	−14.0562	−0.616998
$520$	0	0
$521$	−10.8112	−0.473645	−0.236823	−	0.971553i	$-0.576106\pi$
$521$	−10.8112	−0.473645	−0.236823	+	0.971553i	$0.576106\pi$
$522$	0	0
$523$	−21.1722	−0.925798	−0.462899	−	0.886411i	$-0.653191\pi$
$523$	−21.1722	−0.925798	−0.462899	+	0.886411i	$0.653191\pi$
$524$	0	0
$525$	−6.64358	−0.289950
$526$	0	0
$527$	−1.26807	−0.0552382
$528$	0	0
$529$	42.8745	1.86411
$530$	0	0
$531$	−16.2127	−0.703570
$532$	0	0
$533$	1.22396	0.0530157
$534$	0	0
$535$	26.5170	1.14643
$536$	0	0
$537$	67.1205	2.89646
$538$	0	0
$539$	−13.2021	−0.568653
$540$	0	0
$541$	−15.1179	−0.649969	−0.324984	−	0.945719i	$-0.605359\pi$
$541$	−15.1179	−0.649969	−0.324984	+	0.945719i	$0.605359\pi$
$542$	0	0
$543$	−54.9825	−2.35952
$544$	0	0
$545$	−39.2234	−1.68014
$546$	0	0
$547$	5.51960	0.236001	0.118001	−	0.993014i	$-0.462352\pi$
$547$	5.51960	0.236001	0.118001	+	0.993014i	$0.462352\pi$
$548$	0	0
$549$	79.7493	3.40362
$550$	0	0
$551$	−5.97458	−0.254526
$552$	0	0
$553$	16.0025	0.680497
$554$	0	0
$555$	−94.9881	−4.03202
$556$	0	0
$557$	18.9708	0.803820	0.401910	−	0.915679i	$-0.368346\pi$
$557$	18.9708	0.803820	0.401910	+	0.915679i	$0.368346\pi$
$558$	0	0
$559$	−9.62330	−0.407022
$560$	0	0
$561$	1.75312	0.0740167
$562$	0	0
$563$	15.5505	0.655374	0.327687	−	0.944786i	$-0.393731\pi$
$563$	15.5505	0.655374	0.327687	+	0.944786i	$0.393731\pi$
$564$	0	0
$565$	−41.4878	−1.74540
$566$	0	0
$567$	−39.6088	−1.66341
$568$	0	0
$569$	3.32025	0.139192	0.0695960	−	0.997575i	$-0.477829\pi$
$569$	3.32025	0.139192	0.0695960	+	0.997575i	$0.477829\pi$
$570$	0	0
$571$	8.41420	0.352123	0.176062	−	0.984379i	$-0.443664\pi$
$571$	8.41420	0.352123	0.176062	+	0.984379i	$0.443664\pi$
$572$	0	0
$573$	−52.6092	−2.19778
$574$	0	0
$575$	12.3170	0.513653
$576$	0	0
$577$	5.94857	0.247642	0.123821	−	0.992305i	$-0.460485\pi$
$577$	5.94857	0.247642	0.123821	+	0.992305i	$0.460485\pi$
$578$	0	0
$579$	−85.4857	−3.55266
$580$	0	0
$581$	5.97359	0.247826
$582$	0	0
$583$	8.36911	0.346613
$584$	0	0
$585$	−24.8068	−1.02564
$586$	0	0
$587$	27.1449	1.12039	0.560196	−	0.828360i	$-0.310726\pi$
$587$	27.1449	1.12039	0.560196	+	0.828360i	$0.310726\pi$
$588$	0	0
$589$	−6.01598	−0.247884
$590$	0	0
$591$	60.4071	2.48482
$592$	0	0
$593$	−24.3801	−1.00117	−0.500585	−	0.865687i	$-0.666882\pi$
$593$	−24.3801	−1.00117	−0.500585	+	0.865687i	$0.666882\pi$
$594$	0	0
$595$	0.713090	0.0292338
$596$	0	0
$597$	33.3032	1.36301
$598$	0	0
$599$	18.7108	0.764502	0.382251	−	0.924059i	$-0.375149\pi$
$599$	18.7108	0.764502	0.382251	+	0.924059i	$0.375149\pi$
$600$	0	0
$601$	−44.2967	−1.80690	−0.903451	−	0.428692i	$-0.858975\pi$
$601$	−44.2967	−1.80690	−0.903451	+	0.428692i	$0.858975\pi$
$602$	0	0
$603$	93.0568	3.78957
$604$	0	0
$605$	11.9016	0.483869
$606$	0	0
$607$	−30.5153	−1.23858	−0.619290	−	0.785163i	$-0.712579\pi$
$607$	−30.5153	−1.23858	−0.619290	+	0.785163i	$0.712579\pi$
$608$	0	0
$609$	26.1556	1.05988
$610$	0	0
$611$	1.11845	0.0452478
$612$	0	0
$613$	−21.3377	−0.861822	−0.430911	−	0.902395i	$-0.641808\pi$
$613$	−21.3377	−0.861822	−0.430911	+	0.902395i	$0.641808\pi$
$614$	0	0
$615$	−8.40764	−0.339029
$616$	0	0
$617$	23.5271	0.947163	0.473582	−	0.880750i	$-0.342961\pi$
$617$	23.5271	0.947163	0.473582	+	0.880750i	$0.342961\pi$
$618$	0	0
$619$	18.0716	0.726357	0.363179	−	0.931720i	$-0.381692\pi$
$619$	18.0716	0.726357	0.363179	+	0.931720i	$0.381692\pi$
$620$	0	0
$621$	131.763	5.28745
$622$	0	0
$623$	8.80630	0.352817
$624$	0	0
$625$	−30.2848	−1.21139
$626$	0	0
$627$	8.31712	0.332154
$628$	0	0
$629$	2.37395	0.0946557
$630$	0	0
$631$	−13.2280	−0.526600	−0.263300	−	0.964714i	$-0.584811\pi$
$631$	−13.2280	−0.526600	−0.263300	+	0.964714i	$0.584811\pi$
$632$	0	0
$633$	52.2397	2.07634
$634$	0	0
$635$	−16.7650	−0.665301
$636$	0	0
$637$	−6.43860	−0.255107
$638$	0	0
$639$	94.9245	3.75515
$640$	0	0
$641$	−30.1208	−1.18970	−0.594851	−	0.803836i	$-0.702789\pi$
$641$	−30.1208	−1.18970	−0.594851	+	0.803836i	$0.702789\pi$
$642$	0	0
$643$	−24.7964	−0.977876	−0.488938	−	0.872319i	$-0.662616\pi$
$643$	−24.7964	−0.977876	−0.488938	+	0.872319i	$0.662616\pi$
$644$	0	0
$645$	66.1044	2.60286
$646$	0	0
$647$	−22.8386	−0.897880	−0.448940	−	0.893562i	$-0.648198\pi$
$647$	−22.8386	−0.897880	−0.448940	+	0.893562i	$0.648198\pi$
$648$	0	0
$649$	−5.15744	−0.202447
$650$	0	0
$651$	26.3368	1.03222
$652$	0	0
$653$	−32.4823	−1.27113	−0.635565	−	0.772048i	$-0.719233\pi$
$653$	−32.4823	−1.27113	−0.635565	+	0.772048i	$0.719233\pi$
$654$	0	0
$655$	25.5896	0.999867
$656$	0	0
$657$	102.913	4.01501
$658$	0	0
$659$	17.2055	0.670229	0.335115	−	0.942177i	$-0.391225\pi$
$659$	17.2055	0.670229	0.335115	+	0.942177i	$0.391225\pi$
$660$	0	0
$661$	−13.0942	−0.509305	−0.254653	−	0.967033i	$-0.581961\pi$
$661$	−13.0942	−0.509305	−0.254653	+	0.967033i	$0.581961\pi$
$662$	0	0
$663$	0.854990	0.0332051
$664$	0	0
$665$	3.38303	0.131188
$666$	0	0
$667$	−48.4916	−1.87760
$668$	0	0
$669$	13.7375	0.531121
$670$	0	0
$671$	25.3692	0.979368
$672$	0	0
$673$	−11.3396	−0.437110	−0.218555	−	0.975825i	$-0.570134\pi$
$673$	−11.3396	−0.437110	−0.218555	+	0.975825i	$0.570134\pi$
$674$	0	0
$675$	24.6365	0.948260
$676$	0	0
$677$	43.2663	1.66286	0.831429	−	0.555631i	$-0.187523\pi$
$677$	43.2663	1.66286	0.831429	+	0.555631i	$0.187523\pi$
$678$	0	0
$679$	−13.1554	−0.504859
$680$	0	0
$681$	55.4066	2.12319
$682$	0	0
$683$	−7.63100	−0.291992	−0.145996	−	0.989285i	$-0.546639\pi$
$683$	−7.63100	−0.291992	−0.145996	+	0.989285i	$0.546639\pi$
$684$	0	0
$685$	17.8963	0.683783
$686$	0	0
$687$	4.39051	0.167508
$688$	0	0
$689$	4.08159	0.155496
$690$	0	0
$691$	−20.2177	−0.769117	−0.384558	−	0.923101i	$-0.625646\pi$
$691$	−20.2177	−0.769117	−0.384558	+	0.923101i	$0.625646\pi$
$692$	0	0
$693$	−26.4023	−1.00294
$694$	0	0
$695$	7.99356	0.303213
$696$	0	0
$697$	0.210124	0.00795903
$698$	0	0
$699$	51.0845	1.93219
$700$	0	0
$701$	−13.3108	−0.502743	−0.251372	−	0.967891i	$-0.580882\pi$
$701$	−13.3108	−0.502743	−0.251372	+	0.967891i	$0.580882\pi$
$702$	0	0
$703$	11.2625	0.424772
$704$	0	0
$705$	−7.68288	−0.289354
$706$	0	0
$707$	1.68187	0.0632533
$708$	0	0
$709$	−23.7864	−0.893319	−0.446659	−	0.894704i	$-0.647386\pi$
$709$	−23.7864	−0.893319	−0.446659	+	0.894704i	$0.647386\pi$
$710$	0	0
$711$	−95.5707	−3.58418
$712$	0	0
$713$	−48.8275	−1.82861
$714$	0	0
$715$	−7.89136	−0.295120
$716$	0	0
$717$	51.0626	1.90697
$718$	0	0
$719$	−8.77476	−0.327243	−0.163622	−	0.986523i	$-0.552318\pi$
$719$	−8.77476	−0.327243	−0.163622	+	0.986523i	$0.552318\pi$
$720$	0	0
$721$	−19.1484	−0.713123
$722$	0	0
$723$	−46.2927	−1.72164
$724$	0	0
$725$	−9.06678	−0.336732
$726$	0	0
$727$	−41.1826	−1.52738	−0.763689	−	0.645584i	$-0.776614\pi$
$727$	−41.1826	−1.52738	−0.763689	+	0.645584i	$0.776614\pi$
$728$	0	0
$729$	75.6557	2.80206
$730$	0	0
$731$	−1.65209	−0.0611046
$732$	0	0
$733$	48.2810	1.78330	0.891649	−	0.452727i	$-0.149549\pi$
$733$	48.2810	1.78330	0.891649	+	0.452727i	$0.149549\pi$
$734$	0	0
$735$	44.2280	1.63138
$736$	0	0
$737$	29.6025	1.09042
$738$	0	0
$739$	24.7479	0.910366	0.455183	−	0.890398i	$-0.349574\pi$
$739$	24.7479	0.910366	0.455183	+	0.890398i	$0.349574\pi$
$740$	0	0
$741$	4.05623	0.149009
$742$	0	0
$743$	40.1141	1.47164	0.735822	−	0.677175i	$-0.236796\pi$
$743$	40.1141	1.47164	0.735822	+	0.677175i	$0.236796\pi$
$744$	0	0
$745$	19.0753	0.698865
$746$	0	0
$747$	−35.6756	−1.30530
$748$	0	0
$749$	13.7640	0.502927
$750$	0	0
$751$	31.9595	1.16622	0.583110	−	0.812393i	$-0.301836\pi$
$751$	31.9595	1.16622	0.583110	+	0.812393i	$0.301836\pi$
$752$	0	0
$753$	14.4072	0.525029
$754$	0	0
$755$	46.3036	1.68516
$756$	0	0
$757$	−25.7492	−0.935870	−0.467935	−	0.883763i	$-0.655002\pi$
$757$	−25.7492	−0.935870	−0.467935	+	0.883763i	$0.655002\pi$
$758$	0	0
$759$	67.5043	2.45025
$760$	0	0
$761$	21.4825	0.778740	0.389370	−	0.921082i	$-0.372693\pi$
$761$	21.4825	0.778740	0.389370	+	0.921082i	$0.372693\pi$
$762$	0	0
$763$	−20.3594	−0.737061
$764$	0	0
$765$	−4.25873	−0.153975
$766$	0	0
$767$	−2.51527	−0.0908211
$768$	0	0
$769$	13.9749	0.503949	0.251975	−	0.967734i	$-0.418920\pi$
$769$	13.9749	0.503949	0.251975	+	0.967734i	$0.418920\pi$
$770$	0	0
$771$	70.7762	2.54894
$772$	0	0
$773$	−15.8238	−0.569143	−0.284572	−	0.958655i	$-0.591851\pi$
$773$	−15.8238	−0.569143	−0.284572	+	0.958655i	$0.591851\pi$
$774$	0	0
$775$	−9.12960	−0.327945
$776$	0	0
$777$	−49.3049	−1.76880
$778$	0	0
$779$	0.996870	0.0357166
$780$	0	0
$781$	30.1966	1.08052
$782$	0	0
$783$	−96.9933	−3.46626
$784$	0	0
$785$	17.0890	0.609933
$786$	0	0
$787$	−24.5418	−0.874820	−0.437410	−	0.899262i	$-0.644104\pi$
$787$	−24.5418	−0.874820	−0.437410	+	0.899262i	$0.644104\pi$
$788$	0	0
$789$	−17.7528	−0.632015
$790$	0	0
$791$	−21.5348	−0.765690
$792$	0	0
$793$	12.3725	0.439360
$794$	0	0
$795$	−28.0372	−0.994378
$796$	0	0
$797$	−48.3819	−1.71377	−0.856887	−	0.515505i	$-0.827604\pi$
$797$	−48.3819	−1.71377	−0.856887	+	0.515505i	$0.827604\pi$
$798$	0	0
$799$	0.192011	0.00679287
$800$	0	0
$801$	−52.5932	−1.85829
$802$	0	0
$803$	32.7378	1.15529
$804$	0	0
$805$	27.4577	0.967758
$806$	0	0
$807$	−33.2623	−1.17089
$808$	0	0
$809$	13.1294	0.461604	0.230802	−	0.973001i	$-0.425865\pi$
$809$	13.1294	0.461604	0.230802	+	0.973001i	$0.425865\pi$
$810$	0	0
$811$	−3.76569	−0.132231	−0.0661157	−	0.997812i	$-0.521061\pi$
$811$	−3.76569	−0.132231	−0.0661157	+	0.997812i	$0.521061\pi$
$812$	0	0
$813$	−91.2951	−3.20186
$814$	0	0
$815$	−39.9343	−1.39884
$816$	0	0
$817$	−7.83781	−0.274210
$818$	0	0
$819$	−12.8763	−0.449936
$820$	0	0
$821$	−40.5291	−1.41447	−0.707237	−	0.706977i	$-0.750059\pi$
$821$	−40.5291	−1.41447	−0.707237	+	0.706977i	$0.750059\pi$
$822$	0	0
$823$	−13.7979	−0.480964	−0.240482	−	0.970654i	$-0.577305\pi$
$823$	−13.7979	−0.480964	−0.240482	+	0.970654i	$0.577305\pi$
$824$	0	0
$825$	12.6217	0.439432
$826$	0	0
$827$	47.3483	1.64646	0.823230	−	0.567708i	$-0.192170\pi$
$827$	47.3483	1.64646	0.823230	+	0.567708i	$0.192170\pi$
$828$	0	0
$829$	−2.49303	−0.0865865	−0.0432933	−	0.999062i	$-0.513785\pi$
$829$	−2.49303	−0.0865865	−0.0432933	+	0.999062i	$0.513785\pi$
$830$	0	0
$831$	31.4704	1.09170
$832$	0	0
$833$	−1.10535	−0.0382981
$834$	0	0
$835$	−10.8076	−0.374012
$836$	0	0
$837$	−97.6652	−3.37580
$838$	0	0
$839$	−3.94935	−0.136347	−0.0681733	−	0.997673i	$-0.521717\pi$
$839$	−3.94935	−0.136347	−0.0681733	+	0.997673i	$0.521717\pi$
$840$	0	0
$841$	6.69567	0.230885
$842$	0	0
$843$	−91.2360	−3.14233
$844$	0	0
$845$	29.3398	1.00932
$846$	0	0
$847$	6.17769	0.212268
$848$	0	0
$849$	56.2253	1.92965
$850$	0	0
$851$	91.4097	3.13348
$852$	0	0
$853$	47.6335	1.63094	0.815470	−	0.578799i	$-0.196479\pi$
$853$	47.6335	1.63094	0.815470	+	0.578799i	$0.196479\pi$
$854$	0	0
$855$	−20.2042	−0.690969
$856$	0	0
$857$	50.1888	1.71442	0.857209	−	0.514969i	$-0.172196\pi$
$857$	50.1888	1.71442	0.857209	+	0.514969i	$0.172196\pi$
$858$	0	0
$859$	11.0894	0.378366	0.189183	−	0.981942i	$-0.439416\pi$
$859$	11.0894	0.378366	0.189183	+	0.981942i	$0.439416\pi$
$860$	0	0
$861$	−4.36410	−0.148728
$862$	0	0
$863$	37.7550	1.28520	0.642598	−	0.766204i	$-0.277857\pi$
$863$	37.7550	1.28520	0.642598	+	0.766204i	$0.277857\pi$
$864$	0	0
$865$	10.8622	0.369325
$866$	0	0
$867$	−56.0152	−1.90237
$868$	0	0
$869$	−30.4022	−1.03132
$870$	0	0
$871$	14.4371	0.489181
$872$	0	0
$873$	78.5671	2.65909
$874$	0	0
$875$	−11.7812	−0.398277
$876$	0	0
$877$	43.6274	1.47319	0.736596	−	0.676333i	$-0.236432\pi$
$877$	43.6274	1.47319	0.736596	+	0.676333i	$0.236432\pi$
$878$	0	0
$879$	45.9049	1.54833
$880$	0	0
$881$	−15.5145	−0.522698	−0.261349	−	0.965244i	$-0.584167\pi$
$881$	−15.5145	−0.522698	−0.261349	+	0.965244i	$0.584167\pi$
$882$	0	0
$883$	0.166224	0.00559388	0.00279694	−	0.999996i	$-0.499110\pi$
$883$	0.166224	0.00559388	0.00279694	+	0.999996i	$0.499110\pi$
$884$	0	0
$885$	17.2779	0.580790
$886$	0	0
$887$	−9.04708	−0.303771	−0.151886	−	0.988398i	$-0.548535\pi$
$887$	−9.04708	−0.303771	−0.151886	+	0.988398i	$0.548535\pi$
$888$	0	0
$889$	−8.70213	−0.291860
$890$	0	0
$891$	75.2503	2.52098
$892$	0	0
$893$	0.910938	0.0304834
$894$	0	0
$895$	−51.8686	−1.73378
$896$	0	0
$897$	32.9216	1.09922
$898$	0	0
$899$	35.9430	1.19877
$900$	0	0
$901$	0.700709	0.0233440
$902$	0	0
$903$	34.3124	1.14185
$904$	0	0
$905$	42.4887	1.41237
$906$	0	0
$907$	−16.2748	−0.540395	−0.270198	−	0.962805i	$-0.587089\pi$
$907$	−16.2748	−0.540395	−0.270198	+	0.962805i	$0.587089\pi$
$908$	0	0
$909$	−10.0445	−0.333155
$910$	0	0
$911$	11.6990	0.387604	0.193802	−	0.981041i	$-0.437918\pi$
$911$	11.6990	0.387604	0.193802	+	0.981041i	$0.437918\pi$
$912$	0	0
$913$	−11.3488	−0.375592
$914$	0	0
$915$	−84.9891	−2.80965
$916$	0	0
$917$	13.2826	0.438631
$918$	0	0
$919$	−24.0697	−0.793987	−0.396994	−	0.917821i	$-0.629947\pi$
$919$	−24.0697	−0.793987	−0.396994	+	0.917821i	$0.629947\pi$
$920$	0	0
$921$	−28.3224	−0.933254
$922$	0	0
$923$	14.7268	0.484739
$924$	0	0
$925$	17.0915	0.561964
$926$	0	0
$927$	114.358	3.75602
$928$	0	0
$929$	44.5910	1.46298	0.731491	−	0.681851i	$-0.238825\pi$
$929$	44.5910	1.46298	0.731491	+	0.681851i	$0.238825\pi$
$930$	0	0
$931$	−5.24399	−0.171865
$932$	0	0
$933$	−61.9241	−2.02730
$934$	0	0
$935$	−1.35475	−0.0443052
$936$	0	0
$937$	−46.6674	−1.52456	−0.762278	−	0.647250i	$-0.775919\pi$
$937$	−46.6674	−1.52456	−0.762278	+	0.647250i	$0.775919\pi$
$938$	0	0
$939$	−0.883087	−0.0288185
$940$	0	0
$941$	−38.4624	−1.25384	−0.626919	−	0.779085i	$-0.715684\pi$
$941$	−38.4624	−1.25384	−0.626919	+	0.779085i	$0.715684\pi$
$942$	0	0
$943$	8.09090	0.263476
$944$	0	0
$945$	54.9212	1.78659
$946$	0	0
$947$	−28.6521	−0.931068	−0.465534	−	0.885030i	$-0.654138\pi$
$947$	−28.6521	−0.931068	−0.465534	+	0.885030i	$0.654138\pi$
$948$	0	0
$949$	15.9661	0.518282
$950$	0	0
$951$	80.5642	2.61247
$952$	0	0
$953$	−55.7585	−1.80619	−0.903097	−	0.429437i	$-0.858712\pi$
$953$	−55.7585	−1.80619	−0.903097	+	0.429437i	$0.858712\pi$
$954$	0	0
$955$	40.6547	1.31556
$956$	0	0
$957$	−49.6913	−1.60629
$958$	0	0
$959$	9.28934	0.299968
$960$	0	0
$961$	5.19198	0.167483
$962$	0	0
$963$	−82.2019	−2.64892
$964$	0	0
$965$	66.0606	2.12657
$966$	0	0
$967$	1.84686	0.0593909	0.0296954	−	0.999559i	$-0.490546\pi$
$967$	1.84686	0.0593909	0.0296954	+	0.999559i	$0.490546\pi$
$968$	0	0
$969$	0.696356	0.0223702
$970$	0	0
$971$	6.95804	0.223294	0.111647	−	0.993748i	$-0.464387\pi$
$971$	6.95804	0.223294	0.111647	+	0.993748i	$0.464387\pi$
$972$	0	0
$973$	4.14917	0.133016
$974$	0	0
$975$	6.15557	0.197136
$976$	0	0
$977$	−2.75699	−0.0882039	−0.0441020	−	0.999027i	$-0.514043\pi$
$977$	−2.75699	−0.0882039	−0.0441020	+	0.999027i	$0.514043\pi$
$978$	0	0
$979$	−16.7305	−0.534710
$980$	0	0
$981$	121.591	3.88211
$982$	0	0
$983$	−42.9562	−1.37009	−0.685045	−	0.728500i	$-0.740218\pi$
$983$	−42.9562	−1.37009	−0.685045	+	0.728500i	$0.740218\pi$
$984$	0	0
$985$	−46.6807	−1.48737
$986$	0	0
$987$	−3.98791	−0.126937
$988$	0	0
$989$	−63.6141	−2.02281
$990$	0	0
$991$	45.1884	1.43546	0.717729	−	0.696323i	$-0.245182\pi$
$991$	45.1884	1.43546	0.717729	+	0.696323i	$0.245182\pi$
$992$	0	0
$993$	52.5208	1.66670
$994$	0	0
$995$	−25.7357	−0.815876
$996$	0	0
$997$	−33.7095	−1.06759	−0.533795	−	0.845614i	$-0.679234\pi$
$997$	−33.7095	−1.06759	−0.533795	+	0.845614i	$0.679234\pi$
$998$	0	0
$999$	182.838	5.78475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bt.1.9		10
4.3	odd	2		4864.2.a.bs.1.1		10
8.3	odd	2	inner	4864.2.a.bt.1.10		10
8.5	even	2		4864.2.a.bs.1.2		10
16.3	odd	4		2432.2.c.j.1217.2	yes	20
16.5	even	4		2432.2.c.j.1217.1	✓	20
16.11	odd	4		2432.2.c.j.1217.19	yes	20
16.13	even	4		2432.2.c.j.1217.20	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.j.1217.1	✓	20	16.5	even	4
2432.2.c.j.1217.2	yes	20	16.3	odd	4
2432.2.c.j.1217.19	yes	20	16.11	odd	4
2432.2.c.j.1217.20	yes	20	16.13	even	4
4864.2.a.bs.1.1		10	4.3	odd	2
4864.2.a.bs.1.2		10	8.5	even	2
4864.2.a.bt.1.9		10	1.1	even	1	trivial
4864.2.a.bt.1.10		10	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q+3.30364 q^{3} -2.55295 q^{5} -1.32515 q^{7} +7.91406 q^{9} +2.51756 q^{11} +1.22781 q^{13} -8.43404 q^{15} +0.210784 q^{17} +1.00000 q^{19} -4.37781 q^{21} +8.11631 q^{23} +1.51756 q^{25} +16.2343 q^{27} -5.97458 q^{29} -6.01598 q^{31} +8.31712 q^{33} +3.38303 q^{35} +11.2625 q^{37} +4.05623 q^{39} +0.996870 q^{41} -7.83781 q^{43} -20.2042 q^{45} +0.910938 q^{47} -5.24399 q^{49} +0.696356 q^{51} +3.32429 q^{53} -6.42720 q^{55} +3.30364 q^{57} -2.04859 q^{59} +10.0769 q^{61} -10.4873 q^{63} -3.13453 q^{65} +11.7584 q^{67} +26.8134 q^{69} +11.9944 q^{71} +13.0038 q^{73} +5.01347 q^{75} -3.33613 q^{77} -12.0761 q^{79} +29.8902 q^{81} -4.50788 q^{83} -0.538122 q^{85} -19.7379 q^{87} -6.64553 q^{89} -1.62702 q^{91} -19.8746 q^{93} -2.55295 q^{95} +9.92753 q^{97} +19.9241 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{3} + 14 q^{9} + 20 q^{11} + 4 q^{17} + 10 q^{19} + 10 q^{25} + 28 q^{27} - 8 q^{33} + 36 q^{35} - 12 q^{41} - 4 q^{43} + 26 q^{49} + 36 q^{51} + 4 q^{57} + 52 q^{59} - 24 q^{65} + 12 q^{67}+ \cdots + 60 q^{99}+O(q^{100}) \) 10 * q + 4 * q^3 + 14 * q^9 + 20 * q^11 + 4 * q^17 + 10 * q^19 + 10 * q^25 + 28 * q^27 - 8 * q^33 + 36 * q^35 - 12 * q^41 - 4 * q^43 + 26 * q^49 + 36 * q^51 + 4 * q^57 + 52 * q^59 - 24 * q^65 + 12 * q^67 + 12 * q^73 - 12 * q^75 + 34 * q^81 + 16 * q^83 - 20 * q^89 + 60 * q^91 - 28 * q^97 + 60 * q^99

Introduction

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