LMFDB - Embedded newform 8820.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8820,2,Mod(1,8820)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8820, 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("8820.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8820.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:	$70.4280545828$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 1260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.65544$ of defining polynomial
Character	$\chi$	$=$	8820.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +O(q^{10})$	$q-1.00000 q^{5} -5.70682 q^{11} -6.70682 q^{13} -3.44731 q^{17} +1.25951 q^{19} +7.96633 q^{23} +1.00000 q^{25} +7.18780 q^{29} -1.00000 q^{31} -8.96633 q^{37} -10.2258 q^{41} -4.44731 q^{43} -1.48098 q^{47} -6.89461 q^{53} +5.70682 q^{55} +0.293183 q^{59} -0.259511 q^{61} +6.70682 q^{65} -6.70682 q^{67} +11.7068 q^{71} +0.966327 q^{73} -11.6351 q^{79} +14.8609 q^{83} +3.44731 q^{85} +12.6014 q^{89} -1.25951 q^{95} -13.1541 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5}+O(q^{10}) $ 3 * q - 3 * q^5	$ 3 q - 3 q^{5} - 2 q^{11} - 5 q^{13} + 2 q^{17} + q^{19} + 6 q^{23} + 3 q^{25} + 12 q^{29} - 3 q^{31} - 9 q^{37} - 10 q^{41} - q^{43} - 10 q^{47} + 4 q^{53} + 2 q^{55} + 16 q^{59} + 2 q^{61} + 5 q^{65} - 5 q^{67} + 20 q^{71} - 15 q^{73} - 13 q^{79} + 2 q^{83} - 2 q^{85} - 2 q^{89} - q^{95} - 12 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 2 * q^11 - 5 * q^13 + 2 * q^17 + q^19 + 6 * q^23 + 3 * q^25 + 12 * q^29 - 3 * q^31 - 9 * q^37 - 10 * q^41 - q^43 - 10 * q^47 + 4 * q^53 + 2 * q^55 + 16 * q^59 + 2 * q^61 + 5 * q^65 - 5 * q^67 + 20 * q^71 - 15 * q^73 - 13 * q^79 + 2 * q^83 - 2 * q^85 - 2 * q^89 - q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.70682	−1.72067	−0.860335	−	0.509729i	$-0.829746\pi$
$11$	−5.70682	−1.72067	−0.860335	+	0.509729i	$0.829746\pi$
$12$	0	0
$13$	−6.70682	−1.86014	−0.930068	−	0.367387i	$-0.880252\pi$
$13$	−6.70682	−1.86014	−0.930068	+	0.367387i	$0.880252\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.44731	−0.836095	−0.418047	−	0.908425i	$-0.637285\pi$
$17$	−3.44731	−0.836095	−0.418047	+	0.908425i	$0.637285\pi$
$18$	0	0
$19$	1.25951	0.288952	0.144476	−	0.989508i	$-0.453850\pi$
$19$	1.25951	0.288952	0.144476	+	0.989508i	$0.453850\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.96633	1.66109	0.830547	−	0.556949i	$-0.188028\pi$
$23$	7.96633	1.66109	0.830547	+	0.556949i	$0.188028\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.18780	1.33474	0.667370	−	0.744726i	$-0.267420\pi$
$29$	7.18780	1.33474	0.667370	+	0.744726i	$0.267420\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.96633	−1.47406	−0.737028	−	0.675863i	$-0.763771\pi$
$37$	−8.96633	−1.47406	−0.737028	+	0.675863i	$0.763771\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.2258	−1.59701	−0.798504	−	0.601990i	$-0.794375\pi$
$41$	−10.2258	−1.59701	−0.798504	+	0.601990i	$0.794375\pi$
$42$	0	0
$43$	−4.44731	−0.678208	−0.339104	−	0.940749i	$-0.610124\pi$
$43$	−4.44731	−0.678208	−0.339104	+	0.940749i	$0.610124\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.48098	−0.216023	−0.108011	−	0.994150i	$-0.534448\pi$
$47$	−1.48098	−0.216023	−0.108011	+	0.994150i	$0.534448\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.89461	−0.947048	−0.473524	−	0.880781i	$-0.657018\pi$
$53$	−6.89461	−0.947048	−0.473524	+	0.880781i	$0.657018\pi$
$54$	0	0
$55$	5.70682	0.769507
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.293183	0.0381692	0.0190846	−	0.999818i	$-0.493925\pi$
$59$	0.293183	0.0381692	0.0190846	+	0.999818i	$0.493925\pi$
$60$	0	0
$61$	−0.259511	−0.0332269	−0.0166135	−	0.999862i	$-0.505288\pi$
$61$	−0.259511	−0.0332269	−0.0166135	+	0.999862i	$0.505288\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.70682	0.831878
$66$	0	0
$67$	−6.70682	−0.819368	−0.409684	−	0.912227i	$-0.634361\pi$
$67$	−6.70682	−0.819368	−0.409684	+	0.912227i	$0.634361\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.7068	1.38934	0.694672	−	0.719327i	$-0.255550\pi$
$71$	11.7068	1.38934	0.694672	+	0.719327i	$0.255550\pi$
$72$	0	0
$73$	0.966327	0.113100	0.0565500	−	0.998400i	$-0.481990\pi$
$73$	0.966327	0.113100	0.0565500	+	0.998400i	$0.481990\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.6351	−1.30905	−0.654526	−	0.756040i	$-0.727132\pi$
$79$	−11.6351	−1.30905	−0.654526	+	0.756040i	$0.727132\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.8609	1.63120	0.815600	−	0.578616i	$-0.196407\pi$
$83$	14.8609	1.63120	0.815600	+	0.578616i	$0.196407\pi$
$84$	0	0
$85$	3.44731	0.373913
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.6014	1.33575	0.667874	−	0.744274i	$-0.267204\pi$
$89$	12.6014	1.33575	0.667874	+	0.744274i	$0.267204\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.25951	−0.129223
$96$	0	0
$97$	−13.1541	−1.33560	−0.667799	−	0.744341i	$-0.732764\pi$
$97$	−13.1541	−1.33560	−0.667799	+	0.744341i	$0.732764\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.1204	−1.10653	−0.553263	−	0.833007i	$-0.686618\pi$
$101$	−11.1204	−1.10653	−0.553263	+	0.833007i	$0.686618\pi$
$102$	0	0
$103$	−2.07171	−0.204132	−0.102066	−	0.994778i	$-0.532545\pi$
$103$	−2.07171	−0.204132	−0.102066	+	0.994778i	$0.532545\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.96633	−0.190092	−0.0950460	−	0.995473i	$-0.530300\pi$
$107$	−1.96633	−0.190092	−0.0950460	+	0.995473i	$0.530300\pi$
$108$	0	0
$109$	−8.67314	−0.830737	−0.415368	−	0.909653i	$-0.636347\pi$
$109$	−8.67314	−0.830737	−0.415368	+	0.909653i	$0.636347\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.586367	0.0551607	0.0275804	−	0.999620i	$-0.491220\pi$
$113$	0.586367	0.0551607	0.0275804	+	0.999620i	$0.491220\pi$
$114$	0	0
$115$	−7.96633	−0.742864
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	21.5678	1.96071
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.22584	−0.463718	−0.231859	−	0.972749i	$-0.574481\pi$
$127$	−5.22584	−0.463718	−0.231859	+	0.972749i	$0.574481\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.4136	−1.52144	−0.760718	−	0.649082i	$-0.775153\pi$
$131$	−17.4136	−1.52144	−0.760718	+	0.649082i	$0.775153\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.9663	1.19322	0.596612	−	0.802530i	$-0.296513\pi$
$137$	13.9663	1.19322	0.596612	+	0.802530i	$0.296513\pi$
$138$	0	0
$139$	15.0487	1.27642	0.638209	−	0.769864i	$-0.279676\pi$
$139$	15.0487	1.27642	0.638209	+	0.769864i	$0.279676\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	38.2746	3.20068
$144$	0	0
$145$	−7.18780	−0.596914
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	21.0868	1.71602	0.858009	−	0.513635i	$-0.171701\pi$
$151$	21.0868	1.71602	0.858009	+	0.513635i	$0.171701\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	8.89461	0.709867	0.354934	−	0.934891i	$-0.384503\pi$
$157$	8.89461	0.709867	0.354934	+	0.934891i	$0.384503\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.3756	0.812679	0.406340	−	0.913722i	$-0.366805\pi$
$163$	10.3756	0.812679	0.406340	+	0.913722i	$0.366805\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.96633	0.152159	0.0760795	−	0.997102i	$-0.475760\pi$
$167$	1.96633	0.152159	0.0760795	+	0.997102i	$0.475760\pi$
$168$	0	0
$169$	31.9814	2.46011
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.48098	−0.112597	−0.0562984	−	0.998414i	$-0.517930\pi$
$173$	−1.48098	−0.112597	−0.0562984	+	0.998414i	$0.517930\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.3082	1.36842	0.684211	−	0.729284i	$-0.260147\pi$
$179$	18.3082	1.36842	0.684211	+	0.729284i	$0.260147\pi$
$180$	0	0
$181$	18.6731	1.38796	0.693982	−	0.719992i	$-0.255855\pi$
$181$	18.6731	1.38796	0.693982	+	0.719992i	$0.255855\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.96633	0.659217
$186$	0	0
$187$	19.6731	1.43864
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.30825	−0.456449	−0.228224	−	0.973609i	$-0.573292\pi$
$191$	−6.30825	−0.456449	−0.228224	+	0.973609i	$0.573292\pi$
$192$	0	0
$193$	22.8990	1.64830	0.824152	−	0.566368i	$-0.191652\pi$
$193$	22.8990	1.64830	0.824152	+	0.566368i	$0.191652\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.5527	1.03684	0.518418	−	0.855127i	$-0.326521\pi$
$197$	14.5527	1.03684	0.518418	+	0.855127i	$0.326521\pi$
$198$	0	0
$199$	−17.0868	−1.21125	−0.605625	−	0.795750i	$-0.707077\pi$
$199$	−17.0868	−1.21125	−0.605625	+	0.795750i	$0.707077\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	10.2258	0.714203
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.18780	−0.497190
$210$	0	0
$211$	3.36490	0.231649	0.115825	−	0.993270i	$-0.463049\pi$
$211$	3.36490	0.231649	0.115825	+	0.993270i	$0.463049\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.44731	0.303304
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	23.1204	1.55525
$222$	0	0
$223$	−17.7892	−1.19125	−0.595627	−	0.803261i	$-0.703096\pi$
$223$	−17.7892	−1.19125	−0.595627	+	0.803261i	$0.703096\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.9283	1.12357	0.561785	−	0.827283i	$-0.310115\pi$
$227$	16.9283	1.12357	0.561785	+	0.827283i	$0.310115\pi$
$228$	0	0
$229$	−20.7892	−1.37379	−0.686895	−	0.726756i	$-0.741027\pi$
$229$	−20.7892	−1.37379	−0.686895	+	0.726756i	$0.741027\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.3756	1.33485	0.667425	−	0.744677i	$-0.267397\pi$
$233$	20.3756	1.33485	0.667425	+	0.744677i	$0.267397\pi$
$234$	0	0
$235$	1.48098	0.0966084
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.0380	0.972730	0.486365	−	0.873756i	$-0.338323\pi$
$239$	15.0380	0.972730	0.486365	+	0.873756i	$0.338323\pi$
$240$	0	0
$241$	−2.63510	−0.169742	−0.0848709	−	0.996392i	$-0.527048\pi$
$241$	−2.63510	−0.169742	−0.0848709	+	0.996392i	$0.527048\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.44731	−0.537489
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.5634	−0.982352	−0.491176	−	0.871060i	$-0.663433\pi$
$251$	−15.5634	−0.982352	−0.491176	+	0.871060i	$0.663433\pi$
$252$	0	0
$253$	−45.4624	−2.85819
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.0044	−0.686434	−0.343217	−	0.939256i	$-0.611517\pi$
$257$	−11.0044	−0.686434	−0.343217	+	0.939256i	$0.611517\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.03804	−0.187334	−0.0936669	−	0.995604i	$-0.529859\pi$
$263$	−3.03804	−0.187334	−0.0936669	+	0.995604i	$0.529859\pi$
$264$	0	0
$265$	6.89461	0.423533
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.9327	1.33726	0.668629	−	0.743596i	$-0.266882\pi$
$269$	21.9327	1.33726	0.668629	+	0.743596i	$0.266882\pi$
$270$	0	0
$271$	6.32686	0.384329	0.192165	−	0.981363i	$-0.438449\pi$
$271$	6.32686	0.384329	0.192165	+	0.981363i	$0.438449\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.70682	−0.344134
$276$	0	0
$277$	−3.55269	−0.213461	−0.106730	−	0.994288i	$-0.534038\pi$
$277$	−3.55269	−0.213461	−0.106730	+	0.994288i	$0.534038\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.5190	1.34337	0.671686	−	0.740836i	$-0.265570\pi$
$281$	22.5190	1.34337	0.671686	+	0.740836i	$0.265570\pi$
$282$	0	0
$283$	10.8990	0.647877	0.323939	−	0.946078i	$-0.394993\pi$
$283$	10.8990	0.647877	0.323939	+	0.946078i	$0.394993\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−5.11608	−0.300946
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.8946	−1.45436	−0.727179	−	0.686447i	$-0.759169\pi$
$293$	−24.8946	−1.45436	−0.727179	+	0.686447i	$0.759169\pi$
$294$	0	0
$295$	−0.293183	−0.0170698
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−53.4287	−3.08986
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.259511	0.0148595
$306$	0	0
$307$	0.774162	0.0441838	0.0220919	−	0.999756i	$-0.492967\pi$
$307$	0.774162	0.0441838	0.0220919	+	0.999756i	$0.492967\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.1878	1.08804	0.544020	−	0.839072i	$-0.316902\pi$
$311$	19.1878	1.08804	0.544020	+	0.839072i	$0.316902\pi$
$312$	0	0
$313$	−2.65808	−0.150244	−0.0751218	−	0.997174i	$-0.523935\pi$
$313$	−2.65808	−0.150244	−0.0751218	+	0.997174i	$0.523935\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.0717	0.734181	0.367090	−	0.930185i	$-0.380354\pi$
$317$	13.0717	0.734181	0.367090	+	0.930185i	$0.380354\pi$
$318$	0	0
$319$	−41.0194	−2.29665
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.34192	−0.241591
$324$	0	0
$325$	−6.70682	−0.372027
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.03804	−0.221951	−0.110975	−	0.993823i	$-0.535397\pi$
$331$	−4.03804	−0.221951	−0.110975	+	0.993823i	$0.535397\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.70682	0.366433
$336$	0	0
$337$	2.33123	0.126990	0.0634950	−	0.997982i	$-0.479775\pi$
$337$	2.33123	0.126990	0.0634950	+	0.997982i	$0.479775\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.70682	0.309041
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.5190	−0.886788	−0.443394	−	0.896327i	$-0.646226\pi$
$347$	−16.5190	−0.886788	−0.443394	+	0.896327i	$0.646226\pi$
$348$	0	0
$349$	8.89461	0.476118	0.238059	−	0.971251i	$-0.423489\pi$
$349$	8.89461	0.476118	0.238059	+	0.971251i	$0.423489\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.9663	0.743353	0.371676	−	0.928362i	$-0.378783\pi$
$353$	13.9663	0.743353	0.371676	+	0.928362i	$0.378783\pi$
$354$	0	0
$355$	−11.7068	−0.621333
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.77416	0.410305	0.205152	−	0.978730i	$-0.434231\pi$
$359$	7.77416	0.410305	0.205152	+	0.978730i	$0.434231\pi$
$360$	0	0
$361$	−17.4136	−0.916507
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.966327	−0.0505799
$366$	0	0
$367$	14.3312	0.748084	0.374042	−	0.927412i	$-0.377972\pi$
$367$	14.3312	0.748084	0.374042	+	0.927412i	$0.377972\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.25514	0.116767	0.0583834	−	0.998294i	$-0.481405\pi$
$373$	2.25514	0.116767	0.0583834	+	0.998294i	$0.481405\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−48.2072	−2.48280
$378$	0	0
$379$	2.85657	0.146732	0.0733661	−	0.997305i	$-0.476626\pi$
$379$	2.85657	0.146732	0.0733661	+	0.997305i	$0.476626\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−27.9327	−1.42729	−0.713646	−	0.700507i	$-0.752957\pi$
$383$	−27.9327	−1.42729	−0.713646	+	0.700507i	$0.752957\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.5341	−0.534099	−0.267050	−	0.963683i	$-0.586049\pi$
$389$	−10.5341	−0.534099	−0.267050	+	0.963683i	$0.586049\pi$
$390$	0	0
$391$	−27.4624	−1.38883
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.6351	0.585425
$396$	0	0
$397$	−6.00437	−0.301351	−0.150675	−	0.988583i	$-0.548145\pi$
$397$	−6.00437	−0.301351	−0.150675	+	0.988583i	$0.548145\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.85657	0.492214	0.246107	−	0.969243i	$-0.420849\pi$
$401$	9.85657	0.492214	0.246107	+	0.969243i	$0.420849\pi$
$402$	0	0
$403$	6.70682	0.334090
$404$	0	0
$405$	0	0
$406$	0	0
$407$	51.1692	2.53636
$408$	0	0
$409$	14.8566	0.734610	0.367305	−	0.930101i	$-0.380281\pi$
$409$	14.8566	0.734610	0.367305	+	0.930101i	$0.380281\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−14.8609	−0.729495
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27.9327	−1.36460	−0.682300	−	0.731073i	$-0.739020\pi$
$419$	−27.9327	−1.36460	−0.682300	+	0.731073i	$0.739020\pi$
$420$	0	0
$421$	−5.71119	−0.278346	−0.139173	−	0.990268i	$-0.544444\pi$
$421$	−5.71119	−0.278346	−0.139173	+	0.990268i	$0.544444\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.44731	−0.167219
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.93265	0.478439	0.239220	−	0.970965i	$-0.423108\pi$
$431$	9.93265	0.478439	0.239220	+	0.970965i	$0.423108\pi$
$432$	0	0
$433$	−12.1204	−0.582472	−0.291236	−	0.956651i	$-0.594066\pi$
$433$	−12.1204	−0.582472	−0.291236	+	0.956651i	$0.594066\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.0337	0.479976
$438$	0	0
$439$	23.2302	1.10872	0.554359	−	0.832278i	$-0.312964\pi$
$439$	23.2302	1.10872	0.554359	+	0.832278i	$0.312964\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.0380	0.714479	0.357239	−	0.934013i	$-0.383718\pi$
$443$	15.0380	0.714479	0.357239	+	0.934013i	$0.383718\pi$
$444$	0	0
$445$	−12.6014	−0.597365
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.69175	0.268611	0.134305	−	0.990940i	$-0.457120\pi$
$449$	5.69175	0.268611	0.134305	+	0.990940i	$0.457120\pi$
$450$	0	0
$451$	58.3570	2.74792
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.774162	0.0362138	0.0181069	−	0.999836i	$-0.494236\pi$
$457$	0.774162	0.0362138	0.0181069	+	0.999836i	$0.494236\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.9097	1.43961	0.719804	−	0.694178i	$-0.244232\pi$
$461$	30.9097	1.43961	0.719804	+	0.694178i	$0.244232\pi$
$462$	0	0
$463$	−11.8122	−0.548960	−0.274480	−	0.961593i	$-0.588506\pi$
$463$	−11.8122	−0.548960	−0.274480	+	0.961593i	$0.588506\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.10539	−0.236249	−0.118125	−	0.992999i	$-0.537688\pi$
$467$	−5.10539	−0.236249	−0.118125	+	0.992999i	$0.537688\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	25.3800	1.16697
$474$	0	0
$475$	1.25951	0.0577903
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.8653	−0.633522	−0.316761	−	0.948505i	$-0.602595\pi$
$479$	−13.8653	−0.633522	−0.316761	+	0.948505i	$0.602595\pi$
$480$	0	0
$481$	60.1355	2.74194
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.1541	0.597298
$486$	0	0
$487$	−4.56339	−0.206787	−0.103393	−	0.994641i	$-0.532970\pi$
$487$	−4.56339	−0.206787	−0.103393	+	0.994641i	$0.532970\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9.93265	−0.448254	−0.224127	−	0.974560i	$-0.571953\pi$
$491$	−9.93265	−0.448254	−0.224127	+	0.974560i	$0.571953\pi$
$492$	0	0
$493$	−24.7785	−1.11597
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.89461	−0.353411	−0.176706	−	0.984264i	$-0.556544\pi$
$499$	−7.89461	−0.353411	−0.176706	+	0.984264i	$0.556544\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	11.1204	0.494854
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.9707	1.10681	0.553403	−	0.832913i	$-0.313329\pi$
$509$	24.9707	1.10681	0.553403	+	0.832913i	$0.313329\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.07171	0.0912907
$516$	0	0
$517$	8.45168	0.371704
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.4517	0.896004	0.448002	−	0.894033i	$-0.352136\pi$
$521$	20.4517	0.896004	0.448002	+	0.894033i	$0.352136\pi$
$522$	0	0
$523$	−35.2258	−1.54032	−0.770159	−	0.637852i	$-0.779823\pi$
$523$	−35.2258	−1.54032	−0.770159	+	0.637852i	$0.779823\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.44731	0.150167
$528$	0	0
$529$	40.4624	1.75923
$530$	0	0
$531$	0	0
$532$	0	0
$533$	68.5828	2.97065
$534$	0	0
$535$	1.96633	0.0850117
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−44.2029	−1.90043	−0.950215	−	0.311596i	$-0.899136\pi$
$541$	−44.2029	−1.90043	−0.950215	+	0.311596i	$0.899136\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.67314	0.371517
$546$	0	0
$547$	35.3463	1.51130	0.755649	−	0.654977i	$-0.227322\pi$
$547$	35.3463	1.51130	0.755649	+	0.654977i	$0.227322\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.05310	0.385675
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.82727	−0.204538	−0.102269	−	0.994757i	$-0.532610\pi$
$557$	−4.82727	−0.204538	−0.102269	+	0.994757i	$0.532610\pi$
$558$	0	0
$559$	29.8273	1.26156
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−36.0301	−1.51849	−0.759244	−	0.650806i	$-0.774431\pi$
$563$	−36.0301	−1.51849	−0.759244	+	0.650806i	$0.774431\pi$
$564$	0	0
$565$	−0.586367	−0.0246686
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.7449	1.12120	0.560601	−	0.828086i	$-0.310570\pi$
$569$	26.7449	1.12120	0.560601	+	0.828086i	$0.310570\pi$
$570$	0	0
$571$	−22.7405	−0.951660	−0.475830	−	0.879537i	$-0.657852\pi$
$571$	−22.7405	−0.951660	−0.475830	+	0.879537i	$0.657852\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.96633	0.332219
$576$	0	0
$577$	−3.66877	−0.152733	−0.0763665	−	0.997080i	$-0.524332\pi$
$577$	−3.66877	−0.152733	−0.0763665	+	0.997080i	$0.524332\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	39.3463	1.62956
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.2365	0.959074	0.479537	−	0.877522i	$-0.340805\pi$
$587$	23.2365	0.959074	0.479537	+	0.877522i	$0.340805\pi$
$588$	0	0
$589$	−1.25951	−0.0518972
$590$	0	0
$591$	0	0
$592$	0	0
$593$	34.6502	1.42291	0.711456	−	0.702731i	$-0.248036\pi$
$593$	34.6502	1.42291	0.711456	+	0.702731i	$0.248036\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−38.6838	−1.58058	−0.790289	−	0.612734i	$-0.790070\pi$
$599$	−38.6838	−1.58058	−0.790289	+	0.612734i	$0.790070\pi$
$600$	0	0
$601$	−16.2302	−0.662044	−0.331022	−	0.943623i	$-0.607393\pi$
$601$	−16.2302	−0.662044	−0.331022	+	0.943623i	$0.607393\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−21.5678	−0.876854
$606$	0	0
$607$	−1.60143	−0.0650000	−0.0325000	−	0.999472i	$-0.510347\pi$
$607$	−1.60143	−0.0650000	−0.0325000	+	0.999472i	$0.510347\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.93265	0.401832
$612$	0	0
$613$	6.32686	0.255539	0.127770	−	0.991804i	$-0.459218\pi$
$613$	6.32686	0.255539	0.127770	+	0.991804i	$0.459218\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.6545	−1.35488	−0.677440	−	0.735578i	$-0.736911\pi$
$617$	−33.6545	−1.35488	−0.677440	+	0.735578i	$0.736911\pi$
$618$	0	0
$619$	−41.2409	−1.65761	−0.828806	−	0.559536i	$-0.810979\pi$
$619$	−41.2409	−1.65761	−0.828806	+	0.559536i	$0.810979\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	30.9097	1.23245
$630$	0	0
$631$	37.4897	1.49244	0.746221	−	0.665698i	$-0.231866\pi$
$631$	37.4897	1.49244	0.746221	+	0.665698i	$0.231866\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.22584	0.207381
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.15849	−0.0852553	−0.0426277	−	0.999091i	$-0.513573\pi$
$641$	−2.15849	−0.0852553	−0.0426277	+	0.999091i	$0.513573\pi$
$642$	0	0
$643$	−29.2258	−1.15255	−0.576277	−	0.817254i	$-0.695495\pi$
$643$	−29.2258	−1.15255	−0.576277	+	0.817254i	$0.695495\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.62878	0.339232	0.169616	−	0.985510i	$-0.445747\pi$
$647$	8.62878	0.339232	0.169616	+	0.985510i	$0.445747\pi$
$648$	0	0
$649$	−1.67314	−0.0656766
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17.8229	−0.697464	−0.348732	−	0.937223i	$-0.613388\pi$
$653$	−17.8229	−0.697464	−0.348732	+	0.937223i	$0.613388\pi$
$654$	0	0
$655$	17.4136	0.680407
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−6.07608	−0.236691	−0.118345	−	0.992973i	$-0.537759\pi$
$659$	−6.07608	−0.236691	−0.118345	+	0.992973i	$0.537759\pi$
$660$	0	0
$661$	−11.9433	−0.464542	−0.232271	−	0.972651i	$-0.574616\pi$
$661$	−11.9433	−0.464542	−0.232271	+	0.972651i	$0.574616\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	57.2603	2.21713
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.48098	0.0571726
$672$	0	0
$673$	−43.6014	−1.68071	−0.840356	−	0.542035i	$-0.817654\pi$
$673$	−43.6014	−1.68071	−0.840356	+	0.542035i	$0.817654\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.78486	−0.337629	−0.168815	−	0.985648i	$-0.553994\pi$
$677$	−8.78486	−0.337629	−0.168815	+	0.985648i	$0.553994\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.7936	0.489533	0.244767	−	0.969582i	$-0.421289\pi$
$683$	12.7936	0.489533	0.244767	+	0.969582i	$0.421289\pi$
$684$	0	0
$685$	−13.9663	−0.533626
$686$	0	0
$687$	0	0
$688$	0	0
$689$	46.2409	1.76164
$690$	0	0
$691$	1.65371	0.0629102	0.0314551	−	0.999505i	$-0.489986\pi$
$691$	1.65371	0.0629102	0.0314551	+	0.999505i	$0.489986\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.0487	−0.570831
$696$	0	0
$697$	35.2516	1.33525
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23.0444	0.870374	0.435187	−	0.900340i	$-0.356682\pi$
$701$	23.0444	0.870374	0.435187	+	0.900340i	$0.356682\pi$
$702$	0	0
$703$	−11.2932	−0.425930
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20.4243	0.767052	0.383526	−	0.923530i	$-0.374710\pi$
$709$	20.4243	0.767052	0.383526	+	0.923530i	$0.374710\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.96633	−0.298341
$714$	0	0
$715$	−38.2746	−1.43139
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.62441	−0.135168	−0.0675838	−	0.997714i	$-0.521529\pi$
$719$	−3.62441	−0.135168	−0.0675838	+	0.997714i	$0.521529\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.18780	0.266948
$726$	0	0
$727$	53.1311	1.97053	0.985263	−	0.171049i	$-0.0547157\pi$
$727$	53.1311	1.97053	0.985263	+	0.171049i	$0.0547157\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.3312	0.567046
$732$	0	0
$733$	−33.6688	−1.24358	−0.621792	−	0.783182i	$-0.713595\pi$
$733$	−33.6688	−1.24358	−0.621792	+	0.783182i	$0.713595\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.2746	1.40986
$738$	0	0
$739$	14.2302	0.523467	0.261733	−	0.965140i	$-0.415706\pi$
$739$	14.2302	0.523467	0.261733	+	0.965140i	$0.415706\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.78486	0.102166	0.0510832	−	0.998694i	$-0.483733\pi$
$743$	2.78486	0.102166	0.0510832	+	0.998694i	$0.483733\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−28.9327	−1.05577	−0.527884	−	0.849317i	$-0.677014\pi$
$751$	−28.9327	−1.05577	−0.527884	+	0.849317i	$0.677014\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−21.0868	−0.767426
$756$	0	0
$757$	−15.7219	−0.571421	−0.285711	−	0.958316i	$-0.592230\pi$
$757$	−15.7219	−0.571421	−0.285711	+	0.958316i	$0.592230\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19.1878	0.695557	0.347779	−	0.937577i	$-0.386936\pi$
$761$	19.1878	0.695557	0.347779	+	0.937577i	$0.386936\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.96633	−0.0710000
$768$	0	0
$769$	26.2302	0.945885	0.472943	−	0.881093i	$-0.343192\pi$
$769$	26.2302	0.945885	0.472943	+	0.881093i	$0.343192\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27.8316	−1.00103	−0.500517	−	0.865727i	$-0.666857\pi$
$773$	−27.8316	−1.00103	−0.500517	+	0.865727i	$0.666857\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.8796	−0.461458
$780$	0	0
$781$	−66.8087	−2.39060
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.89461	−0.317462
$786$	0	0
$787$	−35.2789	−1.25756	−0.628779	−	0.777584i	$-0.716445\pi$
$787$	−35.2789	−1.25756	−0.628779	+	0.777584i	$0.716445\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.74049	0.0618066
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.1399	1.42183	0.710914	−	0.703279i	$-0.248282\pi$
$797$	40.1399	1.42183	0.710914	+	0.703279i	$0.248282\pi$
$798$	0	0
$799$	5.10539	0.180616
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.51465	−0.194608
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19.5571	0.687590	0.343795	−	0.939045i	$-0.388288\pi$
$809$	19.5571	0.687590	0.343795	+	0.939045i	$0.388288\pi$
$810$	0	0
$811$	4.33559	0.152243	0.0761217	−	0.997099i	$-0.475746\pi$
$811$	4.33559	0.152243	0.0761217	+	0.997099i	$0.475746\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.3756	−0.363441
$816$	0	0
$817$	−5.60143	−0.195969
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.6775	1.48946	0.744728	−	0.667368i	$-0.232579\pi$
$821$	42.6775	1.48946	0.744728	+	0.667368i	$0.232579\pi$
$822$	0	0
$823$	42.0487	1.46573	0.732863	−	0.680376i	$-0.238183\pi$
$823$	42.0487	1.46573	0.732863	+	0.680376i	$0.238183\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.8566	−0.551387	−0.275693	−	0.961246i	$-0.588907\pi$
$827$	−15.8566	−0.551387	−0.275693	+	0.961246i	$0.588907\pi$
$828$	0	0
$829$	31.6438	1.09904	0.549518	−	0.835482i	$-0.314811\pi$
$829$	31.6438	1.09904	0.549518	+	0.835482i	$0.314811\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.96633	−0.0680476
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.0824	1.31475	0.657375	−	0.753563i	$-0.271667\pi$
$839$	38.0824	1.31475	0.657375	+	0.753563i	$0.271667\pi$
$840$	0	0
$841$	22.6644	0.781531
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−31.9814	−1.10019
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−71.4287	−2.44854
$852$	0	0
$853$	45.1399	1.54556	0.772780	−	0.634674i	$-0.218866\pi$
$853$	45.1399	1.54556	0.772780	+	0.634674i	$0.218866\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.5907	−0.805844	−0.402922	−	0.915234i	$-0.632005\pi$
$857$	−23.5907	−0.805844	−0.402922	+	0.915234i	$0.632005\pi$
$858$	0	0
$859$	−31.2302	−1.06556	−0.532780	−	0.846254i	$-0.678853\pi$
$859$	−31.2302	−1.06556	−0.532780	+	0.846254i	$0.678853\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.0087	−0.544944	−0.272472	−	0.962164i	$-0.587841\pi$
$863$	−16.0087	−0.544944	−0.272472	+	0.962164i	$0.587841\pi$
$864$	0	0
$865$	1.48098	0.0503548
$866$	0	0
$867$	0	0
$868$	0	0
$869$	66.3994	2.25245
$870$	0	0
$871$	44.9814	1.52414
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−0.375591	−0.0126828	−0.00634141	−	0.999980i	$-0.502019\pi$
$877$	−0.375591	−0.0126828	−0.00634141	+	0.999980i	$0.502019\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−31.7892	−1.07101	−0.535503	−	0.844533i	$-0.679878\pi$
$881$	−31.7892	−1.07101	−0.535503	+	0.844533i	$0.679878\pi$
$882$	0	0
$883$	21.0638	0.708853	0.354427	−	0.935084i	$-0.384676\pi$
$883$	21.0638	0.708853	0.354427	+	0.935084i	$0.384676\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.7262	0.964533	0.482267	−	0.876025i	$-0.339814\pi$
$887$	28.7262	0.964533	0.482267	+	0.876025i	$0.339814\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.86531	−0.0624202
$894$	0	0
$895$	−18.3082	−0.611977
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.18780	−0.239726
$900$	0	0
$901$	23.7678	0.791822
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.6731	−0.620716
$906$	0	0
$907$	−2.96633	−0.0984953	−0.0492476	−	0.998787i	$-0.515682\pi$
$907$	−2.96633	−0.0984953	−0.0492476	+	0.998787i	$0.515682\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−15.5634	−0.515638	−0.257819	−	0.966193i	$-0.583004\pi$
$911$	−15.5634	−0.515638	−0.257819	+	0.966193i	$0.583004\pi$
$912$	0	0
$913$	−84.8087	−2.80676
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	25.6838	0.847232	0.423616	−	0.905842i	$-0.360761\pi$
$919$	25.6838	0.847232	0.423616	+	0.905842i	$0.360761\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−78.5155	−2.58437
$924$	0	0
$925$	−8.96633	−0.294811
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.73612	0.155387	0.0776935	−	0.996977i	$-0.475244\pi$
$929$	4.73612	0.155387	0.0776935	+	0.996977i	$0.475244\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19.6731	−0.643381
$936$	0	0
$937$	−23.3419	−0.762547	−0.381274	−	0.924462i	$-0.624514\pi$
$937$	−23.3419	−0.762547	−0.381274	+	0.924462i	$0.624514\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	16.4580	0.536515	0.268258	−	0.963347i	$-0.413552\pi$
$941$	16.4580	0.536515	0.268258	+	0.963347i	$0.413552\pi$
$942$	0	0
$943$	−81.4624	−2.65278
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.2365	0.755086	0.377543	−	0.925992i	$-0.376769\pi$
$947$	23.2365	0.755086	0.377543	+	0.925992i	$0.376769\pi$
$948$	0	0
$949$	−6.48098	−0.210381
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.37559	−0.0769529	−0.0384765	−	0.999260i	$-0.512250\pi$
$953$	−2.37559	−0.0769529	−0.0384765	+	0.999260i	$0.512250\pi$
$954$	0	0
$955$	6.30825	0.204130
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−22.8990	−0.737144
$966$	0	0
$967$	36.6121	1.17737	0.588683	−	0.808364i	$-0.299647\pi$
$967$	36.6121	1.17737	0.588683	+	0.808364i	$0.299647\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.8653	−1.21516	−0.607578	−	0.794260i	$-0.707859\pi$
$971$	−37.8653	−1.21516	−0.607578	+	0.794260i	$0.707859\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.1399	1.86006	0.930030	−	0.367484i	$-0.119781\pi$
$977$	58.1399	1.86006	0.930030	+	0.367484i	$0.119781\pi$
$978$	0	0
$979$	−71.9140	−2.29838
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.3800	−0.809495	−0.404748	−	0.914428i	$-0.632641\pi$
$983$	−25.3800	−0.809495	−0.404748	+	0.914428i	$0.632641\pi$
$984$	0	0
$985$	−14.5527	−0.463687
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−35.4287	−1.12657
$990$	0	0
$991$	31.7999	1.01016	0.505079	−	0.863073i	$-0.331463\pi$
$991$	31.7999	1.01016	0.505079	+	0.863073i	$0.331463\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17.0868	0.541687
$996$	0	0
$997$	54.0258	1.71101	0.855506	−	0.517792i	$-0.173246\pi$
$997$	54.0258	1.71101	0.855506	+	0.517792i	$0.173246\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.a.bn.1.1		3
3.2	odd	2		8820.2.a.bq.1.3		3
7.2	even	3		1260.2.s.h.361.3	yes	6
7.4	even	3		1260.2.s.h.541.3	yes	6
7.6	odd	2		8820.2.a.bp.1.1		3
21.2	odd	6		1260.2.s.g.361.3	✓	6
21.11	odd	6		1260.2.s.g.541.3	yes	6
21.20	even	2		8820.2.a.bo.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.s.g.361.3	✓	6	21.2	odd	6
1260.2.s.g.541.3	yes	6	21.11	odd	6
1260.2.s.h.361.3	yes	6	7.2	even	3
1260.2.s.h.541.3	yes	6	7.4	even	3
8820.2.a.bn.1.1		3	1.1	even	1	trivial
8820.2.a.bo.1.3		3	21.20	even	2
8820.2.a.bp.1.1		3	7.6	odd	2
8820.2.a.bq.1.3		3	3.2	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:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +O(q^{10})\)	\(q-1.00000 q^{5} -5.70682 q^{11} -6.70682 q^{13} -3.44731 q^{17} +1.25951 q^{19} +7.96633 q^{23} +1.00000 q^{25} +7.18780 q^{29} -1.00000 q^{31} -8.96633 q^{37} -10.2258 q^{41} -4.44731 q^{43} -1.48098 q^{47} -6.89461 q^{53} +5.70682 q^{55} +0.293183 q^{59} -0.259511 q^{61} +6.70682 q^{65} -6.70682 q^{67} +11.7068 q^{71} +0.966327 q^{73} -11.6351 q^{79} +14.8609 q^{83} +3.44731 q^{85} +12.6014 q^{89} -1.25951 q^{95} -13.1541 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5}+O(q^{10}) \) 3 * q - 3 * q^5	\( 3 q - 3 q^{5} - 2 q^{11} - 5 q^{13} + 2 q^{17} + q^{19} + 6 q^{23} + 3 q^{25} + 12 q^{29} - 3 q^{31} - 9 q^{37} - 10 q^{41} - q^{43} - 10 q^{47} + 4 q^{53} + 2 q^{55} + 16 q^{59} + 2 q^{61} + 5 q^{65} - 5 q^{67} + 20 q^{71} - 15 q^{73} - 13 q^{79} + 2 q^{83} - 2 q^{85} - 2 q^{89} - q^{95} - 12 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 2 * q^11 - 5 * q^13 + 2 * q^17 + q^19 + 6 * q^23 + 3 * q^25 + 12 * q^29 - 3 * q^31 - 9 * q^37 - 10 * q^41 - q^43 - 10 * q^47 + 4 * q^53 + 2 * q^55 + 16 * q^59 + 2 * q^61 + 5 * q^65 - 5 * q^67 + 20 * q^71 - 15 * q^73 - 13 * q^79 + 2 * q^83 - 2 * q^85 - 2 * q^89 - q^95 - 12 * q^97

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