LMFDB - Embedded newform 4788.2.a.p.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:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1596)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.17741$ 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)

$f(q)$	$=$	$q-2.91852 q^{5} +1.00000 q^{7} -2.00000 q^{11} -4.35482 q^{13} -2.91852 q^{17} -1.00000 q^{19} -2.00000 q^{23} +3.51777 q^{25} -7.79112 q^{29} +6.35482 q^{31} -2.91852 q^{35} -3.83705 q^{37} +4.00000 q^{41} +3.48223 q^{43} +11.6282 q^{47} +1.00000 q^{49} +0.918523 q^{53} +5.83705 q^{55} +0.162955 q^{61} +12.7096 q^{65} -5.83705 q^{67} -0.918523 q^{71} -6.70964 q^{73} -2.00000 q^{77} +10.8726 q^{79} -5.79112 q^{83} +8.51777 q^{85} +4.00000 q^{89} -4.35482 q^{91} +2.91852 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{7} - 6 q^{11} + 2 q^{13} - 3 q^{19} - 6 q^{23} + 13 q^{25} - 2 q^{29} + 4 q^{31} + 6 q^{37} + 12 q^{41} + 8 q^{43} - 4 q^{47} + 3 q^{49} - 6 q^{53} + 18 q^{61} + 8 q^{65} + 6 q^{71} + 10 q^{73}+ \cdots - 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^7 - 6 * q^11 + 2 * q^13 - 3 * q^19 - 6 * q^23 + 13 * q^25 - 2 * q^29 + 4 * q^31 + 6 * q^37 + 12 * q^41 + 8 * q^43 - 4 * q^47 + 3 * q^49 - 6 * q^53 + 18 * q^61 + 8 * q^65 + 6 * q^71 + 10 * q^73 - 6 * q^77 + 20 * q^79 + 4 * q^83 + 28 * q^85 + 12 * q^89 + 2 * q^91 - 6 * 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
$5$	−2.91852	−1.30520	−0.652601	−	0.757701i	$-0.726322\pi$
$5$	−2.91852	−1.30520	−0.652601	+	0.757701i	$0.726322\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.35482	−1.20781	−0.603905	−	0.797056i	$-0.706389\pi$
$13$	−4.35482	−1.20781	−0.603905	+	0.797056i	$0.706389\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.91852	−0.707846	−0.353923	−	0.935275i	$-0.615152\pi$
$17$	−2.91852	−0.707846	−0.353923	+	0.935275i	$0.615152\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	3.51777	0.703555
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.79112	−1.44677	−0.723387	−	0.690443i	$-0.757416\pi$
$29$	−7.79112	−1.44677	−0.723387	+	0.690443i	$0.757416\pi$
$30$	0	0
$31$	6.35482	1.14136	0.570680	−	0.821173i	$-0.306680\pi$
$31$	6.35482	1.14136	0.570680	+	0.821173i	$0.306680\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.91852	−0.493320
$36$	0	0
$37$	−3.83705	−0.630806	−0.315403	−	0.948958i	$-0.602140\pi$
$37$	−3.83705	−0.630806	−0.315403	+	0.948958i	$0.602140\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	3.48223	0.531034	0.265517	−	0.964106i	$-0.414457\pi$
$43$	3.48223	0.531034	0.265517	+	0.964106i	$0.414457\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.6282	1.69614	0.848071	−	0.529883i	$-0.177764\pi$
$47$	11.6282	1.69614	0.848071	+	0.529883i	$0.177764\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.918523	0.126169	0.0630844	−	0.998008i	$-0.479906\pi$
$53$	0.918523	0.126169	0.0630844	+	0.998008i	$0.479906\pi$
$54$	0	0
$55$	5.83705	0.787067
$56$	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
$61$	0.162955	0.0208642	0.0104321	−	0.999946i	$-0.496679\pi$
$61$	0.162955	0.0208642	0.0104321	+	0.999946i	$0.496679\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12.7096	1.57644
$66$	0	0
$67$	−5.83705	−0.713109	−0.356554	−	0.934275i	$-0.616049\pi$
$67$	−5.83705	−0.713109	−0.356554	+	0.934275i	$0.616049\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.918523	−0.109009	−0.0545043	−	0.998514i	$-0.517358\pi$
$71$	−0.918523	−0.109009	−0.0545043	+	0.998514i	$0.517358\pi$
$72$	0	0
$73$	−6.70964	−0.785304	−0.392652	−	0.919687i	$-0.628442\pi$
$73$	−6.70964	−0.785304	−0.392652	+	0.919687i	$0.628442\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	10.8726	1.22326	0.611631	−	0.791143i	$-0.290514\pi$
$79$	10.8726	1.22326	0.611631	+	0.791143i	$0.290514\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.79112	−0.635658	−0.317829	−	0.948148i	$-0.602954\pi$
$83$	−5.79112	−0.635658	−0.317829	+	0.948148i	$0.602954\pi$
$84$	0	0
$85$	8.51777	0.923882
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	−4.35482	−0.456509
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.91852	0.299434
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.08148	−0.107611	−0.0538055	−	0.998551i	$-0.517135\pi$
$101$	−1.08148	−0.107611	−0.0538055	+	0.998551i	$0.517135\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.95407	0.188907	0.0944536	−	0.995529i	$-0.469890\pi$
$107$	1.95407	0.188907	0.0944536	+	0.995529i	$0.469890\pi$
$108$	0	0
$109$	−7.83705	−0.750653	−0.375326	−	0.926893i	$-0.622469\pi$
$109$	−7.83705	−0.750653	−0.375326	+	0.926893i	$0.622469\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.6282	1.28203	0.641015	−	0.767529i	$-0.278514\pi$
$113$	13.6282	1.28203	0.641015	+	0.767529i	$0.278514\pi$
$114$	0	0
$115$	5.83705	0.544307
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.91852	−0.267541
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.32591	0.386921
$126$	0	0
$127$	9.83705	0.872897	0.436448	−	0.899729i	$-0.356236\pi$
$127$	9.83705	0.872897	0.436448	+	0.899729i	$0.356236\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.95407	−0.694950	−0.347475	−	0.937689i	$-0.612961\pi$
$131$	−7.95407	−0.694950	−0.347475	+	0.937689i	$0.612961\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.70964	−0.231500	−0.115750	−	0.993278i	$-0.536927\pi$
$137$	−2.70964	−0.231500	−0.115750	+	0.993278i	$0.536927\pi$
$138$	0	0
$139$	20.3837	1.72893	0.864463	−	0.502697i	$-0.167659\pi$
$139$	20.3837	1.72893	0.864463	+	0.502697i	$0.167659\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.70964	0.728337
$144$	0	0
$145$	22.7385	1.88833
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.872594	−0.0714856	−0.0357428	−	0.999361i	$-0.511380\pi$
$149$	−0.872594	−0.0714856	−0.0357428	+	0.999361i	$0.511380\pi$
$150$	0	0
$151$	−12.7096	−1.03430	−0.517148	−	0.855896i	$-0.673006\pi$
$151$	−12.7096	−1.03430	−0.517148	+	0.855896i	$0.673006\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−18.5467	−1.48971
$156$	0	0
$157$	16.5467	1.32057	0.660285	−	0.751016i	$-0.270436\pi$
$157$	16.5467	1.32057	0.660285	+	0.751016i	$0.270436\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−5.22741	−0.409443	−0.204721	−	0.978820i	$-0.565629\pi$
$163$	−5.22741	−0.409443	−0.204721	+	0.978820i	$0.565629\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.8726	1.46041	0.730203	−	0.683231i	$-0.239426\pi$
$167$	18.8726	1.46041	0.730203	+	0.683231i	$0.239426\pi$
$168$	0	0
$169$	5.96445	0.458804
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.67409	0.583450	0.291725	−	0.956502i	$-0.405771\pi$
$173$	7.67409	0.583450	0.291725	+	0.956502i	$0.405771\pi$
$174$	0	0
$175$	3.51777	0.265919
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.95407	−0.146054	−0.0730271	−	0.997330i	$-0.523266\pi$
$179$	−1.95407	−0.146054	−0.0730271	+	0.997330i	$0.523266\pi$
$180$	0	0
$181$	−18.7096	−1.39068	−0.695338	−	0.718683i	$-0.744745\pi$
$181$	−18.7096	−1.39068	−0.695338	+	0.718683i	$0.744745\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.1985	0.823330
$186$	0	0
$187$	5.83705	0.426847
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.7096	−0.774922	−0.387461	−	0.921886i	$-0.626648\pi$
$191$	−10.7096	−0.774922	−0.387461	+	0.921886i	$0.626648\pi$
$192$	0	0
$193$	−12.5467	−0.903130	−0.451565	−	0.892238i	$-0.649134\pi$
$193$	−12.5467	−0.903130	−0.451565	+	0.892238i	$0.649134\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	10.1630	0.720433	0.360216	−	0.932869i	$-0.382703\pi$
$199$	10.1630	0.720433	0.360216	+	0.932869i	$0.382703\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.79112	−0.546829
$204$	0	0
$205$	−11.6741	−0.815354
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.1630	−0.693108
$216$	0	0
$217$	6.35482	0.431393
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.7096	0.854943
$222$	0	0
$223$	14.7385	0.986966	0.493483	−	0.869755i	$-0.335723\pi$
$223$	14.7385	0.986966	0.493483	+	0.869755i	$0.335723\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.5467	1.76197	0.880983	−	0.473149i	$-0.156883\pi$
$227$	26.5467	1.76197	0.880983	+	0.473149i	$0.156883\pi$
$228$	0	0
$229$	6.80150	0.449456	0.224728	−	0.974422i	$-0.427851\pi$
$229$	6.80150	0.449456	0.224728	+	0.974422i	$0.427851\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.2563	−1.39255	−0.696274	−	0.717776i	$-0.745160\pi$
$233$	−21.2563	−1.39255	−0.696274	+	0.717776i	$0.745160\pi$
$234$	0	0
$235$	−33.9371	−2.21381
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.5467	1.58779	0.793896	−	0.608053i	$-0.208049\pi$
$239$	24.5467	1.58779	0.793896	+	0.608053i	$0.208049\pi$
$240$	0	0
$241$	−0.964452	−0.0621258	−0.0310629	−	0.999517i	$-0.509889\pi$
$241$	−0.964452	−0.0621258	−0.0310629	+	0.999517i	$0.509889\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.91852	−0.186458
$246$	0	0
$247$	4.35482	0.277091
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.6282	1.49140	0.745698	−	0.666284i	$-0.232116\pi$
$251$	23.6282	1.49140	0.745698	+	0.666284i	$0.232116\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.12741	0.319839	0.159919	−	0.987130i	$-0.448877\pi$
$257$	5.12741	0.319839	0.159919	+	0.987130i	$0.448877\pi$
$258$	0	0
$259$	−3.83705	−0.238422
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.83705	0.236602	0.118301	−	0.992978i	$-0.462255\pi$
$263$	3.83705	0.236602	0.118301	+	0.992978i	$0.462255\pi$
$264$	0	0
$265$	−2.68073	−0.164676
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.16295	0.131878	0.0659388	−	0.997824i	$-0.478996\pi$
$269$	2.16295	0.131878	0.0659388	+	0.997824i	$0.478996\pi$
$270$	0	0
$271$	20.8015	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$271$	20.8015	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.03555	−0.424260
$276$	0	0
$277$	23.6111	1.41866	0.709328	−	0.704879i	$-0.248999\pi$
$277$	23.6111	1.41866	0.709328	+	0.704879i	$0.248999\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.08148	0.422446	0.211223	−	0.977438i	$-0.432255\pi$
$281$	7.08148	0.422446	0.211223	+	0.977438i	$0.432255\pi$
$282$	0	0
$283$	10.8726	0.646309	0.323154	−	0.946346i	$-0.395257\pi$
$283$	10.8726	0.646309	0.323154	+	0.946346i	$0.395257\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−8.48223	−0.498954
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.6385	1.32256	0.661279	−	0.750140i	$-0.270014\pi$
$293$	22.6385	1.32256	0.661279	+	0.750140i	$0.270014\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.70964	0.503691
$300$	0	0
$301$	3.48223	0.200712
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.475587	−0.0272320
$306$	0	0
$307$	−33.7030	−1.92353	−0.961766	−	0.273873i	$-0.911695\pi$
$307$	−33.7030	−1.92353	−0.961766	+	0.273873i	$0.911695\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.7200	0.664581	0.332291	−	0.943177i	$-0.392179\pi$
$311$	11.7200	0.664581	0.332291	+	0.943177i	$0.392179\pi$
$312$	0	0
$313$	−1.19850	−0.0677434	−0.0338717	−	0.999426i	$-0.510784\pi$
$313$	−1.19850	−0.0677434	−0.0338717	+	0.999426i	$0.510784\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.0474	1.74380	0.871899	−	0.489686i	$-0.162889\pi$
$317$	31.0474	1.74380	0.871899	+	0.489686i	$0.162889\pi$
$318$	0	0
$319$	15.5822	0.872438
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.91852	0.162391
$324$	0	0
$325$	−15.3193	−0.849760
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.6282	0.641081
$330$	0	0
$331$	0.325910	0.0179136	0.00895681	−	0.999960i	$-0.497149\pi$
$331$	0.325910	0.0179136	0.00895681	+	0.999960i	$0.497149\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	17.0355	0.930751
$336$	0	0
$337$	15.8370	0.862699	0.431349	−	0.902185i	$-0.358038\pi$
$337$	15.8370	0.862699	0.431349	+	0.902185i	$0.358038\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.7096	−0.688266
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.8726	−0.691037	−0.345519	−	0.938412i	$-0.612297\pi$
$347$	−12.8726	−0.691037	−0.345519	+	0.938412i	$0.612297\pi$
$348$	0	0
$349$	−20.6385	−1.10476	−0.552378	−	0.833594i	$-0.686279\pi$
$349$	−20.6385	−1.10476	−0.552378	+	0.833594i	$0.686279\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.9185	1.00693	0.503466	−	0.864015i	$-0.332058\pi$
$353$	18.9185	1.00693	0.503466	+	0.864015i	$0.332058\pi$
$354$	0	0
$355$	2.68073	0.142278
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.6385	−1.30037	−0.650186	−	0.759775i	$-0.725309\pi$
$359$	−24.6385	−1.30037	−0.650186	+	0.759775i	$0.725309\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	19.5822	1.02498
$366$	0	0
$367$	−32.2919	−1.68562	−0.842811	−	0.538210i	$-0.819101\pi$
$367$	−32.2919	−1.68562	−0.842811	+	0.538210i	$0.819101\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.918523	0.0476873
$372$	0	0
$373$	−7.74519	−0.401031	−0.200515	−	0.979691i	$-0.564262\pi$
$373$	−7.74519	−0.401031	−0.200515	+	0.979691i	$0.564262\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	33.9289	1.74743
$378$	0	0
$379$	22.4548	1.15343	0.576713	−	0.816946i	$-0.304335\pi$
$379$	22.4548	1.15343	0.576713	+	0.816946i	$0.304335\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−18.1630	−0.928084	−0.464042	−	0.885813i	$-0.653601\pi$
$383$	−18.1630	−0.928084	−0.464042	+	0.885813i	$0.653601\pi$
$384$	0	0
$385$	5.83705	0.297483
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.90814	0.502363	0.251181	−	0.967940i	$-0.419181\pi$
$389$	9.90814	0.502363	0.251181	+	0.967940i	$0.419181\pi$
$390$	0	0
$391$	5.83705	0.295192
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−31.7319	−1.59661
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.9185	1.04462	0.522311	−	0.852755i	$-0.325070\pi$
$401$	20.9185	1.04462	0.522311	+	0.852755i	$0.325070\pi$
$402$	0	0
$403$	−27.6741	−1.37854
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.67409	0.380391
$408$	0	0
$409$	−20.0289	−0.990366	−0.495183	−	0.868789i	$-0.664899\pi$
$409$	−20.0289	−0.990366	−0.495183	+	0.868789i	$0.664899\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	16.9015	0.829662
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.7200	1.15880	0.579399	−	0.815044i	$-0.303287\pi$
$419$	23.7200	1.15880	0.579399	+	0.815044i	$0.303287\pi$
$420$	0	0
$421$	−2.70964	−0.132060	−0.0660299	−	0.997818i	$-0.521033\pi$
$421$	−2.70964	−0.132060	−0.0660299	+	0.997818i	$0.521033\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−10.2667	−0.498008
$426$	0	0
$427$	0.162955	0.00788594
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.1393	−1.49993	−0.749964	−	0.661479i	$-0.769929\pi$
$431$	−31.1393	−1.49993	−0.749964	+	0.661479i	$0.769929\pi$
$432$	0	0
$433$	−24.6385	−1.18405	−0.592026	−	0.805919i	$-0.701672\pi$
$433$	−24.6385	−1.18405	−0.592026	+	0.805919i	$0.701672\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	−20.1000	−0.959321	−0.479660	−	0.877454i	$-0.659240\pi$
$439$	−20.1000	−0.959321	−0.479660	+	0.877454i	$0.659240\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.8370	−1.32258	−0.661289	−	0.750131i	$-0.729990\pi$
$443$	−27.8370	−1.32258	−0.661289	+	0.750131i	$0.729990\pi$
$444$	0	0
$445$	−11.6741	−0.553405
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.82666	−0.227784	−0.113892	−	0.993493i	$-0.536332\pi$
$449$	−4.82666	−0.227784	−0.113892	+	0.993493i	$0.536332\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.7096	0.595837
$456$	0	0
$457$	1.48223	0.0693356	0.0346678	−	0.999399i	$-0.488963\pi$
$457$	1.48223	0.0693356	0.0346678	+	0.999399i	$0.488963\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	32.3378	1.50612	0.753061	−	0.657951i	$-0.228577\pi$
$461$	32.3378	1.50612	0.753061	+	0.657951i	$0.228577\pi$
$462$	0	0
$463$	29.4193	1.36723	0.683615	−	0.729843i	$-0.260407\pi$
$463$	29.4193	1.36723	0.683615	+	0.729843i	$0.260407\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.4652	0.623095	0.311548	−	0.950231i	$-0.399153\pi$
$467$	13.4652	0.623095	0.311548	+	0.950231i	$0.399153\pi$
$468$	0	0
$469$	−5.83705	−0.269530
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.96445	−0.320226
$474$	0	0
$475$	−3.51777	−0.161407
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.6993	−0.625935	−0.312968	−	0.949764i	$-0.601323\pi$
$479$	−13.6993	−0.625935	−0.312968	+	0.949764i	$0.601323\pi$
$480$	0	0
$481$	16.7096	0.761894
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.83705	0.265047
$486$	0	0
$487$	−25.0355	−1.13447	−0.567234	−	0.823556i	$-0.691987\pi$
$487$	−25.0355	−1.13447	−0.567234	+	0.823556i	$0.691987\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.8015	−0.487465	−0.243732	−	0.969843i	$-0.578372\pi$
$491$	−10.8015	−0.487465	−0.243732	+	0.969843i	$0.578372\pi$
$492$	0	0
$493$	22.7385	1.02409
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.918523	−0.0412014
$498$	0	0
$499$	9.92890	0.444479	0.222239	−	0.974992i	$-0.428663\pi$
$499$	9.92890	0.444479	0.222239	+	0.974992i	$0.428663\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−27.3023	−1.21735	−0.608674	−	0.793421i	$-0.708298\pi$
$503$	−27.3023	−1.21735	−0.608674	+	0.793421i	$0.708298\pi$
$504$	0	0
$505$	3.15632	0.140454
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−23.2563	−1.03082	−0.515409	−	0.856944i	$-0.672360\pi$
$509$	−23.2563	−1.03082	−0.515409	+	0.856944i	$0.672360\pi$
$510$	0	0
$511$	−6.70964	−0.296817
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−23.3482	−1.02884
$516$	0	0
$517$	−23.2563	−1.02281
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.743677	0.0325811	0.0162905	−	0.999867i	$-0.494814\pi$
$521$	0.743677	0.0325811	0.0162905	+	0.999867i	$0.494814\pi$
$522$	0	0
$523$	32.4837	1.42041	0.710207	−	0.703993i	$-0.248601\pi$
$523$	32.4837	1.42041	0.710207	+	0.703993i	$0.248601\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.5467	−0.807906
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−17.4193	−0.754513
$534$	0	0
$535$	−5.70300	−0.246562
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	44.4548	1.91126	0.955631	−	0.294566i	$-0.0951751\pi$
$541$	44.4548	1.91126	0.955631	+	0.294566i	$0.0951751\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.8726	0.979754
$546$	0	0
$547$	−34.5467	−1.47711	−0.738555	−	0.674193i	$-0.764491\pi$
$547$	−34.5467	−1.47711	−0.738555	+	0.674193i	$0.764491\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.79112	0.331913
$552$	0	0
$553$	10.8726	0.462350
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.1274	−0.810454	−0.405227	−	0.914216i	$-0.632808\pi$
$557$	−19.1274	−0.810454	−0.405227	+	0.914216i	$0.632808\pi$
$558$	0	0
$559$	−15.1645	−0.641389
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.87259	−0.289645	−0.144823	−	0.989458i	$-0.546261\pi$
$563$	−6.87259	−0.289645	−0.144823	+	0.989458i	$0.546261\pi$
$564$	0	0
$565$	−39.7741	−1.67331
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.4297	0.940300	0.470150	−	0.882586i	$-0.344200\pi$
$569$	22.4297	0.940300	0.470150	+	0.882586i	$0.344200\pi$
$570$	0	0
$571$	5.74519	0.240429	0.120214	−	0.992748i	$-0.461642\pi$
$571$	5.74519	0.240429	0.120214	+	0.992748i	$0.461642\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.03555	−0.293403
$576$	0	0
$577$	−40.5467	−1.68798	−0.843990	−	0.536358i	$-0.819800\pi$
$577$	−40.5467	−1.68798	−0.843990	+	0.536358i	$0.819800\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.79112	−0.240256
$582$	0	0
$583$	−1.83705	−0.0760826
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.72002	−0.318639	−0.159320	−	0.987227i	$-0.550930\pi$
$587$	−7.72002	−0.318639	−0.159320	+	0.987227i	$0.550930\pi$
$588$	0	0
$589$	−6.35482	−0.261846
$590$	0	0
$591$	0	0
$592$	0	0
$593$	36.2459	1.48844	0.744221	−	0.667933i	$-0.232821\pi$
$593$	36.2459	1.48844	0.744221	+	0.667933i	$0.232821\pi$
$594$	0	0
$595$	8.51777	0.349195
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.1393	1.10888	0.554441	−	0.832223i	$-0.312932\pi$
$599$	27.1393	1.10888	0.554441	+	0.832223i	$0.312932\pi$
$600$	0	0
$601$	15.0934	0.615671	0.307836	−	0.951440i	$-0.400395\pi$
$601$	15.0934	0.615671	0.307836	+	0.951440i	$0.400395\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	20.4297	0.830584
$606$	0	0
$607$	25.4193	1.03174	0.515868	−	0.856668i	$-0.327469\pi$
$607$	25.4193	1.03174	0.515868	+	0.856668i	$0.327469\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−50.6385	−2.04862
$612$	0	0
$613$	26.1919	1.05788	0.528940	−	0.848659i	$-0.322590\pi$
$613$	26.1919	1.05788	0.528940	+	0.848659i	$0.322590\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.34818	0.376344	0.188172	−	0.982136i	$-0.439744\pi$
$617$	9.34818	0.376344	0.188172	+	0.982136i	$0.439744\pi$
$618$	0	0
$619$	−20.3837	−0.819291	−0.409646	−	0.912245i	$-0.634348\pi$
$619$	−20.3837	−0.819291	−0.409646	+	0.912245i	$0.634348\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	−30.2141	−1.20857
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.1985	0.446514
$630$	0	0
$631$	−27.8660	−1.10933	−0.554663	−	0.832075i	$-0.687153\pi$
$631$	−27.8660	−1.10933	−0.554663	+	0.832075i	$0.687153\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−28.7096	−1.13931
$636$	0	0
$637$	−4.35482	−0.172544
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−35.3733	−1.39716	−0.698582	−	0.715530i	$-0.746185\pi$
$641$	−35.3733	−1.39716	−0.698582	+	0.715530i	$0.746185\pi$
$642$	0	0
$643$	−36.2919	−1.43121	−0.715606	−	0.698504i	$-0.753849\pi$
$643$	−36.2919	−1.43121	−0.715606	+	0.698504i	$0.753849\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.6282	1.08618	0.543088	−	0.839676i	$-0.317255\pi$
$647$	27.6282	1.08618	0.543088	+	0.839676i	$0.317255\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.6741	−1.00470	−0.502352	−	0.864663i	$-0.667532\pi$
$653$	−25.6741	−1.00470	−0.502352	+	0.864663i	$0.667532\pi$
$654$	0	0
$655$	23.2141	0.907051
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13.0104	−0.506812	−0.253406	−	0.967360i	$-0.581551\pi$
$659$	−13.0104	−0.506812	−0.253406	+	0.967360i	$0.581551\pi$
$660$	0	0
$661$	−50.1578	−1.95091	−0.975457	−	0.220192i	$-0.929332\pi$
$661$	−50.1578	−1.95091	−0.975457	+	0.220192i	$0.929332\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.91852	0.113175
$666$	0	0
$667$	15.5822	0.603346
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.325910	−0.0125816
$672$	0	0
$673$	48.5467	1.87134	0.935669	−	0.352880i	$-0.114798\pi$
$673$	48.5467	1.87134	0.935669	+	0.352880i	$0.114798\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.76595	0.298470	0.149235	−	0.988802i	$-0.452319\pi$
$677$	7.76595	0.298470	0.149235	+	0.988802i	$0.452319\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20.2089	−0.773271	−0.386636	−	0.922233i	$-0.626363\pi$
$683$	−20.2089	−0.773271	−0.386636	+	0.922233i	$0.626363\pi$
$684$	0	0
$685$	7.90814	0.302155
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−13.1274	−0.499390	−0.249695	−	0.968325i	$-0.580330\pi$
$691$	−13.1274	−0.499390	−0.249695	+	0.968325i	$0.580330\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−59.4904	−2.25660
$696$	0	0
$697$	−11.6741	−0.442188
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.906632	−0.0342430	−0.0171215	−	0.999853i	$-0.505450\pi$
$701$	−0.906632	−0.0342430	−0.0171215	+	0.999853i	$0.505450\pi$
$702$	0	0
$703$	3.83705	0.144717
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.08148	−0.0406731
$708$	0	0
$709$	25.1563	0.944765	0.472383	−	0.881394i	$-0.343394\pi$
$709$	25.1563	0.944765	0.472383	+	0.881394i	$0.343394\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12.7096	−0.475980
$714$	0	0
$715$	−25.4193	−0.950627
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−41.1393	−1.53424	−0.767118	−	0.641505i	$-0.778310\pi$
$719$	−41.1393	−1.53424	−0.767118	+	0.641505i	$0.778310\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−27.4074	−1.01788
$726$	0	0
$727$	25.3274	0.939342	0.469671	−	0.882842i	$-0.344373\pi$
$727$	25.3274	0.939342	0.469671	+	0.882842i	$0.344373\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−10.1630	−0.375890
$732$	0	0
$733$	42.3837	1.56548	0.782739	−	0.622350i	$-0.213822\pi$
$733$	42.3837	1.56548	0.782739	+	0.622350i	$0.213822\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.6741	0.430021
$738$	0	0
$739$	−9.73704	−0.358183	−0.179091	−	0.983832i	$-0.557316\pi$
$739$	−9.73704	−0.358183	−0.179091	+	0.983832i	$0.557316\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	38.7215	1.42056	0.710278	−	0.703922i	$-0.248569\pi$
$743$	38.7215	1.42056	0.710278	+	0.703922i	$0.248569\pi$
$744$	0	0
$745$	2.54668	0.0933033
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.95407	0.0714002
$750$	0	0
$751$	18.6385	0.680130	0.340065	−	0.940402i	$-0.389551\pi$
$751$	18.6385	0.680130	0.340065	+	0.940402i	$0.389551\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	37.0934	1.34997
$756$	0	0
$757$	51.4771	1.87097	0.935483	−	0.353370i	$-0.114964\pi$
$757$	51.4771	1.87097	0.935483	+	0.353370i	$0.114964\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.5008	0.380652	0.190326	−	0.981721i	$-0.439045\pi$
$761$	10.5008	0.380652	0.190326	+	0.981721i	$0.439045\pi$
$762$	0	0
$763$	−7.83705	−0.283720
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	7.45332	0.268773	0.134387	−	0.990929i	$-0.457094\pi$
$769$	7.45332	0.268773	0.134387	+	0.990929i	$0.457094\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−21.4193	−0.770398	−0.385199	−	0.922834i	$-0.625867\pi$
$773$	−21.4193	−0.770398	−0.385199	+	0.922834i	$0.625867\pi$
$774$	0	0
$775$	22.3548	0.803009
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	1.83705	0.0657346
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−48.2919	−1.72361
$786$	0	0
$787$	33.7030	1.20138	0.600691	−	0.799481i	$-0.294892\pi$
$787$	33.7030	1.20138	0.600691	+	0.799481i	$0.294892\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13.6282	0.484562
$792$	0	0
$793$	−0.709639	−0.0252000
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33.4771	−1.18582	−0.592910	−	0.805269i	$-0.702021\pi$
$797$	−33.4771	−1.18582	−0.592910	+	0.805269i	$0.702021\pi$
$798$	0	0
$799$	−33.9371	−1.20061
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.4193	0.473556
$804$	0	0
$805$	5.83705	0.205729
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22.7675	0.800461	0.400231	−	0.916414i	$-0.368930\pi$
$809$	22.7675	0.800461	0.400231	+	0.916414i	$0.368930\pi$
$810$	0	0
$811$	52.7675	1.85292	0.926458	−	0.376398i	$-0.122837\pi$
$811$	52.7675	1.85292	0.926458	+	0.376398i	$0.122837\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	15.2563	0.534406
$816$	0	0
$817$	−3.48223	−0.121828
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.9304	−1.00968	−0.504839	−	0.863213i	$-0.668448\pi$
$821$	−28.9304	−1.00968	−0.504839	+	0.863213i	$0.668448\pi$
$822$	0	0
$823$	−17.6111	−0.613886	−0.306943	−	0.951728i	$-0.599306\pi$
$823$	−17.6111	−0.613886	−0.306943	+	0.951728i	$0.599306\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.82666	0.306933	0.153467	−	0.988154i	$-0.450956\pi$
$827$	8.82666	0.306933	0.153467	+	0.988154i	$0.450956\pi$
$828$	0	0
$829$	11.6030	0.402989	0.201494	−	0.979490i	$-0.435420\pi$
$829$	11.6030	0.402989	0.201494	+	0.979490i	$0.435420\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.91852	−0.101121
$834$	0	0
$835$	−55.0801	−1.90613
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.7452	−0.888823	−0.444411	−	0.895823i	$-0.646587\pi$
$839$	−25.7452	−0.888823	−0.444411	+	0.895823i	$0.646587\pi$
$840$	0	0
$841$	31.7015	1.09315
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−17.4074	−0.598832
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.67409	0.263064
$852$	0	0
$853$	36.1289	1.23703	0.618515	−	0.785773i	$-0.287734\pi$
$853$	36.1289	1.23703	0.618515	+	0.785773i	$0.287734\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44.3837	1.51612	0.758060	−	0.652185i	$-0.226148\pi$
$857$	44.3837	1.51612	0.758060	+	0.652185i	$0.226148\pi$
$858$	0	0
$859$	−24.0578	−0.820842	−0.410421	−	0.911896i	$-0.634618\pi$
$859$	−24.0578	−0.820842	−0.410421	+	0.911896i	$0.634618\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−43.2312	−1.47161	−0.735803	−	0.677196i	$-0.763195\pi$
$863$	−43.2312	−1.47161	−0.735803	+	0.677196i	$0.763195\pi$
$864$	0	0
$865$	−22.3970	−0.761521
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−21.7452	−0.737655
$870$	0	0
$871$	25.4193	0.861299
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.32591	0.146242
$876$	0	0
$877$	49.0223	1.65536	0.827682	−	0.561197i	$-0.189659\pi$
$877$	49.0223	1.65536	0.827682	+	0.561197i	$0.189659\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	32.0119	1.07851	0.539254	−	0.842143i	$-0.318706\pi$
$881$	32.0119	1.07851	0.539254	+	0.842143i	$0.318706\pi$
$882$	0	0
$883$	−25.0934	−0.844459	−0.422230	−	0.906489i	$-0.638752\pi$
$883$	−25.0934	−0.844459	−0.422230	+	0.906489i	$0.638752\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.80150	−0.295525	−0.147763	−	0.989023i	$-0.547207\pi$
$887$	−8.80150	−0.295525	−0.147763	+	0.989023i	$0.547207\pi$
$888$	0	0
$889$	9.83705	0.329924
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.6282	−0.389122
$894$	0	0
$895$	5.70300	0.190630
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−49.5111	−1.65129
$900$	0	0
$901$	−2.68073	−0.0893080
$902$	0	0
$903$	0	0
$904$	0	0
$905$	54.6045	1.81512
$906$	0	0
$907$	−52.3837	−1.73937	−0.869687	−	0.493604i	$-0.835679\pi$
$907$	−52.3837	−1.73937	−0.869687	+	0.493604i	$0.835679\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16.5926	0.549738	0.274869	−	0.961482i	$-0.411366\pi$
$911$	16.5926	0.549738	0.274869	+	0.961482i	$0.411366\pi$
$912$	0	0
$913$	11.5822	0.383316
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.95407	−0.262667
$918$	0	0
$919$	13.9289	0.459472	0.229736	−	0.973253i	$-0.426214\pi$
$919$	13.9289	0.459472	0.229736	+	0.973253i	$0.426214\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.00000	0.131662
$924$	0	0
$925$	−13.4979	−0.443807
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−54.0830	−1.77441	−0.887203	−	0.461380i	$-0.847354\pi$
$929$	−54.0830	−1.77441	−0.887203	+	0.461380i	$0.847354\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−17.0355	−0.557122
$936$	0	0
$937$	4.25481	0.138999	0.0694993	−	0.997582i	$-0.477860\pi$
$937$	4.25481	0.138999	0.0694993	+	0.997582i	$0.477860\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.47559	−0.276296	−0.138148	−	0.990412i	$-0.544115\pi$
$941$	−8.47559	−0.276296	−0.138148	+	0.990412i	$0.544115\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.78074	0.155353	0.0776765	−	0.996979i	$-0.475250\pi$
$947$	4.78074	0.155353	0.0776765	+	0.996979i	$0.475250\pi$
$948$	0	0
$949$	29.2193	0.948497
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−31.0815	−1.00683	−0.503414	−	0.864045i	$-0.667923\pi$
$953$	−31.0815	−1.00683	−0.503414	+	0.864045i	$0.667923\pi$
$954$	0	0
$955$	31.2563	1.01143
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.70964	−0.0874988
$960$	0	0
$961$	9.38373	0.302701
$962$	0	0
$963$	0	0
$964$	0	0
$965$	36.6178	1.17877
$966$	0	0
$967$	21.9371	0.705448	0.352724	−	0.935727i	$-0.385255\pi$
$967$	21.9371	0.705448	0.352724	+	0.935727i	$0.385255\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−39.4904	−1.26731	−0.633653	−	0.773617i	$-0.718445\pi$
$971$	−39.4904	−1.26731	−0.633653	+	0.773617i	$0.718445\pi$
$972$	0	0
$973$	20.3837	0.653472
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.24443	0.295756	0.147878	−	0.989006i	$-0.452756\pi$
$977$	9.24443	0.295756	0.147878	+	0.989006i	$0.452756\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	0	0
$982$	0	0
$983$	19.2904	0.615267	0.307633	−	0.951505i	$-0.400463\pi$
$983$	19.2904	0.615267	0.307633	+	0.951505i	$0.400463\pi$
$984$	0	0
$985$	−17.5111	−0.557951
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.96445	−0.221457
$990$	0	0
$991$	10.4548	0.332108	0.166054	−	0.986117i	$-0.446897\pi$
$991$	10.4548	0.332108	0.166054	+	0.986117i	$0.446897\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−29.6608	−0.940311
$996$	0	0
$997$	34.1497	1.08153	0.540766	−	0.841173i	$-0.318135\pi$
$997$	34.1497	1.08153	0.540766	+	0.841173i	$0.318135\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.p.1.1		3
3.2	odd	2		1596.2.a.j.1.3	✓	3
12.11	even	2		6384.2.a.bt.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.j.1.3	✓	3	3.2	odd	2
4788.2.a.p.1.1		3	1.1	even	1	trivial
6384.2.a.bt.1.3		3	12.11	even	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)

\(f(q)\)	\(=\)	\(q-2.91852 q^{5} +1.00000 q^{7} -2.00000 q^{11} -4.35482 q^{13} -2.91852 q^{17} -1.00000 q^{19} -2.00000 q^{23} +3.51777 q^{25} -7.79112 q^{29} +6.35482 q^{31} -2.91852 q^{35} -3.83705 q^{37} +4.00000 q^{41} +3.48223 q^{43} +11.6282 q^{47} +1.00000 q^{49} +0.918523 q^{53} +5.83705 q^{55} +0.162955 q^{61} +12.7096 q^{65} -5.83705 q^{67} -0.918523 q^{71} -6.70964 q^{73} -2.00000 q^{77} +10.8726 q^{79} -5.79112 q^{83} +8.51777 q^{85} +4.00000 q^{89} -4.35482 q^{91} +2.91852 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{7} - 6 q^{11} + 2 q^{13} - 3 q^{19} - 6 q^{23} + 13 q^{25} - 2 q^{29} + 4 q^{31} + 6 q^{37} + 12 q^{41} + 8 q^{43} - 4 q^{47} + 3 q^{49} - 6 q^{53} + 18 q^{61} + 8 q^{65} + 6 q^{71} + 10 q^{73}+ \cdots - 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^7 - 6 * q^11 + 2 * q^13 - 3 * q^19 - 6 * q^23 + 13 * q^25 - 2 * q^29 + 4 * q^31 + 6 * q^37 + 12 * q^41 + 8 * q^43 - 4 * q^47 + 3 * q^49 - 6 * q^53 + 18 * q^61 + 8 * q^65 + 6 * q^71 + 10 * q^73 - 6 * q^77 + 20 * q^79 + 4 * q^83 + 28 * q^85 + 12 * q^89 + 2 * q^91 - 6 * q^97

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