LMFDB - Embedded newform 4788.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(1,4788)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4788.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4788.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +1.00000 q^{7} -6.09017 q^{11} -2.23607 q^{13} +7.61803 q^{17} -1.00000 q^{19} +8.70820 q^{23} -4.00000 q^{25} -0.854102 q^{29} -2.38197 q^{31} +1.00000 q^{35} -6.70820 q^{37} -0.0901699 q^{41} +8.47214 q^{43} -4.70820 q^{47} +1.00000 q^{49} +14.3262 q^{53} -6.09017 q^{55} +10.7082 q^{59} +5.47214 q^{61} -2.23607 q^{65} -2.09017 q^{67} +11.1803 q^{71} -0.909830 q^{73} -6.09017 q^{77} -6.94427 q^{79} -14.6180 q^{83} +7.61803 q^{85} +12.1803 q^{89} -2.23607 q^{91} -1.00000 q^{95} +8.70820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7} - q^{11} + 13 q^{17} - 2 q^{19} + 4 q^{23} - 8 q^{25} + 5 q^{29} - 7 q^{31} + 2 q^{35} + 11 q^{41} + 8 q^{43} + 4 q^{47} + 2 q^{49} + 13 q^{53} - q^{55} + 8 q^{59} + 2 q^{61} + 7 q^{67}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 - q^11 + 13 * q^17 - 2 * q^19 + 4 * q^23 - 8 * q^25 + 5 * q^29 - 7 * q^31 + 2 * q^35 + 11 * q^41 + 8 * q^43 + 4 * q^47 + 2 * q^49 + 13 * q^53 - q^55 + 8 * q^59 + 2 * q^61 + 7 * q^67 - 13 * q^73 - q^77 + 4 * q^79 - 27 * q^83 + 13 * q^85 + 2 * q^89 - 2 * q^95 + 4 * 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	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.09017	−1.83626	−0.918128	−	0.396285i	$-0.870299\pi$
$11$	−6.09017	−1.83626	−0.918128	+	0.396285i	$0.870299\pi$
$12$	0	0
$13$	−2.23607	−0.620174	−0.310087	−	0.950708i	$-0.600358\pi$
$13$	−2.23607	−0.620174	−0.310087	+	0.950708i	$0.600358\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.61803	1.84764	0.923822	−	0.382822i	$-0.125048\pi$
$17$	7.61803	1.84764	0.923822	+	0.382822i	$0.125048\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.70820	1.81579	0.907893	−	0.419202i	$-0.137690\pi$
$23$	8.70820	1.81579	0.907893	+	0.419202i	$0.137690\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.854102	−0.158603	−0.0793014	−	0.996851i	$-0.525269\pi$
$29$	−0.854102	−0.158603	−0.0793014	+	0.996851i	$0.525269\pi$
$30$	0	0
$31$	−2.38197	−0.427814	−0.213907	−	0.976854i	$-0.568619\pi$
$31$	−2.38197	−0.427814	−0.213907	+	0.976854i	$0.568619\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−6.70820	−1.10282	−0.551411	−	0.834234i	$-0.685910\pi$
$37$	−6.70820	−1.10282	−0.551411	+	0.834234i	$0.685910\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.0901699	−0.0140822	−0.00704109	−	0.999975i	$-0.502241\pi$
$41$	−0.0901699	−0.0140822	−0.00704109	+	0.999975i	$0.502241\pi$
$42$	0	0
$43$	8.47214	1.29199	0.645994	−	0.763342i	$-0.276443\pi$
$43$	8.47214	1.29199	0.645994	+	0.763342i	$0.276443\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.70820	−0.686762	−0.343381	−	0.939196i	$-0.611572\pi$
$47$	−4.70820	−0.686762	−0.343381	+	0.939196i	$0.611572\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	14.3262	1.96786	0.983930	−	0.178554i	$-0.0571420\pi$
$53$	14.3262	1.96786	0.983930	+	0.178554i	$0.0571420\pi$
$54$	0	0
$55$	−6.09017	−0.821198
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.7082	1.39409	0.697045	−	0.717028i	$-0.254498\pi$
$59$	10.7082	1.39409	0.697045	+	0.717028i	$0.254498\pi$
$60$	0	0
$61$	5.47214	0.700635	0.350318	−	0.936631i	$-0.386074\pi$
$61$	5.47214	0.700635	0.350318	+	0.936631i	$0.386074\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.23607	−0.277350
$66$	0	0
$67$	−2.09017	−0.255355	−0.127677	−	0.991816i	$-0.540752\pi$
$67$	−2.09017	−0.255355	−0.127677	+	0.991816i	$0.540752\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.1803	1.32686	0.663431	−	0.748237i	$-0.269100\pi$
$71$	11.1803	1.32686	0.663431	+	0.748237i	$0.269100\pi$
$72$	0	0
$73$	−0.909830	−0.106488	−0.0532438	−	0.998582i	$-0.516956\pi$
$73$	−0.909830	−0.106488	−0.0532438	+	0.998582i	$0.516956\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.09017	−0.694039
$78$	0	0
$79$	−6.94427	−0.781292	−0.390646	−	0.920541i	$-0.627748\pi$
$79$	−6.94427	−0.781292	−0.390646	+	0.920541i	$0.627748\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.6180	−1.60454	−0.802269	−	0.596963i	$-0.796374\pi$
$83$	−14.6180	−1.60454	−0.802269	+	0.596963i	$0.796374\pi$
$84$	0	0
$85$	7.61803	0.826292
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.1803	1.29111	0.645557	−	0.763712i	$-0.276625\pi$
$89$	12.1803	1.29111	0.645557	+	0.763712i	$0.276625\pi$
$90$	0	0
$91$	−2.23607	−0.234404
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	8.70820	0.884184	0.442092	−	0.896970i	$-0.354236\pi$
$97$	8.70820	0.884184	0.442092	+	0.896970i	$0.354236\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.23607	0.819519	0.409760	−	0.912194i	$-0.365613\pi$
$101$	8.23607	0.819519	0.409760	+	0.912194i	$0.365613\pi$
$102$	0	0
$103$	−7.76393	−0.765003	−0.382501	−	0.923955i	$-0.624937\pi$
$103$	−7.76393	−0.765003	−0.382501	+	0.923955i	$0.624937\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.76393	−0.363873	−0.181937	−	0.983310i	$-0.558237\pi$
$107$	−3.76393	−0.363873	−0.181937	+	0.983310i	$0.558237\pi$
$108$	0	0
$109$	18.8885	1.80919	0.904597	−	0.426267i	$-0.140172\pi$
$109$	18.8885	1.80919	0.904597	+	0.426267i	$0.140172\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.6180	1.37515	0.687574	−	0.726114i	$-0.258675\pi$
$113$	14.6180	1.37515	0.687574	+	0.726114i	$0.258675\pi$
$114$	0	0
$115$	8.70820	0.812044
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.61803	0.698344
$120$	0	0
$121$	26.0902	2.37183
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	15.2361	1.35198	0.675991	−	0.736910i	$-0.263716\pi$
$127$	15.2361	1.35198	0.675991	+	0.736910i	$0.263716\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.38197	0.382854	0.191427	−	0.981507i	$-0.438688\pi$
$131$	4.38197	0.382854	0.191427	+	0.981507i	$0.438688\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.52786	−0.301406	−0.150703	−	0.988579i	$-0.548154\pi$
$137$	−3.52786	−0.301406	−0.150703	+	0.988579i	$0.548154\pi$
$138$	0	0
$139$	−5.18034	−0.439391	−0.219695	−	0.975569i	$-0.570506\pi$
$139$	−5.18034	−0.439391	−0.219695	+	0.975569i	$0.570506\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.6180	1.13880
$144$	0	0
$145$	−0.854102	−0.0709293
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.00000	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$150$	0	0
$151$	−4.38197	−0.356599	−0.178300	−	0.983976i	$-0.557060\pi$
$151$	−4.38197	−0.356599	−0.178300	+	0.983976i	$0.557060\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.38197	−0.191324
$156$	0	0
$157$	13.6180	1.08684	0.543419	−	0.839462i	$-0.317130\pi$
$157$	13.6180	1.08684	0.543419	+	0.839462i	$0.317130\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.70820	0.686303
$162$	0	0
$163$	−10.7984	−0.845794	−0.422897	−	0.906178i	$-0.638987\pi$
$163$	−10.7984	−0.845794	−0.422897	+	0.906178i	$0.638987\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.47214	−0.113917	−0.0569587	−	0.998377i	$-0.518140\pi$
$167$	−1.47214	−0.113917	−0.0569587	+	0.998377i	$0.518140\pi$
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.2918	0.782471	0.391235	−	0.920291i	$-0.372048\pi$
$173$	10.2918	0.782471	0.391235	+	0.920291i	$0.372048\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.3820	0.775985	0.387992	−	0.921663i	$-0.373169\pi$
$179$	10.3820	0.775985	0.387992	+	0.921663i	$0.373169\pi$
$180$	0	0
$181$	−13.5066	−1.00394	−0.501968	−	0.864886i	$-0.667390\pi$
$181$	−13.5066	−1.00394	−0.501968	+	0.864886i	$0.667390\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.70820	−0.493197
$186$	0	0
$187$	−46.3951	−3.39275
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.03444	−0.581352	−0.290676	−	0.956822i	$-0.593880\pi$
$191$	−8.03444	−0.581352	−0.290676	+	0.956822i	$0.593880\pi$
$192$	0	0
$193$	−1.43769	−0.103487	−0.0517437	−	0.998660i	$-0.516478\pi$
$193$	−1.43769	−0.103487	−0.0517437	+	0.998660i	$0.516478\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.79837	−0.626858	−0.313429	−	0.949612i	$-0.601478\pi$
$197$	−8.79837	−0.626858	−0.313429	+	0.949612i	$0.601478\pi$
$198$	0	0
$199$	2.23607	0.158511	0.0792553	−	0.996854i	$-0.474746\pi$
$199$	2.23607	0.158511	0.0792553	+	0.996854i	$0.474746\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.854102	−0.0599462
$204$	0	0
$205$	−0.0901699	−0.00629774
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.09017	0.421266
$210$	0	0
$211$	26.9787	1.85729	0.928646	−	0.370968i	$-0.120974\pi$
$211$	26.9787	1.85729	0.928646	+	0.370968i	$0.120974\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.47214	0.577795
$216$	0	0
$217$	−2.38197	−0.161698
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−17.0344	−1.14586
$222$	0	0
$223$	5.94427	0.398058	0.199029	−	0.979994i	$-0.436221\pi$
$223$	5.94427	0.398058	0.199029	+	0.979994i	$0.436221\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.0902	−1.06794	−0.533971	−	0.845503i	$-0.679301\pi$
$227$	−16.0902	−1.06794	−0.533971	+	0.845503i	$0.679301\pi$
$228$	0	0
$229$	−12.4721	−0.824182	−0.412091	−	0.911143i	$-0.635201\pi$
$229$	−12.4721	−0.824182	−0.412091	+	0.911143i	$0.635201\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.32624	0.479958	0.239979	−	0.970778i	$-0.422859\pi$
$233$	7.32624	0.479958	0.239979	+	0.970778i	$0.422859\pi$
$234$	0	0
$235$	−4.70820	−0.307129
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.1246	1.17238	0.586192	−	0.810172i	$-0.300626\pi$
$239$	18.1246	1.17238	0.586192	+	0.810172i	$0.300626\pi$
$240$	0	0
$241$	23.7082	1.52718	0.763590	−	0.645702i	$-0.223435\pi$
$241$	23.7082	1.52718	0.763590	+	0.645702i	$0.223435\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	2.23607	0.142278
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.673762	0.0425275	0.0212637	−	0.999774i	$-0.493231\pi$
$251$	0.673762	0.0425275	0.0212637	+	0.999774i	$0.493231\pi$
$252$	0	0
$253$	−53.0344	−3.33425
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.2705	−0.827792	−0.413896	−	0.910324i	$-0.635832\pi$
$257$	−13.2705	−0.827792	−0.413896	+	0.910324i	$0.635832\pi$
$258$	0	0
$259$	−6.70820	−0.416828
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.7984	−1.15916	−0.579579	−	0.814916i	$-0.696783\pi$
$263$	−18.7984	−1.15916	−0.579579	+	0.814916i	$0.696783\pi$
$264$	0	0
$265$	14.3262	0.880054
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.79837	−0.292562	−0.146281	−	0.989243i	$-0.546730\pi$
$269$	−4.79837	−0.292562	−0.146281	+	0.989243i	$0.546730\pi$
$270$	0	0
$271$	0.562306	0.0341577	0.0170788	−	0.999854i	$-0.494563\pi$
$271$	0.562306	0.0341577	0.0170788	+	0.999854i	$0.494563\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.3607	1.46900
$276$	0	0
$277$	−8.65248	−0.519877	−0.259938	−	0.965625i	$-0.583702\pi$
$277$	−8.65248	−0.519877	−0.259938	+	0.965625i	$0.583702\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.0000	0.656205	0.328102	−	0.944642i	$-0.393591\pi$
$281$	11.0000	0.656205	0.328102	+	0.944642i	$0.393591\pi$
$282$	0	0
$283$	−18.6180	−1.10673	−0.553364	−	0.832940i	$-0.686656\pi$
$283$	−18.6180	−1.10673	−0.553364	+	0.832940i	$0.686656\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.0901699	−0.00532256
$288$	0	0
$289$	41.0344	2.41379
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.4721	1.13757	0.568787	−	0.822485i	$-0.307413\pi$
$293$	19.4721	1.13757	0.568787	+	0.822485i	$0.307413\pi$
$294$	0	0
$295$	10.7082	0.623456
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−19.4721	−1.12610
$300$	0	0
$301$	8.47214	0.488326
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.47214	0.313334
$306$	0	0
$307$	22.7426	1.29799	0.648996	−	0.760792i	$-0.275189\pi$
$307$	22.7426	1.29799	0.648996	+	0.760792i	$0.275189\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.32624	0.188614	0.0943068	−	0.995543i	$-0.469937\pi$
$311$	3.32624	0.188614	0.0943068	+	0.995543i	$0.469937\pi$
$312$	0	0
$313$	14.1803	0.801520	0.400760	−	0.916183i	$-0.368746\pi$
$313$	14.1803	0.801520	0.400760	+	0.916183i	$0.368746\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.47214	−0.251180	−0.125590	−	0.992082i	$-0.540082\pi$
$317$	−4.47214	−0.251180	−0.125590	+	0.992082i	$0.540082\pi$
$318$	0	0
$319$	5.20163	0.291235
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.61803	−0.423879
$324$	0	0
$325$	8.94427	0.496139
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.70820	−0.259572
$330$	0	0
$331$	−0.729490	−0.0400964	−0.0200482	−	0.999799i	$-0.506382\pi$
$331$	−0.729490	−0.0400964	−0.0200482	+	0.999799i	$0.506382\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.09017	−0.114198
$336$	0	0
$337$	−9.32624	−0.508033	−0.254016	−	0.967200i	$-0.581752\pi$
$337$	−9.32624	−0.508033	−0.254016	+	0.967200i	$0.581752\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.5066	0.785575
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.5623	0.620697	0.310349	−	0.950623i	$-0.399554\pi$
$347$	11.5623	0.620697	0.310349	+	0.950623i	$0.399554\pi$
$348$	0	0
$349$	−3.27051	−0.175066	−0.0875332	−	0.996162i	$-0.527898\pi$
$349$	−3.27051	−0.175066	−0.0875332	+	0.996162i	$0.527898\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.6180	1.15061	0.575306	−	0.817938i	$-0.304883\pi$
$353$	21.6180	1.15061	0.575306	+	0.817938i	$0.304883\pi$
$354$	0	0
$355$	11.1803	0.593391
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.43769	0.181435	0.0907173	−	0.995877i	$-0.471084\pi$
$359$	3.43769	0.181435	0.0907173	+	0.995877i	$0.471084\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.909830	−0.0476227
$366$	0	0
$367$	12.8885	0.672777	0.336388	−	0.941723i	$-0.390795\pi$
$367$	12.8885	0.672777	0.336388	+	0.941723i	$0.390795\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14.3262	0.743781
$372$	0	0
$373$	−16.5066	−0.854678	−0.427339	−	0.904091i	$-0.640549\pi$
$373$	−16.5066	−0.854678	−0.427339	+	0.904091i	$0.640549\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.90983	0.0983613
$378$	0	0
$379$	6.52786	0.335314	0.167657	−	0.985845i	$-0.446380\pi$
$379$	6.52786	0.335314	0.167657	+	0.985845i	$0.446380\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.2361	0.829624	0.414812	−	0.909907i	$-0.363847\pi$
$383$	16.2361	0.829624	0.414812	+	0.909907i	$0.363847\pi$
$384$	0	0
$385$	−6.09017	−0.310384
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.61803	−0.284846	−0.142423	−	0.989806i	$-0.545489\pi$
$389$	−5.61803	−0.284846	−0.142423	+	0.989806i	$0.545489\pi$
$390$	0	0
$391$	66.3394	3.35493
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.94427	−0.349404
$396$	0	0
$397$	−15.2361	−0.764676	−0.382338	−	0.924022i	$-0.624881\pi$
$397$	−15.2361	−0.764676	−0.382338	+	0.924022i	$0.624881\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.7426	1.33546	0.667732	−	0.744402i	$-0.267265\pi$
$401$	26.7426	1.33546	0.667732	+	0.744402i	$0.267265\pi$
$402$	0	0
$403$	5.32624	0.265319
$404$	0	0
$405$	0	0
$406$	0	0
$407$	40.8541	2.02506
$408$	0	0
$409$	−1.27051	−0.0628227	−0.0314113	−	0.999507i	$-0.510000\pi$
$409$	−1.27051	−0.0628227	−0.0314113	+	0.999507i	$0.510000\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.7082	0.526916
$414$	0	0
$415$	−14.6180	−0.717571
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.70820	0.425424	0.212712	−	0.977115i	$-0.431770\pi$
$419$	8.70820	0.425424	0.212712	+	0.977115i	$0.431770\pi$
$420$	0	0
$421$	−9.18034	−0.447422	−0.223711	−	0.974655i	$-0.571817\pi$
$421$	−9.18034	−0.447422	−0.223711	+	0.974655i	$0.571817\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−30.4721	−1.47812
$426$	0	0
$427$	5.47214	0.264815
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	31.4721	1.51245	0.756227	−	0.654309i	$-0.227040\pi$
$433$	31.4721	1.51245	0.756227	+	0.654309i	$0.227040\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.70820	−0.416570
$438$	0	0
$439$	28.3607	1.35358	0.676791	−	0.736175i	$-0.263370\pi$
$439$	28.3607	1.35358	0.676791	+	0.736175i	$0.263370\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.7984	−1.65332	−0.826660	−	0.562701i	$-0.809762\pi$
$443$	−34.7984	−1.65332	−0.826660	+	0.562701i	$0.809762\pi$
$444$	0	0
$445$	12.1803	0.577403
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.38197	0.0652190	0.0326095	−	0.999468i	$-0.489618\pi$
$449$	1.38197	0.0652190	0.0326095	+	0.999468i	$0.489618\pi$
$450$	0	0
$451$	0.549150	0.0258585
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.23607	−0.104828
$456$	0	0
$457$	16.2148	0.758495	0.379248	−	0.925295i	$-0.376183\pi$
$457$	16.2148	0.758495	0.379248	+	0.925295i	$0.376183\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	24.0344	1.11940	0.559698	−	0.828697i	$-0.310917\pi$
$461$	24.0344	1.11940	0.559698	+	0.828697i	$0.310917\pi$
$462$	0	0
$463$	−40.7771	−1.89507	−0.947536	−	0.319649i	$-0.896435\pi$
$463$	−40.7771	−1.89507	−0.947536	+	0.319649i	$0.896435\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.8541	−0.918738	−0.459369	−	0.888245i	$-0.651924\pi$
$467$	−19.8541	−0.918738	−0.459369	+	0.888245i	$0.651924\pi$
$468$	0	0
$469$	−2.09017	−0.0965151
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−51.5967	−2.37242
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15.9787	0.730086	0.365043	−	0.930991i	$-0.381054\pi$
$479$	15.9787	0.730086	0.365043	+	0.930991i	$0.381054\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.70820	0.395419
$486$	0	0
$487$	−20.7082	−0.938378	−0.469189	−	0.883098i	$-0.655454\pi$
$487$	−20.7082	−0.938378	−0.469189	+	0.883098i	$0.655454\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−23.0557	−1.04049	−0.520245	−	0.854017i	$-0.674159\pi$
$491$	−23.0557	−1.04049	−0.520245	+	0.854017i	$0.674159\pi$
$492$	0	0
$493$	−6.50658	−0.293042
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.1803	0.501507
$498$	0	0
$499$	6.90983	0.309326	0.154663	−	0.987967i	$-0.450571\pi$
$499$	6.90983	0.309326	0.154663	+	0.987967i	$0.450571\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.944272	−0.0421030	−0.0210515	−	0.999778i	$-0.506701\pi$
$503$	−0.944272	−0.0421030	−0.0210515	+	0.999778i	$0.506701\pi$
$504$	0	0
$505$	8.23607	0.366500
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−33.4164	−1.48116	−0.740578	−	0.671970i	$-0.765448\pi$
$509$	−33.4164	−1.48116	−0.740578	+	0.671970i	$0.765448\pi$
$510$	0	0
$511$	−0.909830	−0.0402485
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7.76393	−0.342120
$516$	0	0
$517$	28.6738	1.26107
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.88854	−0.301793	−0.150896	−	0.988550i	$-0.548216\pi$
$521$	−6.88854	−0.301793	−0.150896	+	0.988550i	$0.548216\pi$
$522$	0	0
$523$	4.41641	0.193116	0.0965580	−	0.995327i	$-0.469217\pi$
$523$	4.41641	0.193116	0.0965580	+	0.995327i	$0.469217\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.1459	−0.790448
$528$	0	0
$529$	52.8328	2.29708
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.201626	0.00873340
$534$	0	0
$535$	−3.76393	−0.162729
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.09017	−0.262322
$540$	0	0
$541$	−10.5836	−0.455024	−0.227512	−	0.973775i	$-0.573059\pi$
$541$	−10.5836	−0.455024	−0.227512	+	0.973775i	$0.573059\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.8885	0.809096
$546$	0	0
$547$	10.6180	0.453994	0.226997	−	0.973895i	$-0.427109\pi$
$547$	10.6180	0.453994	0.226997	+	0.973895i	$0.427109\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.854102	0.0363860
$552$	0	0
$553$	−6.94427	−0.295300
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.0344	0.637030	0.318515	−	0.947918i	$-0.396816\pi$
$557$	15.0344	0.637030	0.318515	+	0.947918i	$0.396816\pi$
$558$	0	0
$559$	−18.9443	−0.801257
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−37.5410	−1.58217	−0.791083	−	0.611709i	$-0.790482\pi$
$563$	−37.5410	−1.58217	−0.791083	+	0.611709i	$0.790482\pi$
$564$	0	0
$565$	14.6180	0.614985
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.2918	0.557221	0.278611	−	0.960404i	$-0.410126\pi$
$569$	13.2918	0.557221	0.278611	+	0.960404i	$0.410126\pi$
$570$	0	0
$571$	−5.94427	−0.248760	−0.124380	−	0.992235i	$-0.539694\pi$
$571$	−5.94427	−0.248760	−0.124380	+	0.992235i	$0.539694\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−34.8328	−1.45263
$576$	0	0
$577$	9.67376	0.402724	0.201362	−	0.979517i	$-0.435463\pi$
$577$	9.67376	0.402724	0.201362	+	0.979517i	$0.435463\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14.6180	−0.606458
$582$	0	0
$583$	−87.2492	−3.61349
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.3607	1.41822	0.709109	−	0.705099i	$-0.249098\pi$
$587$	34.3607	1.41822	0.709109	+	0.705099i	$0.249098\pi$
$588$	0	0
$589$	2.38197	0.0981472
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−27.2918	−1.12074	−0.560370	−	0.828242i	$-0.689341\pi$
$593$	−27.2918	−1.12074	−0.560370	+	0.828242i	$0.689341\pi$
$594$	0	0
$595$	7.61803	0.312309
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−40.1459	−1.64032	−0.820158	−	0.572136i	$-0.806115\pi$
$599$	−40.1459	−1.64032	−0.820158	+	0.572136i	$0.806115\pi$
$600$	0	0
$601$	18.9787	0.774158	0.387079	−	0.922047i	$-0.373484\pi$
$601$	18.9787	0.774158	0.387079	+	0.922047i	$0.373484\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	26.0902	1.06072
$606$	0	0
$607$	−36.5967	−1.48542	−0.742708	−	0.669615i	$-0.766459\pi$
$607$	−36.5967	−1.48542	−0.742708	+	0.669615i	$0.766459\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.5279	0.425912
$612$	0	0
$613$	2.03444	0.0821703	0.0410852	−	0.999156i	$-0.486919\pi$
$613$	2.03444	0.0821703	0.0410852	+	0.999156i	$0.486919\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.8541	1.08111	0.540553	−	0.841310i	$-0.318215\pi$
$617$	26.8541	1.08111	0.540553	+	0.841310i	$0.318215\pi$
$618$	0	0
$619$	−46.7984	−1.88099	−0.940493	−	0.339814i	$-0.889636\pi$
$619$	−46.7984	−1.88099	−0.940493	+	0.339814i	$0.889636\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.1803	0.487995
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−51.1033	−2.03762
$630$	0	0
$631$	0.888544	0.0353724	0.0176862	−	0.999844i	$-0.494370\pi$
$631$	0.888544	0.0353724	0.0176862	+	0.999844i	$0.494370\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15.2361	0.604625
$636$	0	0
$637$	−2.23607	−0.0885962
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.38197	−0.252073	−0.126036	−	0.992026i	$-0.540226\pi$
$641$	−6.38197	−0.252073	−0.126036	+	0.992026i	$0.540226\pi$
$642$	0	0
$643$	−7.47214	−0.294672	−0.147336	−	0.989086i	$-0.547070\pi$
$643$	−7.47214	−0.294672	−0.147336	+	0.989086i	$0.547070\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.6525	1.08713	0.543566	−	0.839367i	$-0.317074\pi$
$647$	27.6525	1.08713	0.543566	+	0.839367i	$0.317074\pi$
$648$	0	0
$649$	−65.2148	−2.55990
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.1246	0.865803	0.432901	−	0.901441i	$-0.357490\pi$
$653$	22.1246	0.865803	0.432901	+	0.901441i	$0.357490\pi$
$654$	0	0
$655$	4.38197	0.171218
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22.0902	−0.860511	−0.430255	−	0.902707i	$-0.641577\pi$
$659$	−22.0902	−0.860511	−0.430255	+	0.902707i	$0.641577\pi$
$660$	0	0
$661$	12.4721	0.485110	0.242555	−	0.970138i	$-0.422015\pi$
$661$	12.4721	0.485110	0.242555	+	0.970138i	$0.422015\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.00000	−0.0387783
$666$	0	0
$667$	−7.43769	−0.287989
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33.3262	−1.28655
$672$	0	0
$673$	−44.5066	−1.71560	−0.857801	−	0.513982i	$-0.828170\pi$
$673$	−44.5066	−1.71560	−0.857801	+	0.513982i	$0.828170\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.9230	1.84183	0.920915	−	0.389764i	$-0.127443\pi$
$677$	47.9230	1.84183	0.920915	+	0.389764i	$0.127443\pi$
$678$	0	0
$679$	8.70820	0.334190
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.29180	−0.279013	−0.139506	−	0.990221i	$-0.544552\pi$
$683$	−7.29180	−0.279013	−0.139506	+	0.990221i	$0.544552\pi$
$684$	0	0
$685$	−3.52786	−0.134793
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−32.0344	−1.22042
$690$	0	0
$691$	−39.1246	−1.48837	−0.744185	−	0.667973i	$-0.767162\pi$
$691$	−39.1246	−1.48837	−0.744185	+	0.667973i	$0.767162\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.18034	−0.196501
$696$	0	0
$697$	−0.686918	−0.0260189
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.47214	0.244449	0.122225	−	0.992502i	$-0.460997\pi$
$701$	6.47214	0.244449	0.122225	+	0.992502i	$0.460997\pi$
$702$	0	0
$703$	6.70820	0.253005
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.23607	0.309749
$708$	0	0
$709$	−22.7082	−0.852824	−0.426412	−	0.904529i	$-0.640223\pi$
$709$	−22.7082	−0.852824	−0.426412	+	0.904529i	$0.640223\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−20.7426	−0.776818
$714$	0	0
$715$	13.6180	0.509286
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−28.3607	−1.05767	−0.528837	−	0.848723i	$-0.677372\pi$
$719$	−28.3607	−1.05767	−0.528837	+	0.848723i	$0.677372\pi$
$720$	0	0
$721$	−7.76393	−0.289144
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.41641	0.126882
$726$	0	0
$727$	−18.5279	−0.687160	−0.343580	−	0.939123i	$-0.611640\pi$
$727$	−18.5279	−0.687160	−0.343580	+	0.939123i	$0.611640\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	64.5410	2.38714
$732$	0	0
$733$	38.8328	1.43432	0.717161	−	0.696907i	$-0.245441\pi$
$733$	38.8328	1.43432	0.717161	+	0.696907i	$0.245441\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.7295	0.468897
$738$	0	0
$739$	3.18034	0.116991	0.0584953	−	0.998288i	$-0.481370\pi$
$739$	3.18034	0.116991	0.0584953	+	0.998288i	$0.481370\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.34752	0.306241	0.153120	−	0.988208i	$-0.451068\pi$
$743$	8.34752	0.306241	0.153120	+	0.988208i	$0.451068\pi$
$744$	0	0
$745$	−7.00000	−0.256460
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.76393	−0.137531
$750$	0	0
$751$	−33.3820	−1.21813	−0.609063	−	0.793122i	$-0.708454\pi$
$751$	−33.3820	−1.21813	−0.609063	+	0.793122i	$0.708454\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.38197	−0.159476
$756$	0	0
$757$	28.8328	1.04795	0.523973	−	0.851735i	$-0.324449\pi$
$757$	28.8328	1.04795	0.523973	+	0.851735i	$0.324449\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.94427	0.215480	0.107740	−	0.994179i	$-0.465639\pi$
$761$	5.94427	0.215480	0.107740	+	0.994179i	$0.465639\pi$
$762$	0	0
$763$	18.8885	0.683811
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23.9443	−0.864578
$768$	0	0
$769$	−28.3607	−1.02271	−0.511356	−	0.859369i	$-0.670857\pi$
$769$	−28.3607	−1.02271	−0.511356	+	0.859369i	$0.670857\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	43.1246	1.55108	0.775542	−	0.631296i	$-0.217477\pi$
$773$	43.1246	1.55108	0.775542	+	0.631296i	$0.217477\pi$
$774$	0	0
$775$	9.52786	0.342251
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.0901699	0.00323067
$780$	0	0
$781$	−68.0902	−2.43646
$782$	0	0
$783$	0	0
$784$	0	0
$785$	13.6180	0.486048
$786$	0	0
$787$	−0.360680	−0.0128568	−0.00642842	−	0.999979i	$-0.502046\pi$
$787$	−0.360680	−0.0128568	−0.00642842	+	0.999979i	$0.502046\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.6180	0.519757
$792$	0	0
$793$	−12.2361	−0.434516
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.7295	0.698854	0.349427	−	0.936964i	$-0.386376\pi$
$797$	19.7295	0.698854	0.349427	+	0.936964i	$0.386376\pi$
$798$	0	0
$799$	−35.8673	−1.26889
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.54102	0.195538
$804$	0	0
$805$	8.70820	0.306924
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.0689	0.529794	0.264897	−	0.964277i	$-0.414662\pi$
$809$	15.0689	0.529794	0.264897	+	0.964277i	$0.414662\pi$
$810$	0	0
$811$	7.88854	0.277004	0.138502	−	0.990362i	$-0.455771\pi$
$811$	7.88854	0.277004	0.138502	+	0.990362i	$0.455771\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.7984	−0.378251
$816$	0	0
$817$	−8.47214	−0.296403
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.4164	−0.817238	−0.408619	−	0.912705i	$-0.633990\pi$
$821$	−23.4164	−0.817238	−0.408619	+	0.912705i	$0.633990\pi$
$822$	0	0
$823$	−33.2361	−1.15854	−0.579268	−	0.815137i	$-0.696662\pi$
$823$	−33.2361	−1.15854	−0.579268	+	0.815137i	$0.696662\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.8197	−0.863064	−0.431532	−	0.902098i	$-0.642027\pi$
$827$	−24.8197	−0.863064	−0.431532	+	0.902098i	$0.642027\pi$
$828$	0	0
$829$	−38.2492	−1.32845	−0.664225	−	0.747533i	$-0.731238\pi$
$829$	−38.2492	−1.32845	−0.664225	+	0.747533i	$0.731238\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.61803	0.263949
$834$	0	0
$835$	−1.47214	−0.0509454
$836$	0	0
$837$	0	0
$838$	0	0
$839$	36.3607	1.25531	0.627655	−	0.778492i	$-0.284015\pi$
$839$	36.3607	1.25531	0.627655	+	0.778492i	$0.284015\pi$
$840$	0	0
$841$	−28.2705	−0.974845
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.00000	−0.275208
$846$	0	0
$847$	26.0902	0.896469
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−58.4164	−2.00249
$852$	0	0
$853$	38.2148	1.30845	0.654225	−	0.756300i	$-0.272995\pi$
$853$	38.2148	1.30845	0.654225	+	0.756300i	$0.272995\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.3820	−0.832872	−0.416436	−	0.909165i	$-0.636721\pi$
$857$	−24.3820	−0.832872	−0.416436	+	0.909165i	$0.636721\pi$
$858$	0	0
$859$	31.9787	1.09110	0.545550	−	0.838078i	$-0.316321\pi$
$859$	31.9787	1.09110	0.545550	+	0.838078i	$0.316321\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.0344	−0.750061	−0.375031	−	0.927012i	$-0.622368\pi$
$863$	−22.0344	−0.750061	−0.375031	+	0.927012i	$0.622368\pi$
$864$	0	0
$865$	10.2918	0.349932
$866$	0	0
$867$	0	0
$868$	0	0
$869$	42.2918	1.43465
$870$	0	0
$871$	4.67376	0.158364
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−37.7082	−1.27332	−0.636658	−	0.771146i	$-0.719684\pi$
$877$	−37.7082	−1.27332	−0.636658	+	0.771146i	$0.719684\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16.5066	−0.556121	−0.278060	−	0.960564i	$-0.589692\pi$
$881$	−16.5066	−0.556121	−0.278060	+	0.960564i	$0.589692\pi$
$882$	0	0
$883$	−38.1246	−1.28300	−0.641498	−	0.767125i	$-0.721687\pi$
$883$	−38.1246	−1.28300	−0.641498	+	0.767125i	$0.721687\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	55.8328	1.87468	0.937341	−	0.348413i	$-0.113279\pi$
$887$	55.8328	1.87468	0.937341	+	0.348413i	$0.113279\pi$
$888$	0	0
$889$	15.2361	0.511001
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.70820	0.157554
$894$	0	0
$895$	10.3820	0.347031
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.03444	0.0678524
$900$	0	0
$901$	109.138	3.63591
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.5066	−0.448974
$906$	0	0
$907$	34.3050	1.13908	0.569539	−	0.821965i	$-0.307122\pi$
$907$	34.3050	1.13908	0.569539	+	0.821965i	$0.307122\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−0.472136	−0.0156426	−0.00782128	−	0.999969i	$-0.502490\pi$
$911$	−0.472136	−0.0156426	−0.00782128	+	0.999969i	$0.502490\pi$
$912$	0	0
$913$	89.0263	2.94634
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.38197	0.144705
$918$	0	0
$919$	31.3607	1.03449	0.517247	−	0.855836i	$-0.326957\pi$
$919$	31.3607	1.03449	0.517247	+	0.855836i	$0.326957\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−25.0000	−0.822885
$924$	0	0
$925$	26.8328	0.882258
$926$	0	0
$927$	0	0
$928$	0	0
$929$	11.2705	0.369773	0.184887	−	0.982760i	$-0.440808\pi$
$929$	11.2705	0.369773	0.184887	+	0.982760i	$0.440808\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−46.3951	−1.51728
$936$	0	0
$937$	−17.1459	−0.560132	−0.280066	−	0.959981i	$-0.590356\pi$
$937$	−17.1459	−0.560132	−0.280066	+	0.959981i	$0.590356\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−47.8885	−1.56112	−0.780561	−	0.625080i	$-0.785066\pi$
$941$	−47.8885	−1.56112	−0.780561	+	0.625080i	$0.785066\pi$
$942$	0	0
$943$	−0.785218	−0.0255702
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.0344	−0.846006	−0.423003	−	0.906128i	$-0.639024\pi$
$947$	−26.0344	−0.846006	−0.423003	+	0.906128i	$0.639024\pi$
$948$	0	0
$949$	2.03444	0.0660408
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−13.2148	−0.428069	−0.214034	−	0.976826i	$-0.568660\pi$
$953$	−13.2148	−0.428069	−0.214034	+	0.976826i	$0.568660\pi$
$954$	0	0
$955$	−8.03444	−0.259988
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.52786	−0.113921
$960$	0	0
$961$	−25.3262	−0.816975
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.43769	−0.0462810
$966$	0	0
$967$	35.9098	1.15478	0.577391	−	0.816468i	$-0.304071\pi$
$967$	35.9098	1.15478	0.577391	+	0.816468i	$0.304071\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.7639	−0.698438	−0.349219	−	0.937041i	$-0.613553\pi$
$971$	−21.7639	−0.698438	−0.349219	+	0.937041i	$0.613553\pi$
$972$	0	0
$973$	−5.18034	−0.166074
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.2361	−0.679402	−0.339701	−	0.940533i	$-0.610326\pi$
$977$	−21.2361	−0.679402	−0.339701	+	0.940533i	$0.610326\pi$
$978$	0	0
$979$	−74.1803	−2.37081
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−45.1803	−1.44103	−0.720515	−	0.693440i	$-0.756094\pi$
$983$	−45.1803	−1.44103	−0.720515	+	0.693440i	$0.756094\pi$
$984$	0	0
$985$	−8.79837	−0.280340
$986$	0	0
$987$	0	0
$988$	0	0
$989$	73.7771	2.34597
$990$	0	0
$991$	−11.1246	−0.353385	−0.176692	−	0.984266i	$-0.556540\pi$
$991$	−11.1246	−0.353385	−0.176692	+	0.984266i	$0.556540\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.23607	0.0708881
$996$	0	0
$997$	−33.2016	−1.05151	−0.525753	−	0.850637i	$-0.676216\pi$
$997$	−33.2016	−1.05151	−0.525753	+	0.850637i	$0.676216\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.l.1.1		2
3.2	odd	2		532.2.a.b.1.1	✓	2
12.11	even	2		2128.2.a.m.1.2		2
21.20	even	2		3724.2.a.g.1.2		2
24.5	odd	2		8512.2.a.bg.1.2		2
24.11	even	2		8512.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.b.1.1	✓	2	3.2	odd	2
2128.2.a.m.1.2		2	12.11	even	2
3724.2.a.g.1.2		2	21.20	even	2
4788.2.a.l.1.1		2	1.1	even	1	trivial
8512.2.a.k.1.1		2	24.11	even	2
8512.2.a.bg.1.2		2	24.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.00000 q^{7} -6.09017 q^{11} -2.23607 q^{13} +7.61803 q^{17} -1.00000 q^{19} +8.70820 q^{23} -4.00000 q^{25} -0.854102 q^{29} -2.38197 q^{31} +1.00000 q^{35} -6.70820 q^{37} -0.0901699 q^{41} +8.47214 q^{43} -4.70820 q^{47} +1.00000 q^{49} +14.3262 q^{53} -6.09017 q^{55} +10.7082 q^{59} +5.47214 q^{61} -2.23607 q^{65} -2.09017 q^{67} +11.1803 q^{71} -0.909830 q^{73} -6.09017 q^{77} -6.94427 q^{79} -14.6180 q^{83} +7.61803 q^{85} +12.1803 q^{89} -2.23607 q^{91} -1.00000 q^{95} +8.70820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7} - q^{11} + 13 q^{17} - 2 q^{19} + 4 q^{23} - 8 q^{25} + 5 q^{29} - 7 q^{31} + 2 q^{35} + 11 q^{41} + 8 q^{43} + 4 q^{47} + 2 q^{49} + 13 q^{53} - q^{55} + 8 q^{59} + 2 q^{61} + 7 q^{67}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 - q^11 + 13 * q^17 - 2 * q^19 + 4 * q^23 - 8 * q^25 + 5 * q^29 - 7 * q^31 + 2 * q^35 + 11 * q^41 + 8 * q^43 + 4 * q^47 + 2 * q^49 + 13 * q^53 - q^55 + 8 * q^59 + 2 * q^61 + 7 * q^67 - 13 * q^73 - q^77 + 4 * q^79 - 27 * q^83 + 13 * q^85 + 2 * q^89 - 2 * q^95 + 4 * q^97

Introduction

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