LMFDB - Embedded newform 5220.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, 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("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.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:	$41.6819098551$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1740)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	5220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +2.37228 q^{7} +0.372281 q^{11} -4.37228 q^{13} +6.37228 q^{17} +6.74456 q^{19} +1.00000 q^{25} -1.00000 q^{29} -6.74456 q^{31} +2.37228 q^{35} +2.00000 q^{41} +4.74456 q^{43} +10.3723 q^{47} -1.37228 q^{49} -6.74456 q^{53} +0.372281 q^{55} +12.0000 q^{59} +6.00000 q^{61} -4.37228 q^{65} -14.3723 q^{67} -8.74456 q^{71} +0.883156 q^{77} +2.74456 q^{79} +12.0000 q^{83} +6.37228 q^{85} +17.1168 q^{89} -10.3723 q^{91} +6.74456 q^{95} +13.4891 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - q^{7} - 5 q^{11} - 3 q^{13} + 7 q^{17} + 2 q^{19} + 2 q^{25} - 2 q^{29} - 2 q^{31} - q^{35} + 4 q^{41} - 2 q^{43} + 15 q^{47} + 3 q^{49} - 2 q^{53} - 5 q^{55} + 24 q^{59} + 12 q^{61} - 3 q^{65}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - q^7 - 5 * q^11 - 3 * q^13 + 7 * q^17 + 2 * q^19 + 2 * q^25 - 2 * q^29 - 2 * q^31 - q^35 + 4 * q^41 - 2 * q^43 + 15 * q^47 + 3 * q^49 - 2 * q^53 - 5 * q^55 + 24 * q^59 + 12 * q^61 - 3 * q^65 - 23 * q^67 - 6 * q^71 + 19 * q^77 - 6 * q^79 + 24 * q^83 + 7 * q^85 + 17 * q^89 - 15 * q^91 + 2 * q^95 + 4 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.372281	0.112247	0.0561235	−	0.998424i	$-0.482126\pi$
$11$	0.372281	0.112247	0.0561235	+	0.998424i	$0.482126\pi$
$12$	0	0
$13$	−4.37228	−1.21265	−0.606326	−	0.795216i	$-0.707357\pi$
$13$	−4.37228	−1.21265	−0.606326	+	0.795216i	$0.707357\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.37228	1.54551	0.772753	−	0.634707i	$-0.218879\pi$
$17$	6.37228	1.54551	0.772753	+	0.634707i	$0.218879\pi$
$18$	0	0
$19$	6.74456	1.54731	0.773654	−	0.633608i	$-0.218427\pi$
$19$	6.74456	1.54731	0.773654	+	0.633608i	$0.218427\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.74456	−1.21136	−0.605680	−	0.795709i	$-0.707099\pi$
$31$	−6.74456	−1.21136	−0.605680	+	0.795709i	$0.707099\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.37228	0.400989
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.74456	0.723539	0.361770	−	0.932268i	$-0.382173\pi$
$43$	4.74456	0.723539	0.361770	+	0.932268i	$0.382173\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.3723	1.51295	0.756476	−	0.654021i	$-0.226919\pi$
$47$	10.3723	1.51295	0.756476	+	0.654021i	$0.226919\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.74456	−0.926437	−0.463218	−	0.886244i	$-0.653305\pi$
$53$	−6.74456	−0.926437	−0.463218	+	0.886244i	$0.653305\pi$
$54$	0	0
$55$	0.372281	0.0501984
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.37228	−0.542315
$66$	0	0
$67$	−14.3723	−1.75585	−0.877927	−	0.478795i	$-0.841074\pi$
$67$	−14.3723	−1.75585	−0.877927	+	0.478795i	$0.841074\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.74456	−1.03779	−0.518894	−	0.854838i	$-0.673656\pi$
$71$	−8.74456	−1.03779	−0.518894	+	0.854838i	$0.673656\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.883156	0.100645
$78$	0	0
$79$	2.74456	0.308787	0.154394	−	0.988009i	$-0.450658\pi$
$79$	2.74456	0.308787	0.154394	+	0.988009i	$0.450658\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	6.37228	0.691171
$86$	0	0
$87$	0	0
$88$	0	0
$89$	17.1168	1.81438	0.907191	−	0.420719i	$-0.138222\pi$
$89$	17.1168	1.81438	0.907191	+	0.420719i	$0.138222\pi$
$90$	0	0
$91$	−10.3723	−1.08731
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.74456	0.691978
$96$	0	0
$97$	13.4891	1.36961	0.684807	−	0.728725i	$-0.259887\pi$
$97$	13.4891	1.36961	0.684807	+	0.728725i	$0.259887\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.11684	−0.111130	−0.0555651	−	0.998455i	$-0.517696\pi$
$101$	−1.11684	−0.111130	−0.0555651	+	0.998455i	$0.517696\pi$
$102$	0	0
$103$	−17.4891	−1.72325	−0.861627	−	0.507541i	$-0.830554\pi$
$103$	−17.4891	−1.72325	−0.861627	+	0.507541i	$0.830554\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.74456	0.845369	0.422684	−	0.906277i	$-0.361088\pi$
$107$	8.74456	0.845369	0.422684	+	0.906277i	$0.361088\pi$
$108$	0	0
$109$	−13.8614	−1.32768	−0.663841	−	0.747874i	$-0.731075\pi$
$109$	−13.8614	−1.32768	−0.663841	+	0.747874i	$0.731075\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.11684	−0.669496	−0.334748	−	0.942308i	$-0.608651\pi$
$113$	−7.11684	−0.669496	−0.334748	+	0.942308i	$0.608651\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	15.1168	1.38576
$120$	0	0
$121$	−10.8614	−0.987401
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.7446	−1.48584	−0.742920	−	0.669380i	$-0.766560\pi$
$127$	−16.7446	−1.48584	−0.742920	+	0.669380i	$0.766560\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.3723	1.43045	0.715226	−	0.698893i	$-0.246324\pi$
$131$	16.3723	1.43045	0.715226	+	0.698893i	$0.246324\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.7446	1.43058	0.715292	−	0.698825i	$-0.246294\pi$
$137$	16.7446	1.43058	0.715292	+	0.698825i	$0.246294\pi$
$138$	0	0
$139$	15.1168	1.28219	0.641097	−	0.767460i	$-0.278480\pi$
$139$	15.1168	1.28219	0.641097	+	0.767460i	$0.278480\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.62772	−0.136117
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−13.4891	−1.09773	−0.548865	−	0.835911i	$-0.684940\pi$
$151$	−13.4891	−1.09773	−0.548865	+	0.835911i	$0.684940\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.74456	−0.541736
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.4891	1.36985	0.684927	−	0.728612i	$-0.259834\pi$
$163$	17.4891	1.36985	0.684927	+	0.728612i	$0.259834\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	6.11684	0.470526
$170$	0	0
$171$	0	0
$172$	0	0
$173$	24.2337	1.84245	0.921227	−	0.389026i	$-0.127189\pi$
$173$	24.2337	1.84245	0.921227	+	0.389026i	$0.127189\pi$
$174$	0	0
$175$	2.37228	0.179328
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.74456	−0.653599	−0.326800	−	0.945094i	$-0.605970\pi$
$179$	−8.74456	−0.653599	−0.326800	+	0.945094i	$0.605970\pi$
$180$	0	0
$181$	20.3723	1.51426	0.757130	−	0.653264i	$-0.226601\pi$
$181$	20.3723	1.51426	0.757130	+	0.653264i	$0.226601\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.37228	0.173478
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.7446	−1.06688	−0.533440	−	0.845838i	$-0.679101\pi$
$191$	−14.7446	−1.06688	−0.533440	+	0.845838i	$0.679101\pi$
$192$	0	0
$193$	−4.74456	−0.341521	−0.170761	−	0.985313i	$-0.554622\pi$
$193$	−4.74456	−0.341521	−0.170761	+	0.985313i	$0.554622\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.4891	−1.10355	−0.551777	−	0.833992i	$-0.686050\pi$
$197$	−15.4891	−1.10355	−0.551777	+	0.833992i	$0.686050\pi$
$198$	0	0
$199$	3.86141	0.273728	0.136864	−	0.990590i	$-0.456298\pi$
$199$	3.86141	0.273728	0.136864	+	0.990590i	$0.456298\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.37228	−0.166502
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.51087	0.173681
$210$	0	0
$211$	15.4891	1.06632	0.533158	−	0.846016i	$-0.321005\pi$
$211$	15.4891	1.06632	0.533158	+	0.846016i	$0.321005\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.74456	0.323576
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−27.8614	−1.87416
$222$	0	0
$223$	−3.86141	−0.258579	−0.129289	−	0.991607i	$-0.541270\pi$
$223$	−3.86141	−0.258579	−0.129289	+	0.991607i	$0.541270\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.4891	−0.895305	−0.447652	−	0.894208i	$-0.647740\pi$
$227$	−13.4891	−0.895305	−0.447652	+	0.894208i	$0.647740\pi$
$228$	0	0
$229$	−19.4891	−1.28788	−0.643939	−	0.765077i	$-0.722701\pi$
$229$	−19.4891	−1.28788	−0.643939	+	0.765077i	$0.722701\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	10.3723	0.676613
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.74456	−0.306900	−0.153450	−	0.988156i	$-0.549038\pi$
$239$	−4.74456	−0.306900	−0.153450	+	0.988156i	$0.549038\pi$
$240$	0	0
$241$	−2.88316	−0.185720	−0.0928602	−	0.995679i	$-0.529601\pi$
$241$	−2.88316	−0.185720	−0.0928602	+	0.995679i	$0.529601\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.37228	−0.0876718
$246$	0	0
$247$	−29.4891	−1.87635
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−26.6060	−1.67935	−0.839677	−	0.543086i	$-0.817256\pi$
$251$	−26.6060	−1.67935	−0.839677	+	0.543086i	$0.817256\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.510875	0.0318675	0.0159337	−	0.999873i	$-0.494928\pi$
$257$	0.510875	0.0318675	0.0159337	+	0.999873i	$0.494928\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.4891	0.831775	0.415888	−	0.909416i	$-0.363471\pi$
$263$	13.4891	0.831775	0.415888	+	0.909416i	$0.363471\pi$
$264$	0	0
$265$	−6.74456	−0.414315
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.62772	−0.465070	−0.232535	−	0.972588i	$-0.574702\pi$
$269$	−7.62772	−0.465070	−0.232535	+	0.972588i	$0.574702\pi$
$270$	0	0
$271$	24.9783	1.51732	0.758660	−	0.651486i	$-0.225854\pi$
$271$	24.9783	1.51732	0.758660	+	0.651486i	$0.225854\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.372281	0.0224494
$276$	0	0
$277$	2.88316	0.173232	0.0866160	−	0.996242i	$-0.472395\pi$
$277$	2.88316	0.173232	0.0866160	+	0.996242i	$0.472395\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.23369	0.252561	0.126280	−	0.991995i	$-0.459696\pi$
$281$	4.23369	0.252561	0.126280	+	0.991995i	$0.459696\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.74456	0.280063
$288$	0	0
$289$	23.6060	1.38859
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.62772	0.562457	0.281229	−	0.959641i	$-0.409258\pi$
$293$	9.62772	0.562457	0.281229	+	0.959641i	$0.409258\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	11.2554	0.648753
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	−22.9783	−1.31144	−0.655719	−	0.755005i	$-0.727634\pi$
$307$	−22.9783	−1.31144	−0.655719	+	0.755005i	$0.727634\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.6277	−0.659347	−0.329674	−	0.944095i	$-0.606939\pi$
$311$	−11.6277	−0.659347	−0.329674	+	0.944095i	$0.606939\pi$
$312$	0	0
$313$	19.6277	1.10942	0.554712	−	0.832042i	$-0.312828\pi$
$313$	19.6277	1.10942	0.554712	+	0.832042i	$0.312828\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.37228	−0.133241	−0.0666203	−	0.997778i	$-0.521222\pi$
$317$	−2.37228	−0.133241	−0.0666203	+	0.997778i	$0.521222\pi$
$318$	0	0
$319$	−0.372281	−0.0208438
$320$	0	0
$321$	0	0
$322$	0	0
$323$	42.9783	2.39137
$324$	0	0
$325$	−4.37228	−0.242531
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.6060	1.35657
$330$	0	0
$331$	−7.48913	−0.411640	−0.205820	−	0.978590i	$-0.565986\pi$
$331$	−7.48913	−0.411640	−0.205820	+	0.978590i	$0.565986\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−14.3723	−0.785242
$336$	0	0
$337$	10.2337	0.557465	0.278732	−	0.960369i	$-0.410086\pi$
$337$	10.2337	0.557465	0.278732	+	0.960369i	$0.410086\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.51087	−0.135971
$342$	0	0
$343$	−19.8614	−1.07242
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.9783	1.66300	0.831500	−	0.555525i	$-0.187483\pi$
$347$	30.9783	1.66300	0.831500	+	0.555525i	$0.187483\pi$
$348$	0	0
$349$	32.9783	1.76529	0.882643	−	0.470045i	$-0.155762\pi$
$349$	32.9783	1.76529	0.882643	+	0.470045i	$0.155762\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−11.4891	−0.611504	−0.305752	−	0.952111i	$-0.598908\pi$
$353$	−11.4891	−0.611504	−0.305752	+	0.952111i	$0.598908\pi$
$354$	0	0
$355$	−8.74456	−0.464113
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.7446	−0.778188	−0.389094	−	0.921198i	$-0.627212\pi$
$359$	−14.7446	−0.778188	−0.389094	+	0.921198i	$0.627212\pi$
$360$	0	0
$361$	26.4891	1.39416
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−34.9783	−1.82585	−0.912925	−	0.408128i	$-0.866182\pi$
$367$	−34.9783	−1.82585	−0.912925	+	0.408128i	$0.866182\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−16.0000	−0.830679
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.37228	0.225184
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.9783	−0.560962	−0.280481	−	0.959860i	$-0.590494\pi$
$383$	−10.9783	−0.560962	−0.280481	+	0.959860i	$0.590494\pi$
$384$	0	0
$385$	0.883156	0.0450098
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.60597	0.334936	0.167468	−	0.985878i	$-0.446441\pi$
$389$	6.60597	0.334936	0.167468	+	0.985878i	$0.446441\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.74456	0.138094
$396$	0	0
$397$	24.9783	1.25362	0.626811	−	0.779171i	$-0.284360\pi$
$397$	24.9783	1.25362	0.626811	+	0.779171i	$0.284360\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	33.7228	1.68404	0.842018	−	0.539449i	$-0.181367\pi$
$401$	33.7228	1.68404	0.842018	+	0.539449i	$0.181367\pi$
$402$	0	0
$403$	29.4891	1.46896
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14.7446	0.729072	0.364536	−	0.931189i	$-0.381228\pi$
$409$	14.7446	0.729072	0.364536	+	0.931189i	$0.381228\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	28.4674	1.40079
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.7446	1.01344	0.506719	−	0.862111i	$-0.330858\pi$
$419$	20.7446	1.01344	0.506719	+	0.862111i	$0.330858\pi$
$420$	0	0
$421$	36.2337	1.76592	0.882961	−	0.469446i	$-0.155546\pi$
$421$	36.2337	1.76592	0.882961	+	0.469446i	$0.155546\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.37228	0.309101
$426$	0	0
$427$	14.2337	0.688816
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.4891	1.03509	0.517547	−	0.855655i	$-0.326845\pi$
$431$	21.4891	1.03509	0.517547	+	0.855655i	$0.326845\pi$
$432$	0	0
$433$	13.4891	0.648246	0.324123	−	0.946015i	$-0.394931\pi$
$433$	13.4891	0.648246	0.324123	+	0.946015i	$0.394931\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	4.88316	0.233060	0.116530	−	0.993187i	$-0.462823\pi$
$439$	4.88316	0.233060	0.116530	+	0.993187i	$0.462823\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−33.3505	−1.58453	−0.792266	−	0.610176i	$-0.791099\pi$
$443$	−33.3505	−1.58453	−0.792266	+	0.610176i	$0.791099\pi$
$444$	0	0
$445$	17.1168	0.811416
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.8832	0.702380	0.351190	−	0.936304i	$-0.385777\pi$
$449$	14.8832	0.702380	0.351190	+	0.936304i	$0.385777\pi$
$450$	0	0
$451$	0.744563	0.0350601
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.3723	−0.486260
$456$	0	0
$457$	−36.8397	−1.72329	−0.861643	−	0.507514i	$-0.830564\pi$
$457$	−36.8397	−1.72329	−0.861643	+	0.507514i	$0.830564\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−0.510875	−0.0237938	−0.0118969	−	0.999929i	$-0.503787\pi$
$461$	−0.510875	−0.0237938	−0.0118969	+	0.999929i	$0.503787\pi$
$462$	0	0
$463$	13.6277	0.633334	0.316667	−	0.948537i	$-0.397436\pi$
$463$	13.6277	0.633334	0.316667	+	0.948537i	$0.397436\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	0	0
$469$	−34.0951	−1.57437
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.76631	0.0812151
$474$	0	0
$475$	6.74456	0.309462
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17.7228	−0.809776	−0.404888	−	0.914366i	$-0.632689\pi$
$479$	−17.7228	−0.809776	−0.404888	+	0.914366i	$0.632689\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.4891	0.612510
$486$	0	0
$487$	2.97825	0.134957	0.0674787	−	0.997721i	$-0.478505\pi$
$487$	2.97825	0.134957	0.0674787	+	0.997721i	$0.478505\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.7228	0.980337	0.490168	−	0.871628i	$-0.336935\pi$
$491$	21.7228	0.980337	0.490168	+	0.871628i	$0.336935\pi$
$492$	0	0
$493$	−6.37228	−0.286993
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.7446	−0.930521
$498$	0	0
$499$	−17.3505	−0.776716	−0.388358	−	0.921508i	$-0.626958\pi$
$499$	−17.3505	−0.776716	−0.388358	+	0.921508i	$0.626958\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.88316	0.217729	0.108865	−	0.994057i	$-0.465279\pi$
$503$	4.88316	0.217729	0.108865	+	0.994057i	$0.465279\pi$
$504$	0	0
$505$	−1.11684	−0.0496989
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−0.233688	−0.0103580	−0.00517902	−	0.999987i	$-0.501649\pi$
$509$	−0.233688	−0.0103580	−0.00517902	+	0.999987i	$0.501649\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−17.4891	−0.770663
$516$	0	0
$517$	3.86141	0.169824
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.9783	−1.44480	−0.722402	−	0.691474i	$-0.756962\pi$
$521$	−32.9783	−1.44480	−0.722402	+	0.691474i	$0.756962\pi$
$522$	0	0
$523$	15.8614	0.693571	0.346785	−	0.937944i	$-0.387273\pi$
$523$	15.8614	0.693571	0.346785	+	0.937944i	$0.387273\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.9783	−1.87216
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.74456	−0.378769
$534$	0	0
$535$	8.74456	0.378060
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.510875	−0.0220049
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−13.8614	−0.593757
$546$	0	0
$547$	31.8614	1.36230	0.681148	−	0.732146i	$-0.261481\pi$
$547$	31.8614	1.36230	0.681148	+	0.732146i	$0.261481\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.74456	−0.287328
$552$	0	0
$553$	6.51087	0.276871
$554$	0	0
$555$	0	0
$556$	0	0
$557$	34.7446	1.47217	0.736087	−	0.676887i	$-0.236671\pi$
$557$	34.7446	1.47217	0.736087	+	0.676887i	$0.236671\pi$
$558$	0	0
$559$	−20.7446	−0.877402
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.3505	−0.899818	−0.449909	−	0.893074i	$-0.648544\pi$
$563$	−21.3505	−0.899818	−0.449909	+	0.893074i	$0.648544\pi$
$564$	0	0
$565$	−7.11684	−0.299408
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−28.3723	−1.18943	−0.594714	−	0.803937i	$-0.702735\pi$
$569$	−28.3723	−1.18943	−0.594714	+	0.803937i	$0.702735\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−25.4891	−1.06113	−0.530563	−	0.847645i	$-0.678019\pi$
$577$	−25.4891	−1.06113	−0.530563	+	0.847645i	$0.678019\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	28.4674	1.18103
$582$	0	0
$583$	−2.51087	−0.103990
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.9783	0.948414	0.474207	−	0.880413i	$-0.342735\pi$
$587$	22.9783	0.948414	0.474207	+	0.880413i	$0.342735\pi$
$588$	0	0
$589$	−45.4891	−1.87435
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.2554	0.708596	0.354298	−	0.935133i	$-0.384720\pi$
$593$	17.2554	0.708596	0.354298	+	0.935133i	$0.384720\pi$
$594$	0	0
$595$	15.1168	0.619730
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−30.6060	−1.25053	−0.625263	−	0.780414i	$-0.715008\pi$
$599$	−30.6060	−1.25053	−0.625263	+	0.780414i	$0.715008\pi$
$600$	0	0
$601$	−19.4891	−0.794978	−0.397489	−	0.917607i	$-0.630118\pi$
$601$	−19.4891	−0.794978	−0.397489	+	0.917607i	$0.630118\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.8614	−0.441579
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−45.3505	−1.83469
$612$	0	0
$613$	18.6060	0.751488	0.375744	−	0.926723i	$-0.377387\pi$
$613$	18.6060	0.751488	0.375744	+	0.926723i	$0.377387\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.2554	−0.614161	−0.307080	−	0.951684i	$-0.599352\pi$
$617$	−15.2554	−0.614161	−0.307080	+	0.951684i	$0.599352\pi$
$618$	0	0
$619$	−26.4674	−1.06381	−0.531907	−	0.846803i	$-0.678524\pi$
$619$	−26.4674	−1.06381	−0.531907	+	0.846803i	$0.678524\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	40.6060	1.62684
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−20.1386	−0.801705	−0.400852	−	0.916143i	$-0.631286\pi$
$631$	−20.1386	−0.801705	−0.400852	+	0.916143i	$0.631286\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16.7446	−0.664488
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19.3505	−0.764300	−0.382150	−	0.924100i	$-0.624816\pi$
$641$	−19.3505	−0.764300	−0.382150	+	0.924100i	$0.624816\pi$
$642$	0	0
$643$	0.138593	0.00546559	0.00273279	−	0.999996i	$-0.499130\pi$
$643$	0.138593	0.00546559	0.00273279	+	0.999996i	$0.499130\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.76631	0.383953	0.191977	−	0.981399i	$-0.438510\pi$
$647$	9.76631	0.383953	0.191977	+	0.981399i	$0.438510\pi$
$648$	0	0
$649$	4.46738	0.175360
$650$	0	0
$651$	0	0
$652$	0	0
$653$	31.1168	1.21770	0.608848	−	0.793287i	$-0.291632\pi$
$653$	31.1168	1.21770	0.608848	+	0.793287i	$0.291632\pi$
$654$	0	0
$655$	16.3723	0.639718
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4.37228	−0.170320	−0.0851599	−	0.996367i	$-0.527140\pi$
$659$	−4.37228	−0.170320	−0.0851599	+	0.996367i	$0.527140\pi$
$660$	0	0
$661$	−34.6060	−1.34602	−0.673008	−	0.739635i	$-0.734998\pi$
$661$	−34.6060	−1.34602	−0.673008	+	0.739635i	$0.734998\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.23369	0.0862306
$672$	0	0
$673$	2.13859	0.0824367	0.0412184	−	0.999150i	$-0.486876\pi$
$673$	2.13859	0.0824367	0.0412184	+	0.999150i	$0.486876\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.35053	0.359370	0.179685	−	0.983724i	$-0.442492\pi$
$677$	9.35053	0.359370	0.179685	+	0.983724i	$0.442492\pi$
$678$	0	0
$679$	32.0000	1.22805
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21.4891	−0.822259	−0.411129	−	0.911577i	$-0.634866\pi$
$683$	−21.4891	−0.822259	−0.411129	+	0.911577i	$0.634866\pi$
$684$	0	0
$685$	16.7446	0.639777
$686$	0	0
$687$	0	0
$688$	0	0
$689$	29.4891	1.12345
$690$	0	0
$691$	−28.6060	−1.08822	−0.544111	−	0.839013i	$-0.683133\pi$
$691$	−28.6060	−1.08822	−0.544111	+	0.839013i	$0.683133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.1168	0.573415
$696$	0	0
$697$	12.7446	0.482735
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−39.4891	−1.49148	−0.745742	−	0.666235i	$-0.767905\pi$
$701$	−39.4891	−1.49148	−0.745742	+	0.666235i	$0.767905\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.64947	−0.0996435
$708$	0	0
$709$	−39.4891	−1.48305	−0.741523	−	0.670928i	$-0.765896\pi$
$709$	−39.4891	−1.48305	−0.741523	+	0.670928i	$0.765896\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−1.62772	−0.0608732
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.7446	0.475292	0.237646	−	0.971352i	$-0.423624\pi$
$719$	12.7446	0.475292	0.237646	+	0.971352i	$0.423624\pi$
$720$	0	0
$721$	−41.4891	−1.54514
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−9.48913	−0.351932	−0.175966	−	0.984396i	$-0.556305\pi$
$727$	−9.48913	−0.351932	−0.175966	+	0.984396i	$0.556305\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	30.2337	1.11823
$732$	0	0
$733$	24.7446	0.913961	0.456981	−	0.889477i	$-0.348931\pi$
$733$	24.7446	0.913961	0.456981	+	0.889477i	$0.348931\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.35053	−0.197089
$738$	0	0
$739$	−40.9783	−1.50741	−0.753705	−	0.657213i	$-0.771735\pi$
$739$	−40.9783	−1.50741	−0.753705	+	0.657213i	$0.771735\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.13859	−0.298576	−0.149288	−	0.988794i	$-0.547698\pi$
$743$	−8.13859	−0.298576	−0.149288	+	0.988794i	$0.547698\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20.7446	0.757990
$750$	0	0
$751$	−34.7446	−1.26785	−0.633924	−	0.773396i	$-0.718557\pi$
$751$	−34.7446	−1.26785	−0.633924	+	0.773396i	$0.718557\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−13.4891	−0.490920
$756$	0	0
$757$	9.48913	0.344888	0.172444	−	0.985019i	$-0.444834\pi$
$757$	9.48913	0.344888	0.172444	+	0.985019i	$0.444834\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−49.7228	−1.80245	−0.901225	−	0.433351i	$-0.857331\pi$
$761$	−49.7228	−1.80245	−0.901225	+	0.433351i	$0.857331\pi$
$762$	0	0
$763$	−32.8832	−1.19045
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−52.4674	−1.89449
$768$	0	0
$769$	−0.510875	−0.0184226	−0.00921131	−	0.999958i	$-0.502932\pi$
$769$	−0.510875	−0.0184226	−0.00921131	+	0.999958i	$0.502932\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−28.7446	−1.03387	−0.516935	−	0.856025i	$-0.672927\pi$
$773$	−28.7446	−1.03387	−0.516935	+	0.856025i	$0.672927\pi$
$774$	0	0
$775$	−6.74456	−0.242272
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.4891	0.483298
$780$	0	0
$781$	−3.25544	−0.116489
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.8832	−0.600296
$792$	0	0
$793$	−26.2337	−0.931586
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.2554	0.398688	0.199344	−	0.979930i	$-0.436119\pi$
$797$	11.2554	0.398688	0.199344	+	0.979930i	$0.436119\pi$
$798$	0	0
$799$	66.0951	2.33828
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	28.3723	0.997516	0.498758	−	0.866741i	$-0.333790\pi$
$809$	28.3723	0.997516	0.498758	+	0.866741i	$0.333790\pi$
$810$	0	0
$811$	−16.1386	−0.566703	−0.283351	−	0.959016i	$-0.591446\pi$
$811$	−16.1386	−0.566703	−0.283351	+	0.959016i	$0.591446\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17.4891	0.612617
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11.4891	−0.400973	−0.200487	−	0.979696i	$-0.564252\pi$
$821$	−11.4891	−0.400973	−0.200487	+	0.979696i	$0.564252\pi$
$822$	0	0
$823$	29.2119	1.01826	0.509132	−	0.860688i	$-0.329966\pi$
$823$	29.2119	1.01826	0.509132	+	0.860688i	$0.329966\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.9783	0.938126	0.469063	−	0.883165i	$-0.344592\pi$
$827$	26.9783	0.938126	0.469063	+	0.883165i	$0.344592\pi$
$828$	0	0
$829$	16.2337	0.563819	0.281910	−	0.959441i	$-0.409032\pi$
$829$	16.2337	0.563819	0.281910	+	0.959441i	$0.409032\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.74456	−0.302981
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−6.13859	−0.211928	−0.105964	−	0.994370i	$-0.533793\pi$
$839$	−6.13859	−0.211928	−0.105964	+	0.994370i	$0.533793\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.11684	0.210426
$846$	0	0
$847$	−25.7663	−0.885341
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	18.2337	0.624310	0.312155	−	0.950031i	$-0.398949\pi$
$853$	18.2337	0.624310	0.312155	+	0.950031i	$0.398949\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.4674	1.04075	0.520373	−	0.853939i	$-0.325793\pi$
$857$	30.4674	1.04075	0.520373	+	0.853939i	$0.325793\pi$
$858$	0	0
$859$	11.7663	0.401461	0.200731	−	0.979646i	$-0.435668\pi$
$859$	11.7663	0.401461	0.200731	+	0.979646i	$0.435668\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.2119	−0.858224	−0.429112	−	0.903251i	$-0.641174\pi$
$863$	−25.2119	−0.858224	−0.429112	+	0.903251i	$0.641174\pi$
$864$	0	0
$865$	24.2337	0.823970
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.02175	0.0346605
$870$	0	0
$871$	62.8397	2.12924
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.37228	0.0801977
$876$	0	0
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.62772	0.256984	0.128492	−	0.991711i	$-0.458986\pi$
$881$	7.62772	0.256984	0.128492	+	0.991711i	$0.458986\pi$
$882$	0	0
$883$	−49.9565	−1.68117	−0.840585	−	0.541680i	$-0.817789\pi$
$883$	−49.9565	−1.68117	−0.840585	+	0.541680i	$0.817789\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.88316	−0.163960	−0.0819802	−	0.996634i	$-0.526124\pi$
$887$	−4.88316	−0.163960	−0.0819802	+	0.996634i	$0.526124\pi$
$888$	0	0
$889$	−39.7228	−1.33226
$890$	0	0
$891$	0	0
$892$	0	0
$893$	69.9565	2.34101
$894$	0	0
$895$	−8.74456	−0.292298
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.74456	0.224944
$900$	0	0
$901$	−42.9783	−1.43181
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.3723	0.677198
$906$	0	0
$907$	−12.4674	−0.413972	−0.206986	−	0.978344i	$-0.566366\pi$
$907$	−12.4674	−0.413972	−0.206986	+	0.978344i	$0.566366\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.11684	−0.0370027	−0.0185013	−	0.999829i	$-0.505889\pi$
$911$	−1.11684	−0.0370027	−0.0185013	+	0.999829i	$0.505889\pi$
$912$	0	0
$913$	4.46738	0.147849
$914$	0	0
$915$	0	0
$916$	0	0
$917$	38.8397	1.28260
$918$	0	0
$919$	−34.0951	−1.12469	−0.562347	−	0.826902i	$-0.690101\pi$
$919$	−34.0951	−1.12469	−0.562347	+	0.826902i	$0.690101\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	38.2337	1.25848
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−32.9783	−1.08198	−0.540991	−	0.841029i	$-0.681951\pi$
$929$	−32.9783	−1.08198	−0.540991	+	0.841029i	$0.681951\pi$
$930$	0	0
$931$	−9.25544	−0.303335
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.37228	0.0775819
$936$	0	0
$937$	−46.6060	−1.52255	−0.761275	−	0.648429i	$-0.775426\pi$
$937$	−46.6060	−1.52255	−0.761275	+	0.648429i	$0.775426\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.6060	−1.96943	−0.984715	−	0.174175i	$-0.944274\pi$
$947$	−60.6060	−1.96943	−0.984715	+	0.174175i	$0.944274\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−0.510875	−0.0165489	−0.00827443	−	0.999966i	$-0.502634\pi$
$953$	−0.510875	−0.0165489	−0.00827443	+	0.999966i	$0.502634\pi$
$954$	0	0
$955$	−14.7446	−0.477123
$956$	0	0
$957$	0	0
$958$	0	0
$959$	39.7228	1.28272
$960$	0	0
$961$	14.4891	0.467391
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.74456	−0.152733
$966$	0	0
$967$	24.4674	0.786818	0.393409	−	0.919364i	$-0.371296\pi$
$967$	24.4674	0.786818	0.393409	+	0.919364i	$0.371296\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.25544	0.168655	0.0843275	−	0.996438i	$-0.473126\pi$
$971$	5.25544	0.168655	0.0843275	+	0.996438i	$0.473126\pi$
$972$	0	0
$973$	35.8614	1.14966
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37.7228	−1.20686	−0.603430	−	0.797416i	$-0.706200\pi$
$977$	−37.7228	−1.20686	−0.603430	+	0.797416i	$0.706200\pi$
$978$	0	0
$979$	6.37228	0.203659
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−18.5109	−0.590405	−0.295203	−	0.955435i	$-0.595387\pi$
$983$	−18.5109	−0.590405	−0.295203	+	0.955435i	$0.595387\pi$
$984$	0	0
$985$	−15.4891	−0.493525
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−7.86141	−0.249726	−0.124863	−	0.992174i	$-0.539849\pi$
$991$	−7.86141	−0.249726	−0.124863	+	0.992174i	$0.539849\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.86141	0.122415
$996$	0	0
$997$	37.9565	1.20209	0.601047	−	0.799214i	$-0.294750\pi$
$997$	37.9565	1.20209	0.601047	+	0.799214i	$0.294750\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.a.s.1.2		2
3.2	odd	2		1740.2.a.i.1.2	✓	2
12.11	even	2		6960.2.a.cd.1.1		2
15.2	even	4		8700.2.g.t.349.4		4
15.8	even	4		8700.2.g.t.349.1		4
15.14	odd	2		8700.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1740.2.a.i.1.2	✓	2	3.2	odd	2
5220.2.a.s.1.2		2	1.1	even	1	trivial
6960.2.a.cd.1.1		2	12.11	even	2
8700.2.a.y.1.1		2	15.14	odd	2
8700.2.g.t.349.1		4	15.8	even	4
8700.2.g.t.349.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +2.37228 q^{7} +0.372281 q^{11} -4.37228 q^{13} +6.37228 q^{17} +6.74456 q^{19} +1.00000 q^{25} -1.00000 q^{29} -6.74456 q^{31} +2.37228 q^{35} +2.00000 q^{41} +4.74456 q^{43} +10.3723 q^{47} -1.37228 q^{49} -6.74456 q^{53} +0.372281 q^{55} +12.0000 q^{59} +6.00000 q^{61} -4.37228 q^{65} -14.3723 q^{67} -8.74456 q^{71} +0.883156 q^{77} +2.74456 q^{79} +12.0000 q^{83} +6.37228 q^{85} +17.1168 q^{89} -10.3723 q^{91} +6.74456 q^{95} +13.4891 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - q^{7} - 5 q^{11} - 3 q^{13} + 7 q^{17} + 2 q^{19} + 2 q^{25} - 2 q^{29} - 2 q^{31} - q^{35} + 4 q^{41} - 2 q^{43} + 15 q^{47} + 3 q^{49} - 2 q^{53} - 5 q^{55} + 24 q^{59} + 12 q^{61} - 3 q^{65}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - q^7 - 5 * q^11 - 3 * q^13 + 7 * q^17 + 2 * q^19 + 2 * q^25 - 2 * q^29 - 2 * q^31 - q^35 + 4 * q^41 - 2 * q^43 + 15 * q^47 + 3 * q^49 - 2 * q^53 - 5 * q^55 + 24 * q^59 + 12 * q^61 - 3 * q^65 - 23 * q^67 - 6 * q^71 + 19 * q^77 - 6 * q^79 + 24 * q^83 + 7 * q^85 + 17 * q^89 - 15 * q^91 + 2 * q^95 + 4 * q^97

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