LMFDB - Embedded newform 4840.2.a.bg.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4840,2,Mod(1,4840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,1,0,8,0,-6,0,19,0,0,0,12] 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:	$38.6475945783$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 $ x^8 - x^7 - 21x^6 + 15x^5 + 140x^4 - 60x^3 - 295x^2 + 50x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.713352$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.713352 q^{3} +1.00000 q^{5} +0.376172 q^{7} -2.49113 q^{9} -2.82831 q^{13} +0.713352 q^{15} -4.83768 q^{17} +2.92990 q^{19} +0.268343 q^{21} +3.77226 q^{23} +1.00000 q^{25} -3.91711 q^{27} +8.44396 q^{29} +8.04011 q^{31} +0.376172 q^{35} +2.83990 q^{37} -2.01758 q^{39} -3.72034 q^{41} -6.48484 q^{43} -2.49113 q^{45} +2.58897 q^{47} -6.85849 q^{49} -3.45097 q^{51} -0.0487626 q^{53} +2.09005 q^{57} -1.64929 q^{59} +8.69705 q^{61} -0.937093 q^{63} -2.82831 q^{65} +11.4395 q^{67} +2.69095 q^{69} +14.1216 q^{71} +0.513225 q^{73} +0.713352 q^{75} +12.9823 q^{79} +4.67911 q^{81} +2.73687 q^{83} -4.83768 q^{85} +6.02351 q^{87} +12.0195 q^{89} -1.06393 q^{91} +5.73543 q^{93} +2.92990 q^{95} +16.0745 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} + 12 q^{13} + q^{15} + 2 q^{17} - 6 q^{19} + 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} + 8 q^{29} + 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} - 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) $ 8 * q + q^3 + 8 * q^5 - 6 * q^7 + 19 * q^9 + 12 * q^13 + q^15 + 2 * q^17 - 6 * q^19 + 6 * q^21 + 10 * q^23 + 8 * q^25 + 13 * q^27 + 8 * q^29 + 19 * q^31 - 6 * q^35 + 12 * q^37 - 21 * q^39 - 3 * q^41 - 8 * q^43 + 19 * q^45 + 10 * q^47 + 22 * q^49 - 7 * q^51 + 28 * q^53 + 25 * q^57 + 25 * q^59 + 10 * q^61 - 64 * q^63 + 12 * q^65 - 2 * q^67 + 18 * q^69 + 25 * q^71 + 38 * q^73 + q^75 - 38 * q^79 + 32 * q^81 - 28 * q^83 + 2 * q^85 + 15 * q^87 + 12 * q^89 + 8 * q^91 - 15 * q^93 - 6 * q^95 + 21 * 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
$3$	0.713352	0.411854	0.205927	−	0.978567i	$-0.433979\pi$
$3$	0.713352	0.411854	0.205927	+	0.978567i	$0.433979\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.376172	0.142180	0.0710899	−	0.997470i	$-0.477352\pi$
$7$	0.376172	0.142180	0.0710899	+	0.997470i	$0.477352\pi$
$8$	0	0
$9$	−2.49113	−0.830376
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.82831	−0.784432	−0.392216	−	0.919873i	$-0.628291\pi$
$13$	−2.82831	−0.784432	−0.392216	+	0.919873i	$0.628291\pi$
$14$	0	0
$15$	0.713352	0.184187
$16$	0	0
$17$	−4.83768	−1.17331	−0.586655	−	0.809837i	$-0.699556\pi$
$17$	−4.83768	−1.17331	−0.586655	+	0.809837i	$0.699556\pi$
$18$	0	0
$19$	2.92990	0.672165	0.336082	−	0.941833i	$-0.390898\pi$
$19$	2.92990	0.672165	0.336082	+	0.941833i	$0.390898\pi$
$20$	0	0
$21$	0.268343	0.0585573
$22$	0	0
$23$	3.77226	0.786571	0.393285	−	0.919416i	$-0.371338\pi$
$23$	3.77226	0.786571	0.393285	+	0.919416i	$0.371338\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.91711	−0.753848
$28$	0	0
$29$	8.44396	1.56800	0.784002	−	0.620759i	$-0.213175\pi$
$29$	8.44396	1.56800	0.784002	+	0.620759i	$0.213175\pi$
$30$	0	0
$31$	8.04011	1.44405	0.722023	−	0.691869i	$-0.243213\pi$
$31$	8.04011	1.44405	0.722023	+	0.691869i	$0.243213\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.376172	0.0635847
$36$	0	0
$37$	2.83990	0.466877	0.233438	−	0.972372i	$-0.425002\pi$
$37$	2.83990	0.466877	0.233438	+	0.972372i	$0.425002\pi$
$38$	0	0
$39$	−2.01758	−0.323071
$40$	0	0
$41$	−3.72034	−0.581020	−0.290510	−	0.956872i	$-0.593825\pi$
$41$	−3.72034	−0.581020	−0.290510	+	0.956872i	$0.593825\pi$
$42$	0	0
$43$	−6.48484	−0.988929	−0.494465	−	0.869198i	$-0.664636\pi$
$43$	−6.48484	−0.988929	−0.494465	+	0.869198i	$0.664636\pi$
$44$	0	0
$45$	−2.49113	−0.371356
$46$	0	0
$47$	2.58897	0.377639	0.188820	−	0.982012i	$-0.439534\pi$
$47$	2.58897	0.377639	0.188820	+	0.982012i	$0.439534\pi$
$48$	0	0
$49$	−6.85849	−0.979785
$50$	0	0
$51$	−3.45097	−0.483233
$52$	0	0
$53$	−0.0487626	−0.00669805	−0.00334903	−	0.999994i	$-0.501066\pi$
$53$	−0.0487626	−0.00669805	−0.00334903	+	0.999994i	$0.501066\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.09005	0.276834
$58$	0	0
$59$	−1.64929	−0.214720	−0.107360	−	0.994220i	$-0.534240\pi$
$59$	−1.64929	−0.214720	−0.107360	+	0.994220i	$0.534240\pi$
$60$	0	0
$61$	8.69705	1.11354	0.556772	−	0.830666i	$-0.312040\pi$
$61$	8.69705	1.11354	0.556772	+	0.830666i	$0.312040\pi$
$62$	0	0
$63$	−0.937093	−0.118063
$64$	0	0
$65$	−2.82831	−0.350809
$66$	0	0
$67$	11.4395	1.39756	0.698779	−	0.715338i	$-0.253727\pi$
$67$	11.4395	1.39756	0.698779	+	0.715338i	$0.253727\pi$
$68$	0	0
$69$	2.69095	0.323952
$70$	0	0
$71$	14.1216	1.67592	0.837962	−	0.545728i	$-0.183747\pi$
$71$	14.1216	1.67592	0.837962	+	0.545728i	$0.183747\pi$
$72$	0	0
$73$	0.513225	0.0600684	0.0300342	−	0.999549i	$-0.490438\pi$
$73$	0.513225	0.0600684	0.0300342	+	0.999549i	$0.490438\pi$
$74$	0	0
$75$	0.713352	0.0823708
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.9823	1.46063	0.730313	−	0.683113i	$-0.239374\pi$
$79$	12.9823	1.46063	0.730313	+	0.683113i	$0.239374\pi$
$80$	0	0
$81$	4.67911	0.519901
$82$	0	0
$83$	2.73687	0.300411	0.150205	−	0.988655i	$-0.452007\pi$
$83$	2.73687	0.300411	0.150205	+	0.988655i	$0.452007\pi$
$84$	0	0
$85$	−4.83768	−0.524720
$86$	0	0
$87$	6.02351	0.645789
$88$	0	0
$89$	12.0195	1.27407	0.637034	−	0.770835i	$-0.280161\pi$
$89$	12.0195	1.27407	0.637034	+	0.770835i	$0.280161\pi$
$90$	0	0
$91$	−1.06393	−0.111530
$92$	0	0
$93$	5.73543	0.594736
$94$	0	0
$95$	2.92990	0.300601
$96$	0	0
$97$	16.0745	1.63212	0.816060	−	0.577967i	$-0.196154\pi$
$97$	16.0745	1.63212	0.816060	+	0.577967i	$0.196154\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.5883	1.55109	0.775546	−	0.631291i	$-0.217475\pi$
$101$	15.5883	1.55109	0.775546	+	0.631291i	$0.217475\pi$
$102$	0	0
$103$	−9.25907	−0.912323	−0.456161	−	0.889897i	$-0.650776\pi$
$103$	−9.25907	−0.912323	−0.456161	+	0.889897i	$0.650776\pi$
$104$	0	0
$105$	0.268343	0.0261876
$106$	0	0
$107$	−2.15917	−0.208735	−0.104368	−	0.994539i	$-0.533282\pi$
$107$	−2.15917	−0.208735	−0.104368	+	0.994539i	$0.533282\pi$
$108$	0	0
$109$	−5.79052	−0.554631	−0.277315	−	0.960779i	$-0.589445\pi$
$109$	−5.79052	−0.554631	−0.277315	+	0.960779i	$0.589445\pi$
$110$	0	0
$111$	2.02585	0.192285
$112$	0	0
$113$	19.0148	1.78876	0.894381	−	0.447306i	$-0.147616\pi$
$113$	19.0148	1.78876	0.894381	+	0.447306i	$0.147616\pi$
$114$	0	0
$115$	3.77226	0.351765
$116$	0	0
$117$	7.04568	0.651373
$118$	0	0
$119$	−1.81980	−0.166821
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−2.65391	−0.239295
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.0973	−0.895993	−0.447996	−	0.894035i	$-0.647862\pi$
$127$	−10.0973	−0.895993	−0.447996	+	0.894035i	$0.647862\pi$
$128$	0	0
$129$	−4.62598	−0.407294
$130$	0	0
$131$	−6.29651	−0.550128	−0.275064	−	0.961426i	$-0.588699\pi$
$131$	−6.29651	−0.550128	−0.275064	+	0.961426i	$0.588699\pi$
$132$	0	0
$133$	1.10215	0.0955682
$134$	0	0
$135$	−3.91711	−0.337131
$136$	0	0
$137$	−8.27329	−0.706835	−0.353418	−	0.935466i	$-0.614981\pi$
$137$	−8.27329	−0.706835	−0.353418	+	0.935466i	$0.614981\pi$
$138$	0	0
$139$	−10.2671	−0.870843	−0.435422	−	0.900227i	$-0.643401\pi$
$139$	−10.2671	−0.870843	−0.435422	+	0.900227i	$0.643401\pi$
$140$	0	0
$141$	1.84684	0.155532
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.44396	0.701232
$146$	0	0
$147$	−4.89252	−0.403528
$148$	0	0
$149$	−4.90160	−0.401554	−0.200777	−	0.979637i	$-0.564347\pi$
$149$	−4.90160	−0.401554	−0.200777	+	0.979637i	$0.564347\pi$
$150$	0	0
$151$	−9.35625	−0.761401	−0.380701	−	0.924698i	$-0.624317\pi$
$151$	−9.35625	−0.761401	−0.380701	+	0.924698i	$0.624317\pi$
$152$	0	0
$153$	12.0513	0.974289
$154$	0	0
$155$	8.04011	0.645797
$156$	0	0
$157$	9.65817	0.770806	0.385403	−	0.922748i	$-0.374062\pi$
$157$	9.65817	0.770806	0.385403	+	0.922748i	$0.374062\pi$
$158$	0	0
$159$	−0.0347849	−0.00275862
$160$	0	0
$161$	1.41902	0.111834
$162$	0	0
$163$	4.48802	0.351529	0.175764	−	0.984432i	$-0.443760\pi$
$163$	4.48802	0.351529	0.175764	+	0.984432i	$0.443760\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.4716	−1.81628	−0.908142	−	0.418662i	$-0.862499\pi$
$167$	−23.4716	−1.81628	−0.908142	+	0.418662i	$0.862499\pi$
$168$	0	0
$169$	−5.00067	−0.384667
$170$	0	0
$171$	−7.29875	−0.558150
$172$	0	0
$173$	−12.9509	−0.984637	−0.492318	−	0.870415i	$-0.663850\pi$
$173$	−12.9509	−0.984637	−0.492318	+	0.870415i	$0.663850\pi$
$174$	0	0
$175$	0.376172	0.0284359
$176$	0	0
$177$	−1.17653	−0.0884332
$178$	0	0
$179$	−3.52933	−0.263795	−0.131897	−	0.991263i	$-0.542107\pi$
$179$	−3.52933	−0.263795	−0.131897	+	0.991263i	$0.542107\pi$
$180$	0	0
$181$	1.79178	0.133182	0.0665910	−	0.997780i	$-0.478788\pi$
$181$	1.79178	0.133182	0.0665910	+	0.997780i	$0.478788\pi$
$182$	0	0
$183$	6.20406	0.458617
$184$	0	0
$185$	2.83990	0.208794
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.47351	−0.107182
$190$	0	0
$191$	3.40113	0.246097	0.123048	−	0.992401i	$-0.460733\pi$
$191$	3.40113	0.246097	0.123048	+	0.992401i	$0.460733\pi$
$192$	0	0
$193$	−7.77543	−0.559688	−0.279844	−	0.960046i	$-0.590283\pi$
$193$	−7.77543	−0.559688	−0.279844	+	0.960046i	$0.590283\pi$
$194$	0	0
$195$	−2.01758	−0.144482
$196$	0	0
$197$	−7.09410	−0.505434	−0.252717	−	0.967540i	$-0.581324\pi$
$197$	−7.09410	−0.505434	−0.252717	+	0.967540i	$0.581324\pi$
$198$	0	0
$199$	12.7734	0.905481	0.452740	−	0.891642i	$-0.350446\pi$
$199$	12.7734	0.905481	0.452740	+	0.891642i	$0.350446\pi$
$200$	0	0
$201$	8.16039	0.575590
$202$	0	0
$203$	3.17638	0.222938
$204$	0	0
$205$	−3.72034	−0.259840
$206$	0	0
$207$	−9.39719	−0.653150
$208$	0	0
$209$	0	0
$210$	0	0
$211$	7.34047	0.505339	0.252669	−	0.967553i	$-0.418692\pi$
$211$	7.34047	0.505339	0.252669	+	0.967553i	$0.418692\pi$
$212$	0	0
$213$	10.0737	0.690236
$214$	0	0
$215$	−6.48484	−0.442263
$216$	0	0
$217$	3.02446	0.205314
$218$	0	0
$219$	0.366110	0.0247394
$220$	0	0
$221$	13.6825	0.920382
$222$	0	0
$223$	6.53875	0.437867	0.218934	−	0.975740i	$-0.429742\pi$
$223$	6.53875	0.437867	0.218934	+	0.975740i	$0.429742\pi$
$224$	0	0
$225$	−2.49113	−0.166075
$226$	0	0
$227$	20.5461	1.36369	0.681847	−	0.731495i	$-0.261177\pi$
$227$	20.5461	1.36369	0.681847	+	0.731495i	$0.261177\pi$
$228$	0	0
$229$	−4.33094	−0.286197	−0.143098	−	0.989708i	$-0.545706\pi$
$229$	−4.33094	−0.286197	−0.143098	+	0.989708i	$0.545706\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.6595	1.87755	0.938774	−	0.344533i	$-0.111963\pi$
$233$	28.6595	1.87755	0.938774	+	0.344533i	$0.111963\pi$
$234$	0	0
$235$	2.58897	0.168886
$236$	0	0
$237$	9.26097	0.601565
$238$	0	0
$239$	−10.4766	−0.677677	−0.338838	−	0.940845i	$-0.610034\pi$
$239$	−10.4766	−0.677677	−0.338838	+	0.940845i	$0.610034\pi$
$240$	0	0
$241$	15.5202	0.999745	0.499873	−	0.866099i	$-0.333380\pi$
$241$	15.5202	0.999745	0.499873	+	0.866099i	$0.333380\pi$
$242$	0	0
$243$	15.0892	0.967971
$244$	0	0
$245$	−6.85849	−0.438173
$246$	0	0
$247$	−8.28666	−0.527267
$248$	0	0
$249$	1.95235	0.123725
$250$	0	0
$251$	−11.9701	−0.755547	−0.377774	−	0.925898i	$-0.623310\pi$
$251$	−11.9701	−0.755547	−0.377774	+	0.925898i	$0.623310\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.45097	−0.216108
$256$	0	0
$257$	−8.47342	−0.528557	−0.264279	−	0.964446i	$-0.585134\pi$
$257$	−8.47342	−0.528557	−0.264279	+	0.964446i	$0.585134\pi$
$258$	0	0
$259$	1.06829	0.0663804
$260$	0	0
$261$	−21.0350	−1.30203
$262$	0	0
$263$	12.5189	0.771950	0.385975	−	0.922509i	$-0.373865\pi$
$263$	12.5189	0.771950	0.385975	+	0.922509i	$0.373865\pi$
$264$	0	0
$265$	−0.0487626	−0.00299546
$266$	0	0
$267$	8.57416	0.524730
$268$	0	0
$269$	−24.0649	−1.46726	−0.733630	−	0.679549i	$-0.762176\pi$
$269$	−24.0649	−1.46726	−0.733630	+	0.679549i	$0.762176\pi$
$270$	0	0
$271$	−9.42221	−0.572358	−0.286179	−	0.958176i	$-0.592385\pi$
$271$	−9.42221	−0.572358	−0.286179	+	0.958176i	$0.592385\pi$
$272$	0	0
$273$	−0.758957	−0.0459342
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.7811	1.42887	0.714434	−	0.699703i	$-0.246684\pi$
$277$	23.7811	1.42887	0.714434	+	0.699703i	$0.246684\pi$
$278$	0	0
$279$	−20.0289	−1.19910
$280$	0	0
$281$	0.496904	0.0296428	0.0148214	−	0.999890i	$-0.495282\pi$
$281$	0.496904	0.0296428	0.0148214	+	0.999890i	$0.495282\pi$
$282$	0	0
$283$	4.68030	0.278215	0.139108	−	0.990277i	$-0.455577\pi$
$283$	4.68030	0.278215	0.139108	+	0.990277i	$0.455577\pi$
$284$	0	0
$285$	2.09005	0.123804
$286$	0	0
$287$	−1.39949	−0.0826092
$288$	0	0
$289$	6.40318	0.376658
$290$	0	0
$291$	11.4668	0.672195
$292$	0	0
$293$	16.9203	0.988493	0.494246	−	0.869322i	$-0.335444\pi$
$293$	16.9203	0.988493	0.494246	+	0.869322i	$0.335444\pi$
$294$	0	0
$295$	−1.64929	−0.0960256
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−10.6691	−0.617011
$300$	0	0
$301$	−2.43942	−0.140606
$302$	0	0
$303$	11.1199	0.638824
$304$	0	0
$305$	8.69705	0.497992
$306$	0	0
$307$	−34.3667	−1.96141	−0.980707	−	0.195486i	$-0.937372\pi$
$307$	−34.3667	−1.96141	−0.980707	+	0.195486i	$0.937372\pi$
$308$	0	0
$309$	−6.60497	−0.375744
$310$	0	0
$311$	7.36790	0.417796	0.208898	−	0.977937i	$-0.433012\pi$
$311$	7.36790	0.417796	0.208898	+	0.977937i	$0.433012\pi$
$312$	0	0
$313$	−25.0606	−1.41651	−0.708253	−	0.705958i	$-0.750517\pi$
$313$	−25.0606	−1.41651	−0.708253	+	0.705958i	$0.750517\pi$
$314$	0	0
$315$	−0.937093	−0.0527992
$316$	0	0
$317$	28.0728	1.57673	0.788364	−	0.615209i	$-0.210929\pi$
$317$	28.0728	1.57673	0.788364	+	0.615209i	$0.210929\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.54025	−0.0859685
$322$	0	0
$323$	−14.1739	−0.788658
$324$	0	0
$325$	−2.82831	−0.156886
$326$	0	0
$327$	−4.13068	−0.228427
$328$	0	0
$329$	0.973897	0.0536927
$330$	0	0
$331$	21.9644	1.20727	0.603637	−	0.797259i	$-0.293718\pi$
$331$	21.9644	1.20727	0.603637	+	0.797259i	$0.293718\pi$
$332$	0	0
$333$	−7.07456	−0.387683
$334$	0	0
$335$	11.4395	0.625007
$336$	0	0
$337$	22.3799	1.21911	0.609555	−	0.792744i	$-0.291348\pi$
$337$	22.3799	1.21911	0.609555	+	0.792744i	$0.291348\pi$
$338$	0	0
$339$	13.5642	0.736709
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−5.21318	−0.281485
$344$	0	0
$345$	2.69095	0.144876
$346$	0	0
$347$	−24.7788	−1.33020	−0.665098	−	0.746756i	$-0.731610\pi$
$347$	−24.7788	−1.33020	−0.665098	+	0.746756i	$0.731610\pi$
$348$	0	0
$349$	−22.3282	−1.19520	−0.597600	−	0.801795i	$-0.703879\pi$
$349$	−22.3282	−1.19520	−0.597600	+	0.801795i	$0.703879\pi$
$350$	0	0
$351$	11.0788	0.591342
$352$	0	0
$353$	−5.79232	−0.308294	−0.154147	−	0.988048i	$-0.549263\pi$
$353$	−5.79232	−0.308294	−0.154147	+	0.988048i	$0.549263\pi$
$354$	0	0
$355$	14.1216	0.749496
$356$	0	0
$357$	−1.29816	−0.0687059
$358$	0	0
$359$	21.0367	1.11027	0.555137	−	0.831759i	$-0.312666\pi$
$359$	21.0367	1.11027	0.555137	+	0.831759i	$0.312666\pi$
$360$	0	0
$361$	−10.4157	−0.548195
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.513225	0.0268634
$366$	0	0
$367$	−23.6235	−1.23314	−0.616569	−	0.787301i	$-0.711478\pi$
$367$	−23.6235	−1.23314	−0.616569	+	0.787301i	$0.711478\pi$
$368$	0	0
$369$	9.26785	0.482465
$370$	0	0
$371$	−0.0183431	−0.000952327 0
$372$	0	0
$373$	−12.7740	−0.661411	−0.330705	−	0.943734i	$-0.607287\pi$
$373$	−12.7740	−0.661411	−0.330705	+	0.943734i	$0.607287\pi$
$374$	0	0
$375$	0.713352	0.0368373
$376$	0	0
$377$	−23.8821	−1.22999
$378$	0	0
$379$	24.7773	1.27273	0.636363	−	0.771390i	$-0.280438\pi$
$379$	24.7773	1.27273	0.636363	+	0.771390i	$0.280438\pi$
$380$	0	0
$381$	−7.20295	−0.369018
$382$	0	0
$383$	38.1710	1.95045	0.975223	−	0.221223i	$-0.0710048\pi$
$383$	38.1710	1.95045	0.975223	+	0.221223i	$0.0710048\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.1546	0.821183
$388$	0	0
$389$	2.40973	0.122178	0.0610891	−	0.998132i	$-0.480543\pi$
$389$	2.40973	0.122178	0.0610891	+	0.998132i	$0.480543\pi$
$390$	0	0
$391$	−18.2490	−0.922892
$392$	0	0
$393$	−4.49163	−0.226573
$394$	0	0
$395$	12.9823	0.653212
$396$	0	0
$397$	−11.5456	−0.579459	−0.289729	−	0.957109i	$-0.593565\pi$
$397$	−11.5456	−0.579459	−0.289729	+	0.957109i	$0.593565\pi$
$398$	0	0
$399$	0.786218	0.0393601
$400$	0	0
$401$	25.6006	1.27844	0.639218	−	0.769026i	$-0.279258\pi$
$401$	25.6006	1.27844	0.639218	+	0.769026i	$0.279258\pi$
$402$	0	0
$403$	−22.7399	−1.13276
$404$	0	0
$405$	4.67911	0.232507
$406$	0	0
$407$	0	0
$408$	0	0
$409$	9.36369	0.463005	0.231502	−	0.972834i	$-0.425636\pi$
$409$	9.36369	0.463005	0.231502	+	0.972834i	$0.425636\pi$
$410$	0	0
$411$	−5.90177	−0.291113
$412$	0	0
$413$	−0.620419	−0.0305288
$414$	0	0
$415$	2.73687	0.134348
$416$	0	0
$417$	−7.32405	−0.358660
$418$	0	0
$419$	−21.3716	−1.04407	−0.522035	−	0.852924i	$-0.674827\pi$
$419$	−21.3716	−1.04407	−0.522035	+	0.852924i	$0.674827\pi$
$420$	0	0
$421$	27.5205	1.34126	0.670632	−	0.741790i	$-0.266023\pi$
$421$	27.5205	1.34126	0.670632	+	0.741790i	$0.266023\pi$
$422$	0	0
$423$	−6.44945	−0.313583
$424$	0	0
$425$	−4.83768	−0.234662
$426$	0	0
$427$	3.27159	0.158323
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.63236	0.126796	0.0633982	−	0.997988i	$-0.479806\pi$
$431$	2.63236	0.126796	0.0633982	+	0.997988i	$0.479806\pi$
$432$	0	0
$433$	9.99044	0.480110	0.240055	−	0.970759i	$-0.422835\pi$
$433$	9.99044	0.480110	0.240055	+	0.970759i	$0.422835\pi$
$434$	0	0
$435$	6.02351	0.288805
$436$	0	0
$437$	11.0523	0.528705
$438$	0	0
$439$	2.85957	0.136480	0.0682400	−	0.997669i	$-0.478262\pi$
$439$	2.85957	0.136480	0.0682400	+	0.997669i	$0.478262\pi$
$440$	0	0
$441$	17.0854	0.813590
$442$	0	0
$443$	−40.3056	−1.91498	−0.957489	−	0.288470i	$-0.906853\pi$
$443$	−40.3056	−1.91498	−0.957489	+	0.288470i	$0.906853\pi$
$444$	0	0
$445$	12.0195	0.569781
$446$	0	0
$447$	−3.49656	−0.165382
$448$	0	0
$449$	−1.95041	−0.0920457	−0.0460228	−	0.998940i	$-0.514655\pi$
$449$	−1.95041	−0.0920457	−0.0460228	+	0.998940i	$0.514655\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−6.67430	−0.313586
$454$	0	0
$455$	−1.06393	−0.0498779
$456$	0	0
$457$	35.7765	1.67355	0.836777	−	0.547543i	$-0.184437\pi$
$457$	35.7765	1.67355	0.836777	+	0.547543i	$0.184437\pi$
$458$	0	0
$459$	18.9497	0.884498
$460$	0	0
$461$	15.1433	0.705295	0.352648	−	0.935756i	$-0.385281\pi$
$461$	15.1433	0.705295	0.352648	+	0.935756i	$0.385281\pi$
$462$	0	0
$463$	−12.6301	−0.586972	−0.293486	−	0.955963i	$-0.594815\pi$
$463$	−12.6301	−0.586972	−0.293486	+	0.955963i	$0.594815\pi$
$464$	0	0
$465$	5.73543	0.265974
$466$	0	0
$467$	31.3546	1.45092	0.725460	−	0.688264i	$-0.241627\pi$
$467$	31.3546	1.45092	0.725460	+	0.688264i	$0.241627\pi$
$468$	0	0
$469$	4.30322	0.198704
$470$	0	0
$471$	6.88968	0.317460
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.92990	0.134433
$476$	0	0
$477$	0.121474	0.00556190
$478$	0	0
$479$	−12.3355	−0.563625	−0.281812	−	0.959470i	$-0.590936\pi$
$479$	−12.3355	−0.563625	−0.281812	+	0.959470i	$0.590936\pi$
$480$	0	0
$481$	−8.03212	−0.366233
$482$	0	0
$483$	1.01226	0.0460595
$484$	0	0
$485$	16.0745	0.729906
$486$	0	0
$487$	−21.9102	−0.992847	−0.496424	−	0.868080i	$-0.665354\pi$
$487$	−21.9102	−0.992847	−0.496424	+	0.868080i	$0.665354\pi$
$488$	0	0
$489$	3.20154	0.144779
$490$	0	0
$491$	−27.3534	−1.23444	−0.617220	−	0.786790i	$-0.711741\pi$
$491$	−27.3534	−1.23444	−0.617220	+	0.786790i	$0.711741\pi$
$492$	0	0
$493$	−40.8492	−1.83976
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.31215	0.238282
$498$	0	0
$499$	−10.3980	−0.465478	−0.232739	−	0.972539i	$-0.574769\pi$
$499$	−10.3980	−0.465478	−0.232739	+	0.972539i	$0.574769\pi$
$500$	0	0
$501$	−16.7435	−0.748044
$502$	0	0
$503$	−10.3868	−0.463123	−0.231561	−	0.972820i	$-0.574383\pi$
$503$	−10.3868	−0.463123	−0.231561	+	0.972820i	$0.574383\pi$
$504$	0	0
$505$	15.5883	0.693670
$506$	0	0
$507$	−3.56724	−0.158427
$508$	0	0
$509$	23.0777	1.02290	0.511450	−	0.859313i	$-0.329109\pi$
$509$	23.0777	1.02290	0.511450	+	0.859313i	$0.329109\pi$
$510$	0	0
$511$	0.193061	0.00854051
$512$	0	0
$513$	−11.4767	−0.506710
$514$	0	0
$515$	−9.25907	−0.408003
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−9.23854	−0.405527
$520$	0	0
$521$	35.4191	1.55174	0.775868	−	0.630895i	$-0.217312\pi$
$521$	35.4191	1.55174	0.775868	+	0.630895i	$0.217312\pi$
$522$	0	0
$523$	−18.1523	−0.793745	−0.396872	−	0.917874i	$-0.629904\pi$
$523$	−18.1523	−0.793745	−0.396872	+	0.917874i	$0.629904\pi$
$524$	0	0
$525$	0.268343	0.0117115
$526$	0	0
$527$	−38.8955	−1.69431
$528$	0	0
$529$	−8.77004	−0.381306
$530$	0	0
$531$	4.10860	0.178298
$532$	0	0
$533$	10.5223	0.455770
$534$	0	0
$535$	−2.15917	−0.0933493
$536$	0	0
$537$	−2.51766	−0.108645
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16.2595	0.699052	0.349526	−	0.936927i	$-0.386343\pi$
$541$	16.2595	0.699052	0.349526	+	0.936927i	$0.386343\pi$
$542$	0	0
$543$	1.27817	0.0548515
$544$	0	0
$545$	−5.79052	−0.248038
$546$	0	0
$547$	15.5972	0.666886	0.333443	−	0.942770i	$-0.391790\pi$
$547$	15.5972	0.666886	0.333443	+	0.942770i	$0.391790\pi$
$548$	0	0
$549$	−21.6655	−0.924660
$550$	0	0
$551$	24.7399	1.05396
$552$	0	0
$553$	4.88359	0.207671
$554$	0	0
$555$	2.02585	0.0859925
$556$	0	0
$557$	10.5251	0.445962	0.222981	−	0.974823i	$-0.428421\pi$
$557$	10.5251	0.445962	0.222981	+	0.974823i	$0.428421\pi$
$558$	0	0
$559$	18.3411	0.775747
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.7231	0.494069	0.247034	−	0.969007i	$-0.420544\pi$
$563$	11.7231	0.494069	0.247034	+	0.969007i	$0.420544\pi$
$564$	0	0
$565$	19.0148	0.799959
$566$	0	0
$567$	1.76015	0.0739194
$568$	0	0
$569$	−9.57510	−0.401409	−0.200705	−	0.979652i	$-0.564323\pi$
$569$	−9.57510	−0.401409	−0.200705	+	0.979652i	$0.564323\pi$
$570$	0	0
$571$	44.0439	1.84318	0.921589	−	0.388166i	$-0.126891\pi$
$571$	44.0439	1.84318	0.921589	+	0.388166i	$0.126891\pi$
$572$	0	0
$573$	2.42620	0.101356
$574$	0	0
$575$	3.77226	0.157314
$576$	0	0
$577$	−28.0466	−1.16760	−0.583798	−	0.811899i	$-0.698434\pi$
$577$	−28.0466	−1.16760	−0.583798	+	0.811899i	$0.698434\pi$
$578$	0	0
$579$	−5.54662	−0.230510
$580$	0	0
$581$	1.02953	0.0427123
$582$	0	0
$583$	0	0
$584$	0	0
$585$	7.04568	0.291303
$586$	0	0
$587$	−2.70939	−0.111828	−0.0559142	−	0.998436i	$-0.517807\pi$
$587$	−2.70939	−0.111828	−0.0559142	+	0.998436i	$0.517807\pi$
$588$	0	0
$589$	23.5567	0.970637
$590$	0	0
$591$	−5.06059	−0.208165
$592$	0	0
$593$	−35.1817	−1.44474	−0.722369	−	0.691508i	$-0.756947\pi$
$593$	−35.1817	−1.44474	−0.722369	+	0.691508i	$0.756947\pi$
$594$	0	0
$595$	−1.81980	−0.0746046
$596$	0	0
$597$	9.11192	0.372926
$598$	0	0
$599$	−47.2602	−1.93100	−0.965500	−	0.260404i	$-0.916144\pi$
$599$	−47.2602	−1.93100	−0.965500	+	0.260404i	$0.916144\pi$
$600$	0	0
$601$	11.6431	0.474932	0.237466	−	0.971396i	$-0.423683\pi$
$601$	11.6431	0.474932	0.237466	+	0.971396i	$0.423683\pi$
$602$	0	0
$603$	−28.4973	−1.16050
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−45.9362	−1.86449	−0.932246	−	0.361825i	$-0.882154\pi$
$607$	−45.9362	−1.86449	−0.932246	+	0.361825i	$0.882154\pi$
$608$	0	0
$609$	2.26588	0.0918180
$610$	0	0
$611$	−7.32239	−0.296232
$612$	0	0
$613$	3.35657	0.135571	0.0677854	−	0.997700i	$-0.478407\pi$
$613$	3.35657	0.135571	0.0677854	+	0.997700i	$0.478407\pi$
$614$	0	0
$615$	−2.65391	−0.107016
$616$	0	0
$617$	−16.4880	−0.663783	−0.331892	−	0.943318i	$-0.607687\pi$
$617$	−16.4880	−0.663783	−0.331892	+	0.943318i	$0.607687\pi$
$618$	0	0
$619$	2.05765	0.0827039	0.0413520	−	0.999145i	$-0.486834\pi$
$619$	2.05765	0.0827039	0.0413520	+	0.999145i	$0.486834\pi$
$620$	0	0
$621$	−14.7764	−0.592955
$622$	0	0
$623$	4.52142	0.181147
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.7385	−0.547791
$630$	0	0
$631$	35.2154	1.40190	0.700951	−	0.713209i	$-0.252759\pi$
$631$	35.2154	1.40190	0.700951	+	0.713209i	$0.252759\pi$
$632$	0	0
$633$	5.23634	0.208126
$634$	0	0
$635$	−10.0973	−0.400700
$636$	0	0
$637$	19.3979	0.768574
$638$	0	0
$639$	−35.1787	−1.39165
$640$	0	0
$641$	−33.6602	−1.32950	−0.664750	−	0.747066i	$-0.731462\pi$
$641$	−33.6602	−1.32950	−0.664750	+	0.747066i	$0.731462\pi$
$642$	0	0
$643$	−2.01326	−0.0793951	−0.0396976	−	0.999212i	$-0.512639\pi$
$643$	−2.01326	−0.0793951	−0.0396976	+	0.999212i	$0.512639\pi$
$644$	0	0
$645$	−4.62598	−0.182148
$646$	0	0
$647$	34.0410	1.33829	0.669145	−	0.743131i	$-0.266660\pi$
$647$	34.0410	1.33829	0.669145	+	0.743131i	$0.266660\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	2.15751	0.0845594
$652$	0	0
$653$	−21.1626	−0.828156	−0.414078	−	0.910241i	$-0.635896\pi$
$653$	−21.1626	−0.828156	−0.414078	+	0.910241i	$0.635896\pi$
$654$	0	0
$655$	−6.29651	−0.246025
$656$	0	0
$657$	−1.27851	−0.0498794
$658$	0	0
$659$	21.3464	0.831539	0.415770	−	0.909470i	$-0.363512\pi$
$659$	21.3464	0.831539	0.415770	+	0.909470i	$0.363512\pi$
$660$	0	0
$661$	11.3318	0.440757	0.220379	−	0.975414i	$-0.429271\pi$
$661$	11.3318	0.440757	0.220379	+	0.975414i	$0.429271\pi$
$662$	0	0
$663$	9.76041	0.379063
$664$	0	0
$665$	1.10215	0.0427394
$666$	0	0
$667$	31.8528	1.23335
$668$	0	0
$669$	4.66443	0.180337
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−19.4852	−0.751100	−0.375550	−	0.926802i	$-0.622546\pi$
$673$	−19.4852	−0.751100	−0.375550	+	0.926802i	$0.622546\pi$
$674$	0	0
$675$	−3.91711	−0.150770
$676$	0	0
$677$	−25.7596	−0.990021	−0.495010	−	0.868887i	$-0.664836\pi$
$677$	−25.7596	−0.990021	−0.495010	+	0.868887i	$0.664836\pi$
$678$	0	0
$679$	6.04679	0.232054
$680$	0	0
$681$	14.6566	0.561643
$682$	0	0
$683$	13.0179	0.498117	0.249059	−	0.968488i	$-0.419879\pi$
$683$	13.0179	0.498117	0.249059	+	0.968488i	$0.419879\pi$
$684$	0	0
$685$	−8.27329	−0.316106
$686$	0	0
$687$	−3.08948	−0.117871
$688$	0	0
$689$	0.137916	0.00525417
$690$	0	0
$691$	−48.7463	−1.85440	−0.927198	−	0.374573i	$-0.877789\pi$
$691$	−48.7463	−1.85440	−0.927198	+	0.374573i	$0.877789\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.2671	−0.389453
$696$	0	0
$697$	17.9978	0.681717
$698$	0	0
$699$	20.4443	0.773276
$700$	0	0
$701$	45.1532	1.70541	0.852706	−	0.522391i	$-0.174960\pi$
$701$	45.1532	1.70541	0.852706	+	0.522391i	$0.174960\pi$
$702$	0	0
$703$	8.32062	0.313818
$704$	0	0
$705$	1.84684	0.0695562
$706$	0	0
$707$	5.86388	0.220534
$708$	0	0
$709$	−7.84526	−0.294635	−0.147318	−	0.989089i	$-0.547064\pi$
$709$	−7.84526	−0.294635	−0.147318	+	0.989089i	$0.547064\pi$
$710$	0	0
$711$	−32.3406	−1.21287
$712$	0	0
$713$	30.3294	1.13584
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.47352	−0.279104
$718$	0	0
$719$	36.4371	1.35888	0.679438	−	0.733733i	$-0.262224\pi$
$719$	36.4371	1.35888	0.679438	+	0.733733i	$0.262224\pi$
$720$	0	0
$721$	−3.48300	−0.129714
$722$	0	0
$723$	11.0714	0.411749
$724$	0	0
$725$	8.44396	0.313601
$726$	0	0
$727$	−15.4970	−0.574753	−0.287376	−	0.957818i	$-0.592783\pi$
$727$	−15.4970	−0.574753	−0.287376	+	0.957818i	$0.592783\pi$
$728$	0	0
$729$	−3.27343	−0.121238
$730$	0	0
$731$	31.3716	1.16032
$732$	0	0
$733$	42.3920	1.56579	0.782893	−	0.622157i	$-0.213743\pi$
$733$	42.3920	1.56579	0.782893	+	0.622157i	$0.213743\pi$
$734$	0	0
$735$	−4.89252	−0.180463
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−30.3196	−1.11532	−0.557661	−	0.830069i	$-0.688301\pi$
$739$	−30.3196	−1.11532	−0.557661	+	0.830069i	$0.688301\pi$
$740$	0	0
$741$	−5.91130	−0.217157
$742$	0	0
$743$	−20.8300	−0.764179	−0.382089	−	0.924125i	$-0.624795\pi$
$743$	−20.8300	−0.764179	−0.382089	+	0.924125i	$0.624795\pi$
$744$	0	0
$745$	−4.90160	−0.179581
$746$	0	0
$747$	−6.81790	−0.249454
$748$	0	0
$749$	−0.812221	−0.0296779
$750$	0	0
$751$	−30.7746	−1.12298	−0.561491	−	0.827483i	$-0.689772\pi$
$751$	−30.7746	−1.12298	−0.561491	+	0.827483i	$0.689772\pi$
$752$	0	0
$753$	−8.53891	−0.311175
$754$	0	0
$755$	−9.35625	−0.340509
$756$	0	0
$757$	−25.3851	−0.922637	−0.461318	−	0.887235i	$-0.652623\pi$
$757$	−25.3851	−0.922637	−0.461318	+	0.887235i	$0.652623\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.6390	1.21941	0.609707	−	0.792627i	$-0.291287\pi$
$761$	33.6390	1.21941	0.609707	+	0.792627i	$0.291287\pi$
$762$	0	0
$763$	−2.17823	−0.0788573
$764$	0	0
$765$	12.0513	0.435715
$766$	0	0
$767$	4.66471	0.168433
$768$	0	0
$769$	−29.4388	−1.06159	−0.530795	−	0.847500i	$-0.678107\pi$
$769$	−29.4388	−1.06159	−0.530795	+	0.847500i	$0.678107\pi$
$770$	0	0
$771$	−6.04453	−0.217688
$772$	0	0
$773$	24.7585	0.890501	0.445251	−	0.895406i	$-0.353115\pi$
$773$	24.7585	0.890501	0.445251	+	0.895406i	$0.353115\pi$
$774$	0	0
$775$	8.04011	0.288809
$776$	0	0
$777$	0.762068	0.0273390
$778$	0	0
$779$	−10.9002	−0.390541
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−33.0759	−1.18204
$784$	0	0
$785$	9.65817	0.344715
$786$	0	0
$787$	37.4598	1.33530	0.667649	−	0.744476i	$-0.267301\pi$
$787$	37.4598	1.33530	0.667649	+	0.744476i	$0.267301\pi$
$788$	0	0
$789$	8.93040	0.317931
$790$	0	0
$791$	7.15284	0.254326
$792$	0	0
$793$	−24.5979	−0.873499
$794$	0	0
$795$	−0.0347849	−0.00123369
$796$	0	0
$797$	−48.1456	−1.70541	−0.852703	−	0.522397i	$-0.825038\pi$
$797$	−48.1456	−1.70541	−0.852703	+	0.522397i	$0.825038\pi$
$798$	0	0
$799$	−12.5246	−0.443088
$800$	0	0
$801$	−29.9422	−1.05796
$802$	0	0
$803$	0	0
$804$	0	0
$805$	1.41902	0.0500139
$806$	0	0
$807$	−17.1667	−0.604297
$808$	0	0
$809$	37.2876	1.31096	0.655481	−	0.755211i	$-0.272466\pi$
$809$	37.2876	1.31096	0.655481	+	0.755211i	$0.272466\pi$
$810$	0	0
$811$	8.87605	0.311680	0.155840	−	0.987782i	$-0.450192\pi$
$811$	8.87605	0.311680	0.155840	+	0.987782i	$0.450192\pi$
$812$	0	0
$813$	−6.72135	−0.235728
$814$	0	0
$815$	4.48802	0.157208
$816$	0	0
$817$	−18.9999	−0.664723
$818$	0	0
$819$	2.65039	0.0926121
$820$	0	0
$821$	−31.1530	−1.08725	−0.543624	−	0.839329i	$-0.682948\pi$
$821$	−31.1530	−1.08725	−0.543624	+	0.839329i	$0.682948\pi$
$822$	0	0
$823$	−32.6349	−1.13758	−0.568790	−	0.822483i	$-0.692588\pi$
$823$	−32.6349	−1.13758	−0.568790	+	0.822483i	$0.692588\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.7594	−1.00006	−0.500031	−	0.866008i	$-0.666678\pi$
$827$	−28.7594	−1.00006	−0.500031	+	0.866008i	$0.666678\pi$
$828$	0	0
$829$	−42.7970	−1.48640	−0.743200	−	0.669069i	$-0.766693\pi$
$829$	−42.7970	−1.48640	−0.743200	+	0.669069i	$0.766693\pi$
$830$	0	0
$831$	16.9643	0.588485
$832$	0	0
$833$	33.1792	1.14959
$834$	0	0
$835$	−23.4716	−0.812267
$836$	0	0
$837$	−31.4940	−1.08859
$838$	0	0
$839$	−28.5327	−0.985059	−0.492529	−	0.870296i	$-0.663928\pi$
$839$	−28.5327	−0.985059	−0.492529	+	0.870296i	$0.663928\pi$
$840$	0	0
$841$	42.3004	1.45863
$842$	0	0
$843$	0.354468	0.0122085
$844$	0	0
$845$	−5.00067	−0.172028
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.33870	0.114584
$850$	0	0
$851$	10.7128	0.367232
$852$	0	0
$853$	33.4101	1.14394	0.571970	−	0.820275i	$-0.306179\pi$
$853$	33.4101	1.14394	0.571970	+	0.820275i	$0.306179\pi$
$854$	0	0
$855$	−7.29875	−0.249612
$856$	0	0
$857$	11.9632	0.408656	0.204328	−	0.978902i	$-0.434499\pi$
$857$	11.9632	0.408656	0.204328	+	0.978902i	$0.434499\pi$
$858$	0	0
$859$	23.4984	0.801755	0.400878	−	0.916132i	$-0.368705\pi$
$859$	23.4984	0.801755	0.400878	+	0.916132i	$0.368705\pi$
$860$	0	0
$861$	−0.998328	−0.0340229
$862$	0	0
$863$	4.09414	0.139366	0.0696830	−	0.997569i	$-0.477801\pi$
$863$	4.09414	0.139366	0.0696830	+	0.997569i	$0.477801\pi$
$864$	0	0
$865$	−12.9509	−0.440343
$866$	0	0
$867$	4.56772	0.155128
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−32.3544	−1.09629
$872$	0	0
$873$	−40.0437	−1.35527
$874$	0	0
$875$	0.376172	0.0127169
$876$	0	0
$877$	44.3667	1.49816	0.749079	−	0.662481i	$-0.230497\pi$
$877$	44.3667	1.49816	0.749079	+	0.662481i	$0.230497\pi$
$878$	0	0
$879$	12.0701	0.407115
$880$	0	0
$881$	−43.0315	−1.44977	−0.724884	−	0.688871i	$-0.758107\pi$
$881$	−43.0315	−1.44977	−0.724884	+	0.688871i	$0.758107\pi$
$882$	0	0
$883$	−36.5444	−1.22982	−0.614909	−	0.788598i	$-0.710807\pi$
$883$	−36.5444	−1.22982	−0.614909	+	0.788598i	$0.710807\pi$
$884$	0	0
$885$	−1.17653	−0.0395485
$886$	0	0
$887$	−1.79439	−0.0602496	−0.0301248	−	0.999546i	$-0.509590\pi$
$887$	−1.79439	−0.0602496	−0.0301248	+	0.999546i	$0.509590\pi$
$888$	0	0
$889$	−3.79833	−0.127392
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.58541	0.253836
$894$	0	0
$895$	−3.52933	−0.117973
$896$	0	0
$897$	−7.61084	−0.254119
$898$	0	0
$899$	67.8903	2.26427
$900$	0	0
$901$	0.235898	0.00785890
$902$	0	0
$903$	−1.74016	−0.0579090
$904$	0	0
$905$	1.79178	0.0595608
$906$	0	0
$907$	−51.2671	−1.70230	−0.851149	−	0.524924i	$-0.824094\pi$
$907$	−51.2671	−1.70230	−0.851149	+	0.524924i	$0.824094\pi$
$908$	0	0
$909$	−38.8324	−1.28799
$910$	0	0
$911$	40.0803	1.32792	0.663960	−	0.747768i	$-0.268875\pi$
$911$	40.0803	1.32792	0.663960	+	0.747768i	$0.268875\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	6.20406	0.205100
$916$	0	0
$917$	−2.36857	−0.0782171
$918$	0	0
$919$	1.08348	0.0357406	0.0178703	−	0.999840i	$-0.494311\pi$
$919$	1.08348	0.0357406	0.0178703	+	0.999840i	$0.494311\pi$
$920$	0	0
$921$	−24.5156	−0.807816
$922$	0	0
$923$	−39.9402	−1.31465
$924$	0	0
$925$	2.83990	0.0933754
$926$	0	0
$927$	23.0655	0.757571
$928$	0	0
$929$	14.2742	0.468320	0.234160	−	0.972198i	$-0.424766\pi$
$929$	14.2742	0.468320	0.234160	+	0.972198i	$0.424766\pi$
$930$	0	0
$931$	−20.0947	−0.658577
$932$	0	0
$933$	5.25591	0.172071
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−27.8901	−0.911130	−0.455565	−	0.890203i	$-0.650563\pi$
$937$	−27.8901	−0.911130	−0.455565	+	0.890203i	$0.650563\pi$
$938$	0	0
$939$	−17.8770	−0.583394
$940$	0	0
$941$	3.62440	0.118152	0.0590760	−	0.998253i	$-0.481185\pi$
$941$	3.62440	0.118152	0.0590760	+	0.998253i	$0.481185\pi$
$942$	0	0
$943$	−14.0341	−0.457013
$944$	0	0
$945$	−1.47351	−0.0479332
$946$	0	0
$947$	−54.9559	−1.78583	−0.892913	−	0.450230i	$-0.851342\pi$
$947$	−54.9559	−1.78583	−0.892913	+	0.450230i	$0.851342\pi$
$948$	0	0
$949$	−1.45156	−0.0471196
$950$	0	0
$951$	20.0258	0.649382
$952$	0	0
$953$	−9.03643	−0.292719	−0.146359	−	0.989231i	$-0.546756\pi$
$953$	−9.03643	−0.292719	−0.146359	+	0.989231i	$0.546756\pi$
$954$	0	0
$955$	3.40113	0.110058
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.11218	−0.100498
$960$	0	0
$961$	33.6433	1.08527
$962$	0	0
$963$	5.37878	0.173329
$964$	0	0
$965$	−7.77543	−0.250300
$966$	0	0
$967$	−44.4619	−1.42980	−0.714900	−	0.699227i	$-0.753528\pi$
$967$	−44.4619	−1.42980	−0.714900	+	0.699227i	$0.753528\pi$
$968$	0	0
$969$	−10.1110	−0.324812
$970$	0	0
$971$	1.66867	0.0535502	0.0267751	−	0.999641i	$-0.491476\pi$
$971$	1.66867	0.0535502	0.0267751	+	0.999641i	$0.491476\pi$
$972$	0	0
$973$	−3.86219	−0.123816
$974$	0	0
$975$	−2.01758	−0.0646143
$976$	0	0
$977$	22.6537	0.724755	0.362377	−	0.932031i	$-0.381965\pi$
$977$	22.6537	0.724755	0.362377	+	0.932031i	$0.381965\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	14.4249	0.460552
$982$	0	0
$983$	46.9510	1.49750	0.748752	−	0.662850i	$-0.230653\pi$
$983$	46.9510	1.49750	0.748752	+	0.662850i	$0.230653\pi$
$984$	0	0
$985$	−7.09410	−0.226037
$986$	0	0
$987$	0.694731	0.0221135
$988$	0	0
$989$	−24.4625	−0.777863
$990$	0	0
$991$	−0.122612	−0.00389489	−0.00194745	−	0.999998i	$-0.500620\pi$
$991$	−0.122612	−0.00389489	−0.00194745	+	0.999998i	$0.500620\pi$
$992$	0	0
$993$	15.6684	0.497221
$994$	0	0
$995$	12.7734	0.404943
$996$	0	0
$997$	−20.1831	−0.639205	−0.319602	−	0.947552i	$-0.603549\pi$
$997$	−20.1831	−0.639205	−0.319602	+	0.947552i	$0.603549\pi$
$998$	0	0
$999$	−11.1242	−0.351954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.bg.1.5		8
4.3	odd	2		9680.2.a.df.1.4		8
11.5	even	5		440.2.y.d.201.3	yes	16
11.9	even	5		440.2.y.d.81.3	✓	16
11.10	odd	2		4840.2.a.bh.1.5		8
44.27	odd	10		880.2.bo.k.641.2		16
44.31	odd	10		880.2.bo.k.81.2		16
44.43	even	2		9680.2.a.de.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.d.81.3	✓	16	11.9	even	5
440.2.y.d.201.3	yes	16	11.5	even	5
880.2.bo.k.81.2		16	44.31	odd	10
880.2.bo.k.641.2		16	44.27	odd	10
4840.2.a.bg.1.5		8	1.1	even	1	trivial
4840.2.a.bh.1.5		8	11.10	odd	2
9680.2.a.de.1.4		8	44.43	even	2
9680.2.a.df.1.4		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q+0.713352 q^{3} +1.00000 q^{5} +0.376172 q^{7} -2.49113 q^{9} -2.82831 q^{13} +0.713352 q^{15} -4.83768 q^{17} +2.92990 q^{19} +0.268343 q^{21} +3.77226 q^{23} +1.00000 q^{25} -3.91711 q^{27} +8.44396 q^{29} +8.04011 q^{31} +0.376172 q^{35} +2.83990 q^{37} -2.01758 q^{39} -3.72034 q^{41} -6.48484 q^{43} -2.49113 q^{45} +2.58897 q^{47} -6.85849 q^{49} -3.45097 q^{51} -0.0487626 q^{53} +2.09005 q^{57} -1.64929 q^{59} +8.69705 q^{61} -0.937093 q^{63} -2.82831 q^{65} +11.4395 q^{67} +2.69095 q^{69} +14.1216 q^{71} +0.513225 q^{73} +0.713352 q^{75} +12.9823 q^{79} +4.67911 q^{81} +2.73687 q^{83} -4.83768 q^{85} +6.02351 q^{87} +12.0195 q^{89} -1.06393 q^{91} +5.73543 q^{93} +2.92990 q^{95} +16.0745 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} + 12 q^{13} + q^{15} + 2 q^{17} - 6 q^{19} + 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} + 8 q^{29} + 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} - 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) \) 8 * q + q^3 + 8 * q^5 - 6 * q^7 + 19 * q^9 + 12 * q^13 + q^15 + 2 * q^17 - 6 * q^19 + 6 * q^21 + 10 * q^23 + 8 * q^25 + 13 * q^27 + 8 * q^29 + 19 * q^31 - 6 * q^35 + 12 * q^37 - 21 * q^39 - 3 * q^41 - 8 * q^43 + 19 * q^45 + 10 * q^47 + 22 * q^49 - 7 * q^51 + 28 * q^53 + 25 * q^57 + 25 * q^59 + 10 * q^61 - 64 * q^63 + 12 * q^65 - 2 * q^67 + 18 * q^69 + 25 * q^71 + 38 * q^73 + q^75 - 38 * q^79 + 32 * q^81 - 28 * q^83 + 2 * q^85 + 15 * q^87 + 12 * q^89 + 8 * q^91 - 15 * q^93 - 6 * q^95 + 21 * q^97

Introduction

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