LMFDB - Embedded newform 8280.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.31662$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -3.00000 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -3.00000 q^{7} -2.31662 q^{11} +4.31662 q^{13} -3.31662 q^{17} -8.31662 q^{19} -1.00000 q^{23} +1.00000 q^{25} +7.31662 q^{29} +3.63325 q^{31} -3.00000 q^{35} -1.63325 q^{37} +3.31662 q^{41} -10.6332 q^{43} +8.94987 q^{47} +2.00000 q^{49} -9.94987 q^{53} -2.31662 q^{55} -7.94987 q^{59} +10.3166 q^{61} +4.31662 q^{65} +15.6332 q^{67} -11.9499 q^{71} +8.31662 q^{73} +6.94987 q^{77} +1.31662 q^{83} -3.31662 q^{85} +4.63325 q^{89} -12.9499 q^{91} -8.31662 q^{95} +12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 6 * q^7	$ 2 q + 2 q^{5} - 6 q^{7} + 2 q^{11} + 2 q^{13} - 10 q^{19} - 2 q^{23} + 2 q^{25} + 8 q^{29} - 6 q^{31} - 6 q^{35} + 10 q^{37} - 8 q^{43} - 2 q^{47} + 4 q^{49} + 2 q^{55} + 4 q^{59} + 14 q^{61} + 2 q^{65} + 18 q^{67} - 4 q^{71} + 10 q^{73} - 6 q^{77} - 4 q^{83} - 4 q^{89} - 6 q^{91} - 10 q^{95} + 24 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 6 * q^7 + 2 * q^11 + 2 * q^13 - 10 * q^19 - 2 * q^23 + 2 * q^25 + 8 * q^29 - 6 * q^31 - 6 * q^35 + 10 * q^37 - 8 * q^43 - 2 * q^47 + 4 * q^49 + 2 * q^55 + 4 * q^59 + 14 * q^61 + 2 * q^65 + 18 * q^67 - 4 * q^71 + 10 * q^73 - 6 * q^77 - 4 * q^83 - 4 * q^89 - 6 * q^91 - 10 * q^95 + 24 * 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
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.31662	−0.698489	−0.349244	−	0.937032i	$-0.613562\pi$
$11$	−2.31662	−0.698489	−0.349244	+	0.937032i	$0.613562\pi$
$12$	0	0
$13$	4.31662	1.19722	0.598608	−	0.801042i	$-0.295721\pi$
$13$	4.31662	1.19722	0.598608	+	0.801042i	$0.295721\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.31662	−0.804400	−0.402200	−	0.915552i	$-0.631754\pi$
$17$	−3.31662	−0.804400	−0.402200	+	0.915552i	$0.631754\pi$
$18$	0	0
$19$	−8.31662	−1.90796	−0.953982	−	0.299863i	$-0.903059\pi$
$19$	−8.31662	−1.90796	−0.953982	+	0.299863i	$0.903059\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.31662	1.35866	0.679332	−	0.733831i	$-0.262270\pi$
$29$	7.31662	1.35866	0.679332	+	0.733831i	$0.262270\pi$
$30$	0	0
$31$	3.63325	0.652551	0.326275	−	0.945275i	$-0.394206\pi$
$31$	3.63325	0.652551	0.326275	+	0.945275i	$0.394206\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−1.63325	−0.268505	−0.134252	−	0.990947i	$-0.542863\pi$
$37$	−1.63325	−0.268505	−0.134252	+	0.990947i	$0.542863\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.31662	0.517970	0.258985	−	0.965881i	$-0.416612\pi$
$41$	3.31662	0.517970	0.258985	+	0.965881i	$0.416612\pi$
$42$	0	0
$43$	−10.6332	−1.62156	−0.810778	−	0.585354i	$-0.800955\pi$
$43$	−10.6332	−1.62156	−0.810778	+	0.585354i	$0.800955\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.94987	1.30547	0.652737	−	0.757585i	$-0.273621\pi$
$47$	8.94987	1.30547	0.652737	+	0.757585i	$0.273621\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.94987	−1.36672	−0.683360	−	0.730081i	$-0.739482\pi$
$53$	−9.94987	−1.36672	−0.683360	+	0.730081i	$0.739482\pi$
$54$	0	0
$55$	−2.31662	−0.312374
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.94987	−1.03499	−0.517493	−	0.855688i	$-0.673135\pi$
$59$	−7.94987	−1.03499	−0.517493	+	0.855688i	$0.673135\pi$
$60$	0	0
$61$	10.3166	1.32091	0.660454	−	0.750866i	$-0.270364\pi$
$61$	10.3166	1.32091	0.660454	+	0.750866i	$0.270364\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.31662	0.535411
$66$	0	0
$67$	15.6332	1.90991	0.954953	−	0.296758i	$-0.0959055\pi$
$67$	15.6332	1.90991	0.954953	+	0.296758i	$0.0959055\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.9499	−1.41819	−0.709095	−	0.705113i	$-0.750896\pi$
$71$	−11.9499	−1.41819	−0.709095	+	0.705113i	$0.750896\pi$
$72$	0	0
$73$	8.31662	0.973387	0.486694	−	0.873573i	$-0.338203\pi$
$73$	8.31662	0.973387	0.486694	+	0.873573i	$0.338203\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.94987	0.792012
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.31662	0.144518	0.0722592	−	0.997386i	$-0.476979\pi$
$83$	1.31662	0.144518	0.0722592	+	0.997386i	$0.476979\pi$
$84$	0	0
$85$	−3.31662	−0.359738
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.63325	0.491123	0.245562	−	0.969381i	$-0.421028\pi$
$89$	4.63325	0.491123	0.245562	+	0.969381i	$0.421028\pi$
$90$	0	0
$91$	−12.9499	−1.35752
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.31662	−0.853268
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.31662	0.131009	0.0655045	−	0.997852i	$-0.479134\pi$
$101$	1.31662	0.131009	0.0655045	+	0.997852i	$0.479134\pi$
$102$	0	0
$103$	18.0000	1.77359	0.886796	−	0.462160i	$-0.152926\pi$
$103$	18.0000	1.77359	0.886796	+	0.462160i	$0.152926\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.94987	−0.575196	−0.287598	−	0.957751i	$-0.592857\pi$
$107$	−5.94987	−0.575196	−0.287598	+	0.957751i	$0.592857\pi$
$108$	0	0
$109$	8.94987	0.857242	0.428621	−	0.903484i	$-0.358999\pi$
$109$	8.94987	0.857242	0.428621	+	0.903484i	$0.358999\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.68338	−0.816863	−0.408432	−	0.912789i	$-0.633924\pi$
$113$	−8.68338	−0.816863	−0.408432	+	0.912789i	$0.633924\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.94987	0.912103
$120$	0	0
$121$	−5.63325	−0.512114
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.9499	1.32659	0.663293	−	0.748359i	$-0.269158\pi$
$127$	14.9499	1.32659	0.663293	+	0.748359i	$0.269158\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	24.9499	2.16343
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.2665	−1.13343	−0.566717	−	0.823913i	$-0.691787\pi$
$137$	−13.2665	−1.13343	−0.566717	+	0.823913i	$0.691787\pi$
$138$	0	0
$139$	11.6332	0.986719	0.493360	−	0.869825i	$-0.335769\pi$
$139$	11.6332	0.986719	0.493360	+	0.869825i	$0.335769\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−10.0000	−0.836242
$144$	0	0
$145$	7.31662	0.607613
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.31662	0.353632	0.176816	−	0.984244i	$-0.443420\pi$
$149$	4.31662	0.353632	0.176816	+	0.984244i	$0.443420\pi$
$150$	0	0
$151$	−14.6332	−1.19084	−0.595418	−	0.803416i	$-0.703014\pi$
$151$	−14.6332	−1.19084	−0.595418	+	0.803416i	$0.703014\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.63325	0.291830
$156$	0	0
$157$	19.6332	1.56690	0.783452	−	0.621452i	$-0.213457\pi$
$157$	19.6332	1.56690	0.783452	+	0.621452i	$0.213457\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	−23.2665	−1.82237	−0.911186	−	0.411994i	$-0.864832\pi$
$163$	−23.2665	−1.82237	−0.911186	+	0.411994i	$0.864832\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.5831	1.05109	0.525547	−	0.850765i	$-0.323861\pi$
$167$	13.5831	1.05109	0.525547	+	0.850765i	$0.323861\pi$
$168$	0	0
$169$	5.63325	0.433327
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.6332	1.69169	0.845844	−	0.533430i	$-0.179097\pi$
$179$	22.6332	1.69169	0.845844	+	0.533430i	$0.179097\pi$
$180$	0	0
$181$	7.36675	0.547566	0.273783	−	0.961791i	$-0.411725\pi$
$181$	7.36675	0.547566	0.273783	+	0.961791i	$0.411725\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.63325	−0.120079
$186$	0	0
$187$	7.68338	0.561864
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.5831	0.982840	0.491420	−	0.870923i	$-0.336478\pi$
$191$	13.5831	0.982840	0.491420	+	0.870923i	$0.336478\pi$
$192$	0	0
$193$	−0.633250	−0.0455823	−0.0227912	−	0.999740i	$-0.507255\pi$
$193$	−0.633250	−0.0455823	−0.0227912	+	0.999740i	$0.507255\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.8997	−0.847822	−0.423911	−	0.905704i	$-0.639343\pi$
$197$	−11.8997	−0.847822	−0.423911	+	0.905704i	$0.639343\pi$
$198$	0	0
$199$	14.6332	1.03732	0.518662	−	0.854980i	$-0.326430\pi$
$199$	14.6332	1.03732	0.518662	+	0.854980i	$0.326430\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−21.9499	−1.54058
$204$	0	0
$205$	3.31662	0.231643
$206$	0	0
$207$	0	0
$208$	0	0
$209$	19.2665	1.33269
$210$	0	0
$211$	−3.63325	−0.250123	−0.125062	−	0.992149i	$-0.539913\pi$
$211$	−3.63325	−0.250123	−0.125062	+	0.992149i	$0.539913\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.6332	−0.725182
$216$	0	0
$217$	−10.8997	−0.739923
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−14.3166	−0.963040
$222$	0	0
$223$	23.2665	1.55804	0.779020	−	0.626999i	$-0.215717\pi$
$223$	23.2665	1.55804	0.779020	+	0.626999i	$0.215717\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	0	0
$229$	3.36675	0.222481	0.111241	−	0.993794i	$-0.464518\pi$
$229$	3.36675	0.222481	0.111241	+	0.993794i	$0.464518\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	0	0
$235$	8.94987	0.583825
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.9499	1.03171	0.515856	−	0.856675i	$-0.327474\pi$
$239$	15.9499	1.03171	0.515856	+	0.856675i	$0.327474\pi$
$240$	0	0
$241$	−26.2164	−1.68875	−0.844373	−	0.535756i	$-0.820027\pi$
$241$	−26.2164	−1.68875	−0.844373	+	0.535756i	$0.820027\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	−35.8997	−2.28425
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.2665	−1.46857	−0.734284	−	0.678842i	$-0.762482\pi$
$251$	−23.2665	−1.46857	−0.734284	+	0.678842i	$0.762482\pi$
$252$	0	0
$253$	2.31662	0.145645
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.949874	−0.0592515	−0.0296258	−	0.999561i	$-0.509432\pi$
$257$	−0.949874	−0.0592515	−0.0296258	+	0.999561i	$0.509432\pi$
$258$	0	0
$259$	4.89975	0.304456
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−28.5831	−1.76251	−0.881255	−	0.472640i	$-0.843301\pi$
$263$	−28.5831	−1.76251	−0.881255	+	0.472640i	$0.843301\pi$
$264$	0	0
$265$	−9.94987	−0.611216
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.5831	1.37692	0.688459	−	0.725276i	$-0.258288\pi$
$269$	22.5831	1.37692	0.688459	+	0.725276i	$0.258288\pi$
$270$	0	0
$271$	24.2665	1.47408	0.737042	−	0.675846i	$-0.236222\pi$
$271$	24.2665	1.47408	0.737042	+	0.675846i	$0.236222\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.31662	−0.139698
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.94987	−0.175975	−0.0879874	−	0.996122i	$-0.528044\pi$
$281$	−2.94987	−0.175975	−0.0879874	+	0.996122i	$0.528044\pi$
$282$	0	0
$283$	−5.63325	−0.334862	−0.167431	−	0.985884i	$-0.553547\pi$
$283$	−5.63325	−0.334862	−0.167431	+	0.985884i	$0.553547\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.94987	−0.587323
$288$	0	0
$289$	−6.00000	−0.352941
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.94987	0.464437	0.232218	−	0.972664i	$-0.425402\pi$
$293$	7.94987	0.464437	0.232218	+	0.972664i	$0.425402\pi$
$294$	0	0
$295$	−7.94987	−0.462860
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.31662	−0.249637
$300$	0	0
$301$	31.8997	1.83867
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.3166	0.590728
$306$	0	0
$307$	−24.9499	−1.42396	−0.711982	−	0.702197i	$-0.752202\pi$
$307$	−24.9499	−1.42396	−0.711982	+	0.702197i	$0.752202\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.73350	−0.155003	−0.0775013	−	0.996992i	$-0.524694\pi$
$311$	−2.73350	−0.155003	−0.0775013	+	0.996992i	$0.524694\pi$
$312$	0	0
$313$	33.6332	1.90106	0.950532	−	0.310627i	$-0.100539\pi$
$313$	33.6332	1.90106	0.950532	+	0.310627i	$0.100539\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.3166	0.916433	0.458216	−	0.888841i	$-0.348488\pi$
$317$	16.3166	0.916433	0.458216	+	0.888841i	$0.348488\pi$
$318$	0	0
$319$	−16.9499	−0.949011
$320$	0	0
$321$	0	0
$322$	0	0
$323$	27.5831	1.53477
$324$	0	0
$325$	4.31662	0.239443
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−26.8496	−1.48027
$330$	0	0
$331$	21.5330	1.18356	0.591780	−	0.806099i	$-0.298425\pi$
$331$	21.5330	1.18356	0.591780	+	0.806099i	$0.298425\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	15.6332	0.854136
$336$	0	0
$337$	4.73350	0.257850	0.128925	−	0.991654i	$-0.458847\pi$
$337$	4.73350	0.257850	0.128925	+	0.991654i	$0.458847\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8.41688	−0.455799
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.6332	1.32238	0.661191	−	0.750218i	$-0.270051\pi$
$347$	24.6332	1.32238	0.661191	+	0.750218i	$0.270051\pi$
$348$	0	0
$349$	−8.36675	−0.447862	−0.223931	−	0.974605i	$-0.571889\pi$
$349$	−8.36675	−0.447862	−0.223931	+	0.974605i	$0.571889\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−26.2164	−1.39536	−0.697678	−	0.716411i	$-0.745784\pi$
$353$	−26.2164	−1.39536	−0.697678	+	0.716411i	$0.745784\pi$
$354$	0	0
$355$	−11.9499	−0.634233
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.8496	−1.62818	−0.814090	−	0.580738i	$-0.802764\pi$
$359$	−30.8496	−1.62818	−0.814090	+	0.580738i	$0.802764\pi$
$360$	0	0
$361$	50.1662	2.64033
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.31662	0.435312
$366$	0	0
$367$	35.5330	1.85481	0.927404	−	0.374062i	$-0.122035\pi$
$367$	35.5330	1.85481	0.927404	+	0.374062i	$0.122035\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	29.8496	1.54972
$372$	0	0
$373$	−9.89975	−0.512590	−0.256295	−	0.966599i	$-0.582502\pi$
$373$	−9.89975	−0.512590	−0.256295	+	0.966599i	$0.582502\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.5831	1.62661
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12.6834	−0.648090	−0.324045	−	0.946042i	$-0.605043\pi$
$383$	−12.6834	−0.648090	−0.324045	+	0.946042i	$0.605043\pi$
$384$	0	0
$385$	6.94987	0.354198
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−15.8997	−0.806149	−0.403075	−	0.915167i	$-0.632058\pi$
$389$	−15.8997	−0.806149	−0.403075	+	0.915167i	$0.632058\pi$
$390$	0	0
$391$	3.31662	0.167729
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.8997	−0.993746	−0.496873	−	0.867823i	$-0.665519\pi$
$401$	−19.8997	−0.993746	−0.496873	+	0.867823i	$0.665519\pi$
$402$	0	0
$403$	15.6834	0.781245
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.78363	0.187547
$408$	0	0
$409$	13.6332	0.674121	0.337060	−	0.941483i	$-0.390567\pi$
$409$	13.6332	0.674121	0.337060	+	0.941483i	$0.390567\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	23.8496	1.17356
$414$	0	0
$415$	1.31662	0.0646306
$416$	0	0
$417$	0	0
$418$	0	0
$419$	18.3166	0.894826	0.447413	−	0.894328i	$-0.352345\pi$
$419$	18.3166	0.894826	0.447413	+	0.894328i	$0.352345\pi$
$420$	0	0
$421$	−9.58312	−0.467053	−0.233526	−	0.972350i	$-0.575027\pi$
$421$	−9.58312	−0.467053	−0.233526	+	0.972350i	$0.575027\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.31662	−0.160880
$426$	0	0
$427$	−30.9499	−1.49777
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.36675	0.162171	0.0810853	−	0.996707i	$-0.474161\pi$
$431$	3.36675	0.162171	0.0810853	+	0.996707i	$0.474161\pi$
$432$	0	0
$433$	−23.6332	−1.13574	−0.567871	−	0.823118i	$-0.692233\pi$
$433$	−23.6332	−1.13574	−0.567871	+	0.823118i	$0.692233\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.31662	0.397838
$438$	0	0
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	28.3166	1.34536	0.672682	−	0.739932i	$-0.265142\pi$
$443$	28.3166	1.34536	0.672682	+	0.739932i	$0.265142\pi$
$444$	0	0
$445$	4.63325	0.219637
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.31662	0.345293	0.172646	−	0.984984i	$-0.444768\pi$
$449$	7.31662	0.345293	0.172646	+	0.984984i	$0.444768\pi$
$450$	0	0
$451$	−7.68338	−0.361796
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−12.9499	−0.607099
$456$	0	0
$457$	25.6332	1.19907	0.599536	−	0.800347i	$-0.295352\pi$
$457$	25.6332	1.19907	0.599536	+	0.800347i	$0.295352\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.63325	−0.122643	−0.0613213	−	0.998118i	$-0.519531\pi$
$461$	−2.63325	−0.122643	−0.0613213	+	0.998118i	$0.519531\pi$
$462$	0	0
$463$	38.2164	1.77607	0.888033	−	0.459780i	$-0.152072\pi$
$463$	38.2164	1.77607	0.888033	+	0.459780i	$0.152072\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.0501	0.465064	0.232532	−	0.972589i	$-0.425299\pi$
$467$	10.0501	0.465064	0.232532	+	0.972589i	$0.425299\pi$
$468$	0	0
$469$	−46.8997	−2.16563
$470$	0	0
$471$	0	0
$472$	0	0
$473$	24.6332	1.13264
$474$	0	0
$475$	−8.31662	−0.381593
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11.6834	−0.533827	−0.266914	−	0.963720i	$-0.586004\pi$
$479$	−11.6834	−0.533827	−0.266914	+	0.963720i	$0.586004\pi$
$480$	0	0
$481$	−7.05013	−0.321458
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.0000	0.544892
$486$	0	0
$487$	−19.5831	−0.887396	−0.443698	−	0.896176i	$-0.646334\pi$
$487$	−19.5831	−0.887396	−0.443698	+	0.896176i	$0.646334\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.94987	0.449032	0.224516	−	0.974470i	$-0.427920\pi$
$491$	9.94987	0.449032	0.224516	+	0.974470i	$0.427920\pi$
$492$	0	0
$493$	−24.2665	−1.09291
$494$	0	0
$495$	0	0
$496$	0	0
$497$	35.8496	1.60808
$498$	0	0
$499$	−8.26650	−0.370059	−0.185030	−	0.982733i	$-0.559238\pi$
$499$	−8.26650	−0.370059	−0.185030	+	0.982733i	$0.559238\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.8496	0.974227	0.487113	−	0.873339i	$-0.338050\pi$
$503$	21.8496	0.974227	0.487113	+	0.873339i	$0.338050\pi$
$504$	0	0
$505$	1.31662	0.0585890
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	0	0
$511$	−24.9499	−1.10372
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.0000	0.793175
$516$	0	0
$517$	−20.7335	−0.911858
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.31662	−0.276736	−0.138368	−	0.990381i	$-0.544186\pi$
$521$	−6.31662	−0.276736	−0.138368	+	0.990381i	$0.544186\pi$
$522$	0	0
$523$	30.5330	1.33511	0.667557	−	0.744558i	$-0.267340\pi$
$523$	30.5330	1.33511	0.667557	+	0.744558i	$0.267340\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.0501	−0.524912
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14.3166	0.620122
$534$	0	0
$535$	−5.94987	−0.257236
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.63325	−0.199568
$540$	0	0
$541$	−5.26650	−0.226424	−0.113212	−	0.993571i	$-0.536114\pi$
$541$	−5.26650	−0.226424	−0.113212	+	0.993571i	$0.536114\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.94987	0.383370
$546$	0	0
$547$	1.26650	0.0541516	0.0270758	−	0.999633i	$-0.491380\pi$
$547$	1.26650	0.0541516	0.0270758	+	0.999633i	$0.491380\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−60.8496	−2.59228
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.05013	0.256352	0.128176	−	0.991751i	$-0.459088\pi$
$557$	6.05013	0.256352	0.128176	+	0.991751i	$0.459088\pi$
$558$	0	0
$559$	−45.8997	−1.94135
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−43.3166	−1.82558	−0.912789	−	0.408431i	$-0.866076\pi$
$563$	−43.3166	−1.82558	−0.912789	+	0.408431i	$0.866076\pi$
$564$	0	0
$565$	−8.68338	−0.365312
$566$	0	0
$567$	0	0
$568$	0	0
$569$	35.8997	1.50500	0.752498	−	0.658595i	$-0.228849\pi$
$569$	35.8997	1.50500	0.752498	+	0.658595i	$0.228849\pi$
$570$	0	0
$571$	5.05013	0.211341	0.105671	−	0.994401i	$-0.466301\pi$
$571$	5.05013	0.211341	0.105671	+	0.994401i	$0.466301\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	17.2665	0.718814	0.359407	−	0.933181i	$-0.382979\pi$
$577$	17.2665	0.718814	0.359407	+	0.933181i	$0.382979\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.94987	−0.163868
$582$	0	0
$583$	23.0501	0.954639
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	0	0
$589$	−30.2164	−1.24504
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.6834	0.644039	0.322020	−	0.946733i	$-0.395638\pi$
$593$	15.6834	0.644039	0.322020	+	0.946733i	$0.395638\pi$
$594$	0	0
$595$	9.94987	0.407905
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−15.3668	−0.627868	−0.313934	−	0.949445i	$-0.601647\pi$
$599$	−15.3668	−0.627868	−0.313934	+	0.949445i	$0.601647\pi$
$600$	0	0
$601$	−3.63325	−0.148203	−0.0741017	−	0.997251i	$-0.523609\pi$
$601$	−3.63325	−0.148203	−0.0741017	+	0.997251i	$0.523609\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5.63325	−0.229024
$606$	0	0
$607$	3.58312	0.145435	0.0727173	−	0.997353i	$-0.476833\pi$
$607$	3.58312	0.145435	0.0727173	+	0.997353i	$0.476833\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	38.6332	1.56293
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.0501	−0.485120	−0.242560	−	0.970136i	$-0.577987\pi$
$617$	−12.0501	−0.485120	−0.242560	+	0.970136i	$0.577987\pi$
$618$	0	0
$619$	−35.7995	−1.43890	−0.719452	−	0.694543i	$-0.755607\pi$
$619$	−35.7995	−1.43890	−0.719452	+	0.694543i	$0.755607\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.8997	−0.556882
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.41688	0.215985
$630$	0	0
$631$	−2.94987	−0.117433	−0.0587163	−	0.998275i	$-0.518701\pi$
$631$	−2.94987	−0.117433	−0.0587163	+	0.998275i	$0.518701\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.9499	0.593268
$636$	0	0
$637$	8.63325	0.342062
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23.5831	−0.931477	−0.465739	−	0.884922i	$-0.654211\pi$
$641$	−23.5831	−0.931477	−0.465739	+	0.884922i	$0.654211\pi$
$642$	0	0
$643$	8.26650	0.325999	0.162999	−	0.986626i	$-0.447883\pi$
$643$	8.26650	0.325999	0.162999	+	0.986626i	$0.447883\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.5831	0.612636	0.306318	−	0.951929i	$-0.400903\pi$
$647$	15.5831	0.612636	0.306318	+	0.951929i	$0.400903\pi$
$648$	0	0
$649$	18.4169	0.722926
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.78363	0.148065	0.0740324	−	0.997256i	$-0.476413\pi$
$653$	3.78363	0.148065	0.0740324	+	0.997256i	$0.476413\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.41688	0.172057	0.0860285	−	0.996293i	$-0.472582\pi$
$659$	4.41688	0.172057	0.0860285	+	0.996293i	$0.472582\pi$
$660$	0	0
$661$	−21.3668	−0.831070	−0.415535	−	0.909577i	$-0.636406\pi$
$661$	−21.3668	−0.831070	−0.415535	+	0.909577i	$0.636406\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	24.9499	0.967515
$666$	0	0
$667$	−7.31662	−0.283301
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−23.8997	−0.922640
$672$	0	0
$673$	−22.9499	−0.884653	−0.442326	−	0.896854i	$-0.645847\pi$
$673$	−22.9499	−0.884653	−0.442326	+	0.896854i	$0.645847\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.9499	1.76600	0.882999	−	0.469376i	$-0.155521\pi$
$677$	45.9499	1.76600	0.882999	+	0.469376i	$0.155521\pi$
$678$	0	0
$679$	−36.0000	−1.38155
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20.3166	−0.777394	−0.388697	−	0.921366i	$-0.627075\pi$
$683$	−20.3166	−0.777394	−0.388697	+	0.921366i	$0.627075\pi$
$684$	0	0
$685$	−13.2665	−0.506887
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−42.9499	−1.63626
$690$	0	0
$691$	−2.63325	−0.100174	−0.0500868	−	0.998745i	$-0.515950\pi$
$691$	−2.63325	−0.100174	−0.0500868	+	0.998745i	$0.515950\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.6332	0.441274
$696$	0	0
$697$	−11.0000	−0.416655
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.68338	0.139119	0.0695596	−	0.997578i	$-0.477841\pi$
$701$	3.68338	0.139119	0.0695596	+	0.997578i	$0.477841\pi$
$702$	0	0
$703$	13.5831	0.512297
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.94987	−0.148550
$708$	0	0
$709$	12.8496	0.482578	0.241289	−	0.970453i	$-0.422430\pi$
$709$	12.8496	0.482578	0.241289	+	0.970453i	$0.422430\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.63325	−0.136066
$714$	0	0
$715$	−10.0000	−0.373979
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.8496	−0.814853	−0.407427	−	0.913238i	$-0.633574\pi$
$719$	−21.8496	−0.814853	−0.407427	+	0.913238i	$0.633574\pi$
$720$	0	0
$721$	−54.0000	−2.01107
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.31662	0.271733
$726$	0	0
$727$	23.6332	0.876509	0.438254	−	0.898851i	$-0.355597\pi$
$727$	23.6332	0.876509	0.438254	+	0.898851i	$0.355597\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	35.2665	1.30438
$732$	0	0
$733$	−15.7335	−0.581130	−0.290565	−	0.956855i	$-0.593843\pi$
$733$	−15.7335	−0.581130	−0.290565	+	0.956855i	$0.593843\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−36.2164	−1.33405
$738$	0	0
$739$	−37.0000	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.7995	1.16661	0.583305	−	0.812253i	$-0.301759\pi$
$743$	31.7995	1.16661	0.583305	+	0.812253i	$0.301759\pi$
$744$	0	0
$745$	4.31662	0.158149
$746$	0	0
$747$	0	0
$748$	0	0
$749$	17.8496	0.652211
$750$	0	0
$751$	−10.8496	−0.395908	−0.197954	−	0.980211i	$-0.563430\pi$
$751$	−10.8496	−0.395908	−0.197954	+	0.980211i	$0.563430\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−14.6332	−0.532558
$756$	0	0
$757$	−23.5330	−0.855321	−0.427661	−	0.903939i	$-0.640662\pi$
$757$	−23.5330	−0.855321	−0.427661	+	0.903939i	$0.640662\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.41688	−0.196362	−0.0981808	−	0.995169i	$-0.531302\pi$
$761$	−5.41688	−0.196362	−0.0981808	+	0.995169i	$0.531302\pi$
$762$	0	0
$763$	−26.8496	−0.972022
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−34.3166	−1.23910
$768$	0	0
$769$	11.5831	0.417698	0.208849	−	0.977948i	$-0.433028\pi$
$769$	11.5831	0.417698	0.208849	+	0.977948i	$0.433028\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	45.1662	1.62452	0.812259	−	0.583298i	$-0.198238\pi$
$773$	45.1662	1.62452	0.812259	+	0.583298i	$0.198238\pi$
$774$	0	0
$775$	3.63325	0.130510
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−27.5831	−0.988268
$780$	0	0
$781$	27.6834	0.990589
$782$	0	0
$783$	0	0
$784$	0	0
$785$	19.6332	0.700741
$786$	0	0
$787$	0.467002	0.0166468	0.00832341	−	0.999965i	$-0.497351\pi$
$787$	0.467002	0.0166468	0.00832341	+	0.999965i	$0.497351\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	26.0501	0.926236
$792$	0	0
$793$	44.5330	1.58141
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.8496	−0.632266	−0.316133	−	0.948715i	$-0.602385\pi$
$797$	−17.8496	−0.632266	−0.316133	+	0.948715i	$0.602385\pi$
$798$	0	0
$799$	−29.6834	−1.05012
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−19.2665	−0.679900
$804$	0	0
$805$	3.00000	0.105736
$806$	0	0
$807$	0	0
$808$	0	0
$809$	51.2164	1.80067	0.900336	−	0.435196i	$-0.143321\pi$
$809$	51.2164	1.80067	0.900336	+	0.435196i	$0.143321\pi$
$810$	0	0
$811$	24.1662	0.848592	0.424296	−	0.905524i	$-0.360522\pi$
$811$	24.1662	0.848592	0.424296	+	0.905524i	$0.360522\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23.2665	−0.814990
$816$	0	0
$817$	88.4327	3.09387
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.73350	−0.0953998	−0.0476999	−	0.998862i	$-0.515189\pi$
$821$	−2.73350	−0.0953998	−0.0476999	+	0.998862i	$0.515189\pi$
$822$	0	0
$823$	26.7335	0.931871	0.465936	−	0.884819i	$-0.345718\pi$
$823$	26.7335	0.931871	0.465936	+	0.884819i	$0.345718\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.5831	0.507105	0.253552	−	0.967322i	$-0.418401\pi$
$827$	14.5831	0.507105	0.253552	+	0.967322i	$0.418401\pi$
$828$	0	0
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.63325	−0.229828
$834$	0	0
$835$	13.5831	0.470063
$836$	0	0
$837$	0	0
$838$	0	0
$839$	19.2665	0.665153	0.332577	−	0.943076i	$-0.392082\pi$
$839$	19.2665	0.665153	0.332577	+	0.943076i	$0.392082\pi$
$840$	0	0
$841$	24.5330	0.845965
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.63325	0.193790
$846$	0	0
$847$	16.8997	0.580682
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.63325	0.0559871
$852$	0	0
$853$	13.3668	0.457669	0.228834	−	0.973465i	$-0.426509\pi$
$853$	13.3668	0.457669	0.228834	+	0.973465i	$0.426509\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.5330	0.974669	0.487334	−	0.873215i	$-0.337969\pi$
$857$	28.5330	0.974669	0.487334	+	0.873215i	$0.337969\pi$
$858$	0	0
$859$	33.6332	1.14755	0.573776	−	0.819012i	$-0.305478\pi$
$859$	33.6332	1.14755	0.573776	+	0.819012i	$0.305478\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.6332	0.498122	0.249061	−	0.968488i	$-0.419878\pi$
$863$	14.6332	0.498122	0.249061	+	0.968488i	$0.419878\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	67.4829	2.28657
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	11.8997	0.401826	0.200913	−	0.979609i	$-0.435609\pi$
$877$	11.8997	0.401826	0.200913	+	0.979609i	$0.435609\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41.5831	1.40097	0.700486	−	0.713667i	$-0.252967\pi$
$881$	41.5831	1.40097	0.700486	+	0.713667i	$0.252967\pi$
$882$	0	0
$883$	−18.3166	−0.616404	−0.308202	−	0.951321i	$-0.599727\pi$
$883$	−18.3166	−0.616404	−0.308202	+	0.951321i	$0.599727\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−46.6332	−1.56579	−0.782896	−	0.622153i	$-0.786258\pi$
$887$	−46.6332	−1.56579	−0.782896	+	0.622153i	$0.786258\pi$
$888$	0	0
$889$	−44.8496	−1.50421
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−74.4327	−2.49080
$894$	0	0
$895$	22.6332	0.756546
$896$	0	0
$897$	0	0
$898$	0	0
$899$	26.5831	0.886597
$900$	0	0
$901$	33.0000	1.09939
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.36675	0.244879
$906$	0	0
$907$	−5.63325	−0.187049	−0.0935245	−	0.995617i	$-0.529813\pi$
$907$	−5.63325	−0.187049	−0.0935245	+	0.995617i	$0.529813\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.7995	1.05356	0.526782	−	0.850000i	$-0.323398\pi$
$911$	31.7995	1.05356	0.526782	+	0.850000i	$0.323398\pi$
$912$	0	0
$913$	−3.05013	−0.100944
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	9.26650	0.305674	0.152837	−	0.988251i	$-0.451159\pi$
$919$	9.26650	0.305674	0.152837	+	0.988251i	$0.451159\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−51.5831	−1.69788
$924$	0	0
$925$	−1.63325	−0.0537009
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.6834	0.350510	0.175255	−	0.984523i	$-0.443925\pi$
$929$	10.6834	0.350510	0.175255	+	0.984523i	$0.443925\pi$
$930$	0	0
$931$	−16.6332	−0.545133
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.68338	0.251273
$936$	0	0
$937$	40.0000	1.30674	0.653372	−	0.757037i	$-0.273354\pi$
$937$	40.0000	1.30674	0.653372	+	0.757037i	$0.273354\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−37.5831	−1.22517	−0.612587	−	0.790403i	$-0.709871\pi$
$941$	−37.5831	−1.22517	−0.612587	+	0.790403i	$0.709871\pi$
$942$	0	0
$943$	−3.31662	−0.108004
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.6332	0.410525	0.205263	−	0.978707i	$-0.434195\pi$
$947$	12.6332	0.410525	0.205263	+	0.978707i	$0.434195\pi$
$948$	0	0
$949$	35.8997	1.16536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	52.5330	1.70171	0.850855	−	0.525400i	$-0.176084\pi$
$953$	52.5330	1.70171	0.850855	+	0.525400i	$0.176084\pi$
$954$	0	0
$955$	13.5831	0.439540
$956$	0	0
$957$	0	0
$958$	0	0
$959$	39.7995	1.28519
$960$	0	0
$961$	−17.7995	−0.574177
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.633250	−0.0203850
$966$	0	0
$967$	18.9499	0.609387	0.304693	−	0.952450i	$-0.401446\pi$
$967$	18.9499	0.609387	0.304693	+	0.952450i	$0.401446\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.63325	0.148688	0.0743440	−	0.997233i	$-0.476314\pi$
$971$	4.63325	0.148688	0.0743440	+	0.997233i	$0.476314\pi$
$972$	0	0
$973$	−34.8997	−1.11883
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.8496	1.46686	0.733430	−	0.679765i	$-0.237918\pi$
$977$	45.8496	1.46686	0.733430	+	0.679765i	$0.237918\pi$
$978$	0	0
$979$	−10.7335	−0.343044
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−54.4829	−1.73773	−0.868867	−	0.495046i	$-0.835151\pi$
$983$	−54.4829	−1.73773	−0.868867	+	0.495046i	$0.835151\pi$
$984$	0	0
$985$	−11.8997	−0.379158
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.6332	0.338118
$990$	0	0
$991$	−14.1662	−0.450006	−0.225003	−	0.974358i	$-0.572239\pi$
$991$	−14.1662	−0.450006	−0.225003	+	0.974358i	$0.572239\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.6332	0.463905
$996$	0	0
$997$	11.3668	0.359989	0.179994	−	0.983668i	$-0.442392\pi$
$997$	11.3668	0.359989	0.179994	+	0.983668i	$0.442392\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bc.1.1		2
3.2	odd	2		2760.2.a.l.1.2	✓	2
12.11	even	2		5520.2.a.bo.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.l.1.2	✓	2	3.2	odd	2
5520.2.a.bo.1.1		2	12.11	even	2
8280.2.a.bc.1.1		2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.00000 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -3.00000 q^{7} -2.31662 q^{11} +4.31662 q^{13} -3.31662 q^{17} -8.31662 q^{19} -1.00000 q^{23} +1.00000 q^{25} +7.31662 q^{29} +3.63325 q^{31} -3.00000 q^{35} -1.63325 q^{37} +3.31662 q^{41} -10.6332 q^{43} +8.94987 q^{47} +2.00000 q^{49} -9.94987 q^{53} -2.31662 q^{55} -7.94987 q^{59} +10.3166 q^{61} +4.31662 q^{65} +15.6332 q^{67} -11.9499 q^{71} +8.31662 q^{73} +6.94987 q^{77} +1.31662 q^{83} -3.31662 q^{85} +4.63325 q^{89} -12.9499 q^{91} -8.31662 q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 6 * q^7	\( 2 q + 2 q^{5} - 6 q^{7} + 2 q^{11} + 2 q^{13} - 10 q^{19} - 2 q^{23} + 2 q^{25} + 8 q^{29} - 6 q^{31} - 6 q^{35} + 10 q^{37} - 8 q^{43} - 2 q^{47} + 4 q^{49} + 2 q^{55} + 4 q^{59} + 14 q^{61} + 2 q^{65} + 18 q^{67} - 4 q^{71} + 10 q^{73} - 6 q^{77} - 4 q^{83} - 4 q^{89} - 6 q^{91} - 10 q^{95} + 24 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 6 * q^7 + 2 * q^11 + 2 * q^13 - 10 * q^19 - 2 * q^23 + 2 * q^25 + 8 * q^29 - 6 * q^31 - 6 * q^35 + 10 * q^37 - 8 * q^43 - 2 * q^47 + 4 * q^49 + 2 * q^55 + 4 * q^59 + 14 * q^61 + 2 * q^65 + 18 * q^67 - 4 * q^71 + 10 * q^73 - 6 * q^77 - 4 * q^83 - 4 * q^89 - 6 * q^91 - 10 * q^95 + 24 * q^97

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