LMFDB - Embedded newform 4788.2.a.t.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4788,2,Mod(1,4788)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4788.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [6,0,0,0,0,0,-6,0,0,0,0,0,4] 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:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.56310016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 14x^{2} - 4 $ x^6 - 9x^4 + 14x^2 - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.23277$ 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)

$f(q)$	$=$	$q+3.51120 q^{5} -1.00000 q^{7} +0.779187 q^{11} -1.36794 q^{13} +6.75592 q^{17} -1.00000 q^{19} -5.71027 q^{23} +7.32850 q^{25} +4.29038 q^{29} +4.32850 q^{31} -3.51120 q^{35} -6.65699 q^{37} +8.70874 q^{41} -4.32850 q^{43} +7.66309 q^{47} +1.00000 q^{49} +7.53511 q^{53} +2.73588 q^{55} -6.65699 q^{61} -4.80310 q^{65} -0.960558 q^{67} -5.97674 q^{71} +11.9211 q^{73} -0.779187 q^{77} +8.96056 q^{79} -3.88542 q^{83} +23.7214 q^{85} +3.77767 q^{89} +1.36794 q^{91} -3.51120 q^{95} +12.4323 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} + 4 q^{13} - 6 q^{19} + 14 q^{25} - 4 q^{31} + 20 q^{37} + 4 q^{43} + 6 q^{49} - 8 q^{55} + 20 q^{61} + 12 q^{67} + 36 q^{73} + 36 q^{79} + 28 q^{85} - 4 q^{91} + 8 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 + 4 * q^13 - 6 * q^19 + 14 * q^25 - 4 * q^31 + 20 * q^37 + 4 * q^43 + 6 * q^49 - 8 * q^55 + 20 * q^61 + 12 * q^67 + 36 * q^73 + 36 * q^79 + 28 * q^85 - 4 * q^91 + 8 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.51120	1.57025	0.785127	−	0.619334i	$-0.212598\pi$
$5$	3.51120	1.57025	0.785127	+	0.619334i	$0.212598\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.779187	0.234934	0.117467	−	0.993077i	$-0.462523\pi$
$11$	0.779187	0.234934	0.117467	+	0.993077i	$0.462523\pi$
$12$	0	0
$13$	−1.36794	−0.379398	−0.189699	−	0.981842i	$-0.560751\pi$
$13$	−1.36794	−0.379398	−0.189699	+	0.981842i	$0.560751\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.75592	1.63855	0.819276	−	0.573400i	$-0.194376\pi$
$17$	6.75592	1.63855	0.819276	+	0.573400i	$0.194376\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.71027	−1.19067	−0.595336	−	0.803477i	$-0.702981\pi$
$23$	−5.71027	−1.19067	−0.595336	+	0.803477i	$0.702981\pi$
$24$	0	0
$25$	7.32850	1.46570
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.29038	0.796704	0.398352	−	0.917233i	$-0.369582\pi$
$29$	4.29038	0.796704	0.398352	+	0.917233i	$0.369582\pi$
$30$	0	0
$31$	4.32850	0.777421	0.388710	−	0.921360i	$-0.372921\pi$
$31$	4.32850	0.777421	0.388710	+	0.921360i	$0.372921\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.51120	−0.593500
$36$	0	0
$37$	−6.65699	−1.09440	−0.547201	−	0.837001i	$-0.684307\pi$
$37$	−6.65699	−1.09440	−0.547201	+	0.837001i	$0.684307\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.70874	1.36008	0.680039	−	0.733176i	$-0.261963\pi$
$41$	8.70874	1.36008	0.680039	+	0.733176i	$0.261963\pi$
$42$	0	0
$43$	−4.32850	−0.660089	−0.330045	−	0.943965i	$-0.607064\pi$
$43$	−4.32850	−0.660089	−0.330045	+	0.943965i	$0.607064\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.66309	1.11778	0.558888	−	0.829243i	$-0.311228\pi$
$47$	7.66309	1.11778	0.558888	+	0.829243i	$0.311228\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.53511	1.03503	0.517513	−	0.855675i	$-0.326858\pi$
$53$	7.53511	1.03503	0.517513	+	0.855675i	$0.326858\pi$
$54$	0	0
$55$	2.73588	0.368906
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−6.65699	−0.852340	−0.426170	−	0.904643i	$-0.640138\pi$
$61$	−6.65699	−0.852340	−0.426170	+	0.904643i	$0.640138\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.80310	−0.595751
$66$	0	0
$67$	−0.960558	−0.117351	−0.0586754	−	0.998277i	$-0.518688\pi$
$67$	−0.960558	−0.117351	−0.0586754	+	0.998277i	$0.518688\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.97674	−0.709308	−0.354654	−	0.934998i	$-0.615401\pi$
$71$	−5.97674	−0.709308	−0.354654	+	0.934998i	$0.615401\pi$
$72$	0	0
$73$	11.9211	1.39526	0.697630	−	0.716458i	$-0.254238\pi$
$73$	11.9211	1.39526	0.697630	+	0.716458i	$0.254238\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.779187	−0.0887965
$78$	0	0
$79$	8.96056	1.00814	0.504071	−	0.863662i	$-0.331835\pi$
$79$	8.96056	1.00814	0.504071	+	0.863662i	$0.331835\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.88542	−0.426481	−0.213240	−	0.977000i	$-0.568402\pi$
$83$	−3.88542	−0.426481	−0.213240	+	0.977000i	$0.568402\pi$
$84$	0	0
$85$	23.7214	2.57294
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.77767	0.400432	0.200216	−	0.979752i	$-0.435836\pi$
$89$	3.77767	0.400432	0.200216	+	0.979752i	$0.435836\pi$
$90$	0	0
$91$	1.36794	0.143399
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.51120	−0.360241
$96$	0	0
$97$	12.4323	1.26231	0.631155	−	0.775657i	$-0.282581\pi$
$97$	12.4323	1.26231	0.631155	+	0.775657i	$0.282581\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.82484	−0.181579	−0.0907893	−	0.995870i	$-0.528939\pi$
$101$	−1.82484	−0.181579	−0.0907893	+	0.995870i	$0.528939\pi$
$102$	0	0
$103$	9.92112	0.977557	0.488778	−	0.872408i	$-0.337443\pi$
$103$	9.92112	0.977557	0.488778	+	0.872408i	$0.337443\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.75440	−0.942994	−0.471497	−	0.881868i	$-0.656286\pi$
$107$	−9.75440	−0.942994	−0.471497	+	0.881868i	$0.656286\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.13015	−0.670748	−0.335374	−	0.942085i	$-0.608863\pi$
$113$	−7.13015	−0.670748	−0.335374	+	0.942085i	$0.608863\pi$
$114$	0	0
$115$	−20.0499	−1.86966
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.75592	−0.619314
$120$	0	0
$121$	−10.3929	−0.944806
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.17580	0.731266
$126$	0	0
$127$	10.4323	0.925718	0.462859	−	0.886432i	$-0.346824\pi$
$127$	10.4323	0.925718	0.462859	+	0.886432i	$0.346824\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.06805	0.704909	0.352454	−	0.935829i	$-0.385347\pi$
$131$	8.06805	0.704909	0.352454	+	0.935829i	$0.385347\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.37271	0.288150	0.144075	−	0.989567i	$-0.453979\pi$
$137$	3.37271	0.288150	0.144075	+	0.989567i	$0.453979\pi$
$138$	0	0
$139$	−15.3929	−1.30561	−0.652803	−	0.757528i	$-0.726407\pi$
$139$	−15.3929	−1.30561	−0.652803	+	0.757528i	$0.726407\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.06588	−0.0891333
$144$	0	0
$145$	15.0644	1.25103
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.9535	−0.979266	−0.489633	−	0.871928i	$-0.662869\pi$
$149$	−11.9535	−0.979266	−0.489633	+	0.871928i	$0.662869\pi$
$150$	0	0
$151$	22.4323	1.82552	0.912758	−	0.408501i	$-0.133948\pi$
$151$	22.4323	1.82552	0.912758	+	0.408501i	$0.133948\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.1982	1.22075
$156$	0	0
$157$	11.9211	0.951409	0.475704	−	0.879605i	$-0.342193\pi$
$157$	11.9211	0.951409	0.475704	+	0.879605i	$0.342193\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.71027	0.450032
$162$	0	0
$163$	19.7214	1.54470	0.772348	−	0.635199i	$-0.219082\pi$
$163$	19.7214	1.54470	0.772348	+	0.635199i	$0.219082\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.4429	1.42716	0.713578	−	0.700576i	$-0.247073\pi$
$167$	18.4429	1.42716	0.713578	+	0.700576i	$0.247073\pi$
$168$	0	0
$169$	−11.1287	−0.856057
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.77767	0.287211	0.143605	−	0.989635i	$-0.454130\pi$
$173$	3.77767	0.287211	0.143605	+	0.989635i	$0.454130\pi$
$174$	0	0
$175$	−7.32850	−0.553982
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.3352	−1.37043	−0.685217	−	0.728339i	$-0.740293\pi$
$179$	−18.3352	−1.37043	−0.685217	+	0.728339i	$0.740293\pi$
$180$	0	0
$181$	−17.0893	−1.27024	−0.635119	−	0.772414i	$-0.719049\pi$
$181$	−17.0893	−1.27024	−0.635119	+	0.772414i	$0.719049\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−23.3740	−1.71849
$186$	0	0
$187$	5.26412	0.384951
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.40344	−0.246264	−0.123132	−	0.992390i	$-0.539294\pi$
$191$	−3.40344	−0.246264	−0.123132	+	0.992390i	$0.539294\pi$
$192$	0	0
$193$	−11.3140	−0.814398	−0.407199	−	0.913339i	$-0.633495\pi$
$193$	−11.3140	−0.814398	−0.407199	+	0.913339i	$0.633495\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.11675	0.222059	0.111029	−	0.993817i	$-0.464585\pi$
$197$	3.11675	0.222059	0.111029	+	0.993817i	$0.464585\pi$
$198$	0	0
$199$	−17.3140	−1.22736	−0.613678	−	0.789556i	$-0.710311\pi$
$199$	−17.3140	−1.22736	−0.613678	+	0.789556i	$0.710311\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.29038	−0.301126
$204$	0	0
$205$	30.5781	2.13567
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.779187	−0.0538975
$210$	0	0
$211$	−14.4323	−0.993561	−0.496781	−	0.867876i	$-0.665485\pi$
$211$	−14.4323	−0.993561	−0.496781	+	0.867876i	$0.665485\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−15.1982	−1.03651
$216$	0	0
$217$	−4.32850	−0.293837
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.24168	−0.621663
$222$	0	0
$223$	20.9855	1.40529	0.702646	−	0.711540i	$-0.252002\pi$
$223$	20.9855	1.40529	0.702646	+	0.711540i	$0.252002\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.83672	−0.586514	−0.293257	−	0.956034i	$-0.594739\pi$
$227$	−8.83672	−0.586514	−0.293257	+	0.956034i	$0.594739\pi$
$228$	0	0
$229$	16.7359	1.10594	0.552969	−	0.833202i	$-0.313495\pi$
$229$	16.7359	1.10594	0.552969	+	0.833202i	$0.313495\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.6255	−1.48225	−0.741124	−	0.671368i	$-0.765707\pi$
$233$	−22.6255	−1.48225	−0.741124	+	0.671368i	$0.765707\pi$
$234$	0	0
$235$	26.9066	1.75519
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.17733	0.334893	0.167447	−	0.985881i	$-0.446448\pi$
$239$	5.17733	0.334893	0.167447	+	0.985881i	$0.446448\pi$
$240$	0	0
$241$	−0.224681	−0.0144730	−0.00723649	−	0.999974i	$-0.502303\pi$
$241$	−0.224681	−0.0144730	−0.00723649	+	0.999974i	$0.502303\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.51120	0.224322
$246$	0	0
$247$	1.36794	0.0870398
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.68852	−0.548415	−0.274207	−	0.961671i	$-0.588415\pi$
$251$	−8.68852	−0.548415	−0.274207	+	0.961671i	$0.588415\pi$
$252$	0	0
$253$	−4.44936	−0.279729
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.4063	1.27291	0.636454	−	0.771315i	$-0.280401\pi$
$257$	20.4063	1.27291	0.636454	+	0.771315i	$0.280401\pi$
$258$	0	0
$259$	6.65699	0.413645
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−26.7774	−1.65117	−0.825584	−	0.564279i	$-0.809154\pi$
$263$	−26.7774	−1.65117	−0.825584	+	0.564279i	$0.809154\pi$
$264$	0	0
$265$	26.4572	1.62526
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.61743	−0.403472	−0.201736	−	0.979440i	$-0.564658\pi$
$269$	−6.61743	−0.403472	−0.201736	+	0.979440i	$0.564658\pi$
$270$	0	0
$271$	−11.3929	−0.692067	−0.346034	−	0.938222i	$-0.612472\pi$
$271$	−11.3929	−0.692067	−0.346034	+	0.938222i	$0.612472\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.71027	0.344342
$276$	0	0
$277$	6.93563	0.416721	0.208361	−	0.978052i	$-0.433187\pi$
$277$	6.93563	0.416721	0.208361	+	0.978052i	$0.433187\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.1128	1.31914	0.659570	−	0.751643i	$-0.270738\pi$
$281$	22.1128	1.31914	0.659570	+	0.751643i	$0.270738\pi$
$282$	0	0
$283$	−3.34301	−0.198721	−0.0993606	−	0.995051i	$-0.531680\pi$
$283$	−3.34301	−0.198721	−0.0993606	+	0.995051i	$0.531680\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.70874	−0.514061
$288$	0	0
$289$	28.6425	1.68485
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.8255	−0.690853	−0.345426	−	0.938446i	$-0.612266\pi$
$293$	−11.8255	−0.690853	−0.345426	+	0.938446i	$0.612266\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.81129	0.451739
$300$	0	0
$301$	4.32850	0.249490
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−23.3740	−1.33839
$306$	0	0
$307$	3.51373	0.200539	0.100270	−	0.994960i	$-0.468029\pi$
$307$	3.51373	0.200539	0.100270	+	0.994960i	$0.468029\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.04566	−0.0592937	−0.0296469	−	0.999560i	$-0.509438\pi$
$311$	−1.04566	−0.0592937	−0.0296469	+	0.999560i	$0.509438\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.7557	1.67125	0.835623	−	0.549303i	$-0.185107\pi$
$317$	29.7557	1.67125	0.835623	+	0.549303i	$0.185107\pi$
$318$	0	0
$319$	3.34301	0.187173
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.75592	−0.375910
$324$	0	0
$325$	−10.0249	−0.556083
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.66309	−0.422480
$330$	0	0
$331$	−13.6175	−0.748488	−0.374244	−	0.927330i	$-0.622098\pi$
$331$	−13.6175	−0.748488	−0.374244	+	0.927330i	$0.622098\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.37271	−0.184271
$336$	0	0
$337$	−3.92112	−0.213597	−0.106798	−	0.994281i	$-0.534060\pi$
$337$	−3.92112	−0.213597	−0.106798	+	0.994281i	$0.534060\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.37271	0.182642
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.98725	−0.321412	−0.160706	−	0.987002i	$-0.551377\pi$
$347$	−5.98725	−0.321412	−0.160706	+	0.987002i	$0.551377\pi$
$348$	0	0
$349$	4.73588	0.253506	0.126753	−	0.991934i	$-0.459544\pi$
$349$	4.73588	0.253506	0.126753	+	0.991934i	$0.459544\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.2311	−1.18324	−0.591621	−	0.806216i	$-0.701512\pi$
$353$	−22.2311	−1.18324	−0.591621	+	0.806216i	$0.701512\pi$
$354$	0	0
$355$	−20.9855	−1.11379
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.3134	−1.12488	−0.562440	−	0.826838i	$-0.690137\pi$
$359$	−21.3134	−1.12488	−0.562440	+	0.826838i	$0.690137\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	41.8574	2.19091
$366$	0	0
$367$	14.1287	0.737514	0.368757	−	0.929526i	$-0.379783\pi$
$367$	14.1287	0.737514	0.368757	+	0.929526i	$0.379783\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.53511	−0.391203
$372$	0	0
$373$	−29.2351	−1.51374	−0.756868	−	0.653568i	$-0.773271\pi$
$373$	−29.2351	−1.51374	−0.756868	+	0.653568i	$0.773271\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.86898	−0.302268
$378$	0	0
$379$	−10.4323	−0.535872	−0.267936	−	0.963437i	$-0.586342\pi$
$379$	−10.4323	−0.535872	−0.267936	+	0.963437i	$0.586342\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	36.6703	1.87377	0.936883	−	0.349643i	$-0.113697\pi$
$383$	36.6703	1.87377	0.936883	+	0.349643i	$0.113697\pi$
$384$	0	0
$385$	−2.73588	−0.139433
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.27835	−0.369027	−0.184514	−	0.982830i	$-0.559071\pi$
$389$	−7.27835	−0.369027	−0.184514	+	0.982830i	$0.559071\pi$
$390$	0	0
$391$	−38.5781	−1.95098
$392$	0	0
$393$	0	0
$394$	0	0
$395$	31.4623	1.58304
$396$	0	0
$397$	26.8646	1.34830	0.674148	−	0.738596i	$-0.264511\pi$
$397$	26.8646	1.34830	0.674148	+	0.738596i	$0.264511\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.9078	0.544710	0.272355	−	0.962197i	$-0.412197\pi$
$401$	10.9078	0.544710	0.272355	+	0.962197i	$0.412197\pi$
$402$	0	0
$403$	−5.92112	−0.294952
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.18704	−0.257112
$408$	0	0
$409$	2.02493	0.100126	0.0500632	−	0.998746i	$-0.484058\pi$
$409$	2.02493	0.100126	0.0500632	+	0.998746i	$0.484058\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−13.6425	−0.669683
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.7992	1.16267	0.581333	−	0.813666i	$-0.302531\pi$
$419$	23.7992	1.16267	0.581333	+	0.813666i	$0.302531\pi$
$420$	0	0
$421$	25.2351	1.22988	0.614942	−	0.788573i	$-0.289180\pi$
$421$	25.2351	1.22988	0.614942	+	0.788573i	$0.289180\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	49.5107	2.40162
$426$	0	0
$427$	6.65699	0.322154
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.2469	−0.493576	−0.246788	−	0.969070i	$-0.579375\pi$
$431$	−10.2469	−0.493576	−0.246788	+	0.969070i	$0.579375\pi$
$432$	0	0
$433$	−34.4033	−1.65332	−0.826658	−	0.562704i	$-0.809761\pi$
$433$	−34.4033	−1.65332	−0.826658	+	0.562704i	$0.809761\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.71027	0.273159
$438$	0	0
$439$	36.3784	1.73624	0.868122	−	0.496351i	$-0.165327\pi$
$439$	36.3784	1.73624	0.868122	+	0.496351i	$0.165327\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.80462	0.0857401	0.0428700	−	0.999081i	$-0.486350\pi$
$443$	1.80462	0.0857401	0.0428700	+	0.999081i	$0.486350\pi$
$444$	0	0
$445$	13.2641	0.628780
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.28356	0.390925	0.195463	−	0.980711i	$-0.437379\pi$
$449$	8.28356	0.390925	0.195463	+	0.980711i	$0.437379\pi$
$450$	0	0
$451$	6.78574	0.319528
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.80310	0.225173
$456$	0	0
$457$	26.9855	1.26233	0.631164	−	0.775649i	$-0.282578\pi$
$457$	26.9855	1.26233	0.631164	+	0.775649i	$0.282578\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.6150	−1.05329	−0.526644	−	0.850086i	$-0.676550\pi$
$461$	−22.6150	−1.05329	−0.526644	+	0.850086i	$0.676550\pi$
$462$	0	0
$463$	−22.9435	−1.06628	−0.533138	−	0.846029i	$-0.678987\pi$
$463$	−22.9435	−1.06628	−0.533138	+	0.846029i	$0.678987\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.3973	−0.805049	−0.402525	−	0.915409i	$-0.631867\pi$
$467$	−17.3973	−0.805049	−0.402525	+	0.915409i	$0.631867\pi$
$468$	0	0
$469$	0.960558	0.0443544
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.37271	−0.155077
$474$	0	0
$475$	−7.32850	−0.336254
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.1271	0.599793	0.299896	−	0.953972i	$-0.403048\pi$
$479$	13.1271	0.599793	0.299896	+	0.953972i	$0.403048\pi$
$480$	0	0
$481$	9.10635	0.415214
$482$	0	0
$483$	0	0
$484$	0	0
$485$	43.6523	1.98215
$486$	0	0
$487$	−3.69643	−0.167501	−0.0837507	−	0.996487i	$-0.526690\pi$
$487$	−3.69643	−0.167501	−0.0837507	+	0.996487i	$0.526690\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	35.0812	1.58319	0.791597	−	0.611044i	$-0.209250\pi$
$491$	35.0812	1.58319	0.791597	+	0.611044i	$0.209250\pi$
$492$	0	0
$493$	28.9855	1.30544
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.97674	0.268093
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19.7656	−0.881303	−0.440651	−	0.897678i	$-0.645252\pi$
$503$	−19.7656	−0.881303	−0.440651	+	0.897678i	$0.645252\pi$
$504$	0	0
$505$	−6.40738	−0.285125
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.8927	1.45794	0.728971	−	0.684545i	$-0.239999\pi$
$509$	32.8927	1.45794	0.728971	+	0.684545i	$0.239999\pi$
$510$	0	0
$511$	−11.9211	−0.527359
$512$	0	0
$513$	0	0
$514$	0	0
$515$	34.8350	1.53501
$516$	0	0
$517$	5.97098	0.262603
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.8957	−1.17832	−0.589161	−	0.808016i	$-0.700542\pi$
$521$	−26.8957	−1.17832	−0.589161	+	0.808016i	$0.700542\pi$
$522$	0	0
$523$	18.9066	0.826728	0.413364	−	0.910566i	$-0.364354\pi$
$523$	18.9066	0.826728	0.413364	+	0.910566i	$0.364354\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	29.2430	1.27384
$528$	0	0
$529$	9.60713	0.417701
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.9130	−0.516010
$534$	0	0
$535$	−34.2496	−1.48074
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.779187	0.0335619
$540$	0	0
$541$	10.8148	0.464963	0.232481	−	0.972601i	$-0.425316\pi$
$541$	10.8148	0.464963	0.232481	+	0.972601i	$0.425316\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	35.1120	1.50403
$546$	0	0
$547$	1.11833	0.0478162	0.0239081	−	0.999714i	$-0.492389\pi$
$547$	1.11833	0.0478162	0.0239081	+	0.999714i	$0.492389\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.29038	−0.182776
$552$	0	0
$553$	−8.96056	−0.381042
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17.9504	−0.760584	−0.380292	−	0.924866i	$-0.624177\pi$
$557$	−17.9504	−0.760584	−0.380292	+	0.924866i	$0.624177\pi$
$558$	0	0
$559$	5.92112	0.250436
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.77085	−0.327502	−0.163751	−	0.986502i	$-0.552359\pi$
$563$	−7.77085	−0.327502	−0.163751	+	0.986502i	$0.552359\pi$
$564$	0	0
$565$	−25.0353	−1.05324
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.8359	−0.915406	−0.457703	−	0.889105i	$-0.651328\pi$
$569$	−21.8359	−0.915406	−0.457703	+	0.889105i	$0.651328\pi$
$570$	0	0
$571$	31.2351	1.30715	0.653574	−	0.756863i	$-0.273269\pi$
$571$	31.2351	1.30715	0.653574	+	0.756863i	$0.273269\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−41.8477	−1.74517
$576$	0	0
$577$	−36.6280	−1.52484	−0.762421	−	0.647081i	$-0.775989\pi$
$577$	−36.6280	−1.52484	−0.762421	+	0.647081i	$0.775989\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.88542	0.161194
$582$	0	0
$583$	5.87126	0.243163
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.47301	0.349718	0.174859	−	0.984593i	$-0.444053\pi$
$587$	8.47301	0.349718	0.174859	+	0.984593i	$0.444053\pi$
$588$	0	0
$589$	−4.32850	−0.178353
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.7416	0.646432	0.323216	−	0.946325i	$-0.395236\pi$
$593$	15.7416	0.646432	0.323216	+	0.946325i	$0.395236\pi$
$594$	0	0
$595$	−23.7214	−0.972481
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.1314	−1.10856	−0.554280	−	0.832330i	$-0.687006\pi$
$599$	−27.1314	−1.10856	−0.554280	+	0.832330i	$0.687006\pi$
$600$	0	0
$601$	−13.0394	−0.531890	−0.265945	−	0.963988i	$-0.585684\pi$
$601$	−13.0394	−0.531890	−0.265945	+	0.963988i	$0.585684\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−36.4914	−1.48359
$606$	0	0
$607$	−7.84223	−0.318306	−0.159153	−	0.987254i	$-0.550876\pi$
$607$	−7.84223	−0.318306	−0.159153	+	0.987254i	$0.550876\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.4826	−0.424082
$612$	0	0
$613$	36.2496	1.46411	0.732054	−	0.681247i	$-0.238562\pi$
$613$	36.2496	1.46411	0.732054	+	0.681247i	$0.238562\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17.1615	−0.690897	−0.345449	−	0.938438i	$-0.612273\pi$
$617$	−17.1615	−0.690897	−0.345449	+	0.938438i	$0.612273\pi$
$618$	0	0
$619$	−14.0789	−0.565878	−0.282939	−	0.959138i	$-0.591309\pi$
$619$	−14.0789	−0.565878	−0.282939	+	0.959138i	$0.591309\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.77767	−0.151349
$624$	0	0
$625$	−7.93563	−0.317425
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−44.9741	−1.79324
$630$	0	0
$631$	−22.4074	−0.892024	−0.446012	−	0.895027i	$-0.647156\pi$
$631$	−22.4074	−0.892024	−0.446012	+	0.895027i	$0.647156\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	36.6299	1.45361
$636$	0	0
$637$	−1.36794	−0.0541997
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.4818	0.927474	0.463737	−	0.885973i	$-0.346508\pi$
$641$	23.4818	0.927474	0.463737	+	0.885973i	$0.346508\pi$
$642$	0	0
$643$	−3.34301	−0.131835	−0.0659177	−	0.997825i	$-0.520997\pi$
$643$	−3.34301	−0.131835	−0.0659177	+	0.997825i	$0.520997\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.6209	1.32177	0.660887	−	0.750486i	$-0.270180\pi$
$647$	33.6209	1.32177	0.660887	+	0.750486i	$0.270180\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.60620	−0.375920	−0.187960	−	0.982177i	$-0.560187\pi$
$653$	−9.60620	−0.375920	−0.187960	+	0.982177i	$0.560187\pi$
$654$	0	0
$655$	28.3285	1.10689
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.1594	0.395753	0.197876	−	0.980227i	$-0.436596\pi$
$659$	10.1594	0.395753	0.197876	+	0.980227i	$0.436596\pi$
$660$	0	0
$661$	35.9959	1.40008	0.700039	−	0.714104i	$-0.253166\pi$
$661$	35.9959	1.40008	0.700039	+	0.714104i	$0.253166\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.51120	0.136158
$666$	0	0
$667$	−24.4992	−0.948614
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.18704	−0.200243
$672$	0	0
$673$	4.07888	0.157229	0.0786147	−	0.996905i	$-0.474950\pi$
$673$	4.07888	0.157229	0.0786147	+	0.996905i	$0.474950\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.9630	−1.34374	−0.671868	−	0.740671i	$-0.734508\pi$
$677$	−34.9630	−1.34374	−0.671868	+	0.740671i	$0.734508\pi$
$678$	0	0
$679$	−12.4323	−0.477108
$680$	0	0
$681$	0	0
$682$	0	0
$683$	42.4576	1.62460	0.812298	−	0.583243i	$-0.198216\pi$
$683$	42.4576	1.62460	0.812298	+	0.583243i	$0.198216\pi$
$684$	0	0
$685$	11.8422	0.452468
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.3076	−0.392687
$690$	0	0
$691$	−43.2351	−1.64474	−0.822370	−	0.568953i	$-0.807349\pi$
$691$	−43.2351	−1.64474	−0.822370	+	0.568953i	$0.807349\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−54.0474	−2.05013
$696$	0	0
$697$	58.8356	2.22856
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.60620	−0.362821	−0.181411	−	0.983407i	$-0.558066\pi$
$701$	−9.60620	−0.362821	−0.181411	+	0.983407i	$0.558066\pi$
$702$	0	0
$703$	6.65699	0.251073
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.82484	0.0686303
$708$	0	0
$709$	−23.8002	−0.893837	−0.446919	−	0.894575i	$-0.647479\pi$
$709$	−23.8002	−0.893837	−0.446919	+	0.894575i	$0.647479\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−24.7169	−0.925654
$714$	0	0
$715$	−3.74251	−0.139962
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−49.3925	−1.84203	−0.921014	−	0.389529i	$-0.872638\pi$
$719$	−49.3925	−1.84203	−0.921014	+	0.389529i	$0.872638\pi$
$720$	0	0
$721$	−9.92112	−0.369482
$722$	0	0
$723$	0	0
$724$	0	0
$725$	31.4420	1.16773
$726$	0	0
$727$	−30.5781	−1.13408	−0.567040	−	0.823691i	$-0.691911\pi$
$727$	−30.5781	−1.13408	−0.567040	+	0.823691i	$0.691911\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−29.2430	−1.08159
$732$	0	0
$733$	−13.8422	−0.511274	−0.255637	−	0.966773i	$-0.582285\pi$
$733$	−13.8422	−0.511274	−0.255637	+	0.966773i	$0.582285\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.748454	−0.0275696
$738$	0	0
$739$	−26.4572	−0.973245	−0.486623	−	0.873612i	$-0.661771\pi$
$739$	−26.4572	−0.973245	−0.486623	+	0.873612i	$0.661771\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.4295	−0.529368	−0.264684	−	0.964335i	$-0.585268\pi$
$743$	−14.4295	−0.529368	−0.264684	+	0.964335i	$0.585268\pi$
$744$	0	0
$745$	−41.9710	−1.53770
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.75440	0.356418
$750$	0	0
$751$	17.6674	0.644693	0.322346	−	0.946622i	$-0.395528\pi$
$751$	17.6674	0.644693	0.322346	+	0.946622i	$0.395528\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	78.7642	2.86652
$756$	0	0
$757$	−31.5216	−1.14567	−0.572836	−	0.819670i	$-0.694157\pi$
$757$	−31.5216	−1.14567	−0.572836	+	0.819670i	$0.694157\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.22980	−0.0808303	−0.0404151	−	0.999183i	$-0.512868\pi$
$761$	−2.22980	−0.0808303	−0.0404151	+	0.999183i	$0.512868\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−35.3140	−1.27345	−0.636727	−	0.771089i	$-0.719712\pi$
$769$	−35.3140	−1.27345	−0.636727	+	0.771089i	$0.719712\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.6846	0.995746	0.497873	−	0.867250i	$-0.334115\pi$
$773$	27.6846	0.995746	0.497873	+	0.867250i	$0.334115\pi$
$774$	0	0
$775$	31.7214	1.13947
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.70874	−0.312023
$780$	0	0
$781$	−4.65699	−0.166640
$782$	0	0
$783$	0	0
$784$	0	0
$785$	41.8574	1.49395
$786$	0	0
$787$	−25.5926	−0.912278	−0.456139	−	0.889908i	$-0.650768\pi$
$787$	−25.5926	−0.912278	−0.456139	+	0.889908i	$0.650768\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.13015	0.253519
$792$	0	0
$793$	9.10635	0.323376
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−46.4045	−1.64373	−0.821866	−	0.569681i	$-0.807067\pi$
$797$	−46.4045	−1.64373	−0.821866	+	0.569681i	$0.807067\pi$
$798$	0	0
$799$	51.7712	1.83153
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.28877	0.327794
$804$	0	0
$805$	20.0499	0.706665
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36.3934	−1.27952	−0.639761	−	0.768574i	$-0.720967\pi$
$809$	−36.3934	−1.27952	−0.639761	+	0.768574i	$0.720967\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	69.2456	2.42557
$816$	0	0
$817$	4.32850	0.151435
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.8845	0.589275	0.294637	−	0.955609i	$-0.404801\pi$
$821$	16.8845	0.589275	0.294637	+	0.955609i	$0.404801\pi$
$822$	0	0
$823$	−2.85674	−0.0995798	−0.0497899	−	0.998760i	$-0.515855\pi$
$823$	−2.85674	−0.0995798	−0.0497899	+	0.998760i	$0.515855\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−45.4868	−1.58173	−0.790866	−	0.611989i	$-0.790370\pi$
$827$	−45.4868	−1.58173	−0.790866	+	0.611989i	$0.790370\pi$
$828$	0	0
$829$	−32.2745	−1.12094	−0.560471	−	0.828174i	$-0.689380\pi$
$829$	−32.2745	−1.12094	−0.560471	+	0.828174i	$0.689380\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.75592	0.234079
$834$	0	0
$835$	64.7567	2.24100
$836$	0	0
$837$	0	0
$838$	0	0
$839$	41.3244	1.42668	0.713339	−	0.700819i	$-0.247182\pi$
$839$	41.3244	1.42668	0.713339	+	0.700819i	$0.247182\pi$
$840$	0	0
$841$	−10.5926	−0.365263
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−39.0752	−1.34423
$846$	0	0
$847$	10.3929	0.357103
$848$	0	0
$849$	0	0
$850$	0	0
$851$	38.0132	1.30308
$852$	0	0
$853$	−18.6570	−0.638803	−0.319402	−	0.947619i	$-0.603482\pi$
$853$	−18.6570	−0.638803	−0.319402	+	0.947619i	$0.603482\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.0051	−0.751679	−0.375840	−	0.926685i	$-0.622646\pi$
$857$	−22.0051	−0.751679	−0.375840	+	0.926685i	$0.622646\pi$
$858$	0	0
$859$	−36.6570	−1.25072	−0.625360	−	0.780336i	$-0.715048\pi$
$859$	−36.6570	−1.25072	−0.625360	+	0.780336i	$0.715048\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.94752	0.100335	0.0501674	−	0.998741i	$-0.484025\pi$
$863$	2.94752	0.100335	0.0501674	+	0.998741i	$0.484025\pi$
$864$	0	0
$865$	13.2641	0.450994
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.98195	0.236846
$870$	0	0
$871$	1.31398	0.0445226
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.17580	−0.276393
$876$	0	0
$877$	−43.3140	−1.46261	−0.731305	−	0.682051i	$-0.761088\pi$
$877$	−43.3140	−1.46261	−0.731305	+	0.682051i	$0.761088\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.90934	0.266472	0.133236	−	0.991084i	$-0.457463\pi$
$881$	7.90934	0.266472	0.133236	+	0.991084i	$0.457463\pi$
$882$	0	0
$883$	−21.4718	−0.722582	−0.361291	−	0.932453i	$-0.617664\pi$
$883$	−21.4718	−0.722582	−0.361291	+	0.932453i	$0.617664\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.8815	−0.768286	−0.384143	−	0.923274i	$-0.625503\pi$
$887$	−22.8815	−0.768286	−0.384143	+	0.923274i	$0.625503\pi$
$888$	0	0
$889$	−10.4323	−0.349888
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7.66309	−0.256435
$894$	0	0
$895$	−64.3784	−2.15193
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18.5709	0.619374
$900$	0	0
$901$	50.9066	1.69594
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−60.0039	−1.99460
$906$	0	0
$907$	−3.53866	−0.117499	−0.0587497	−	0.998273i	$-0.518711\pi$
$907$	−3.53866	−0.117499	−0.0587497	+	0.998273i	$0.518711\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.35248	0.111073	0.0555364	−	0.998457i	$-0.482313\pi$
$911$	3.35248	0.111073	0.0555364	+	0.998457i	$0.482313\pi$
$912$	0	0
$913$	−3.02747	−0.100195
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.06805	−0.266430
$918$	0	0
$919$	36.2575	1.19602	0.598012	−	0.801487i	$-0.295957\pi$
$919$	36.2575	1.19602	0.598012	+	0.801487i	$0.295957\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.17580	0.269110
$924$	0	0
$925$	−48.7857	−1.60407
$926$	0	0
$927$	0	0
$928$	0	0
$929$	23.1075	0.758134	0.379067	−	0.925369i	$-0.376245\pi$
$929$	23.1075	0.758134	0.379067	+	0.925369i	$0.376245\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18.4834	0.604471
$936$	0	0
$937$	10.7069	0.349778	0.174889	−	0.984588i	$-0.444043\pi$
$937$	10.7069	0.349778	0.174889	+	0.984588i	$0.444043\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.7548	−1.39377	−0.696884	−	0.717184i	$-0.745431\pi$
$941$	−42.7548	−1.39377	−0.696884	+	0.717184i	$0.745431\pi$
$942$	0	0
$943$	−49.7292	−1.61941
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.10103	−0.0682741	−0.0341371	−	0.999417i	$-0.510868\pi$
$947$	−2.10103	−0.0682741	−0.0341371	+	0.999417i	$0.510868\pi$
$948$	0	0
$949$	−16.3073	−0.529359
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11.4408	−0.370602	−0.185301	−	0.982682i	$-0.559326\pi$
$953$	−11.4408	−0.370602	−0.185301	+	0.982682i	$0.559326\pi$
$954$	0	0
$955$	−11.9501	−0.386697
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.37271	−0.108910
$960$	0	0
$961$	−12.2641	−0.395617
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−39.7256	−1.27881
$966$	0	0
$967$	−11.2221	−0.360880	−0.180440	−	0.983586i	$-0.557752\pi$
$967$	−11.2221	−0.360880	−0.180440	+	0.983586i	$0.557752\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.29937	−0.234248	−0.117124	−	0.993117i	$-0.537367\pi$
$971$	−7.29937	−0.234248	−0.117124	+	0.993117i	$0.537367\pi$
$972$	0	0
$973$	15.3929	0.493473
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−36.3941	−1.16435	−0.582176	−	0.813063i	$-0.697799\pi$
$977$	−36.3941	−1.16435	−0.582176	+	0.813063i	$0.697799\pi$
$978$	0	0
$979$	2.94351	0.0940749
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−43.6313	−1.39162	−0.695810	−	0.718225i	$-0.744955\pi$
$983$	−43.6313	−1.39162	−0.695810	+	0.718225i	$0.744955\pi$
$984$	0	0
$985$	10.9435	0.348689
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.7169	0.785950
$990$	0	0
$991$	−29.6175	−0.940832	−0.470416	−	0.882445i	$-0.655896\pi$
$991$	−29.6175	−0.940832	−0.470416	+	0.882445i	$0.655896\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−60.7928	−1.92726
$996$	0	0
$997$	33.3929	1.05756	0.528781	−	0.848758i	$-0.322649\pi$
$997$	33.3929	1.05756	0.528781	+	0.848758i	$0.322649\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.t.1.6	yes	6
3.2	odd	2	inner	4788.2.a.t.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4788.2.a.t.1.1	✓	6	3.2	odd	2	inner
4788.2.a.t.1.6	yes	6	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+3.51120 q^{5} -1.00000 q^{7} +0.779187 q^{11} -1.36794 q^{13} +6.75592 q^{17} -1.00000 q^{19} -5.71027 q^{23} +7.32850 q^{25} +4.29038 q^{29} +4.32850 q^{31} -3.51120 q^{35} -6.65699 q^{37} +8.70874 q^{41} -4.32850 q^{43} +7.66309 q^{47} +1.00000 q^{49} +7.53511 q^{53} +2.73588 q^{55} -6.65699 q^{61} -4.80310 q^{65} -0.960558 q^{67} -5.97674 q^{71} +11.9211 q^{73} -0.779187 q^{77} +8.96056 q^{79} -3.88542 q^{83} +23.7214 q^{85} +3.77767 q^{89} +1.36794 q^{91} -3.51120 q^{95} +12.4323 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} + 4 q^{13} - 6 q^{19} + 14 q^{25} - 4 q^{31} + 20 q^{37} + 4 q^{43} + 6 q^{49} - 8 q^{55} + 20 q^{61} + 12 q^{67} + 36 q^{73} + 36 q^{79} + 28 q^{85} - 4 q^{91} + 8 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 + 4 * q^13 - 6 * q^19 + 14 * q^25 - 4 * q^31 + 20 * q^37 + 4 * q^43 + 6 * q^49 - 8 * q^55 + 20 * q^61 + 12 * q^67 + 36 * q^73 + 36 * q^79 + 28 * q^85 - 4 * q^91 + 8 * q^97

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