LMFDB - Embedded newform 5712.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5712,2,Mod(1,5712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5712, 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("5712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5712.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:	$45.6105496346$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 357)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{13} +3.00000 q^{15} -1.00000 q^{17} +7.00000 q^{19} +1.00000 q^{21} -1.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} -4.00000 q^{31} -3.00000 q^{33} +3.00000 q^{35} -10.0000 q^{37} -1.00000 q^{39} +3.00000 q^{41} +11.0000 q^{43} -3.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} +1.00000 q^{51} -4.00000 q^{53} -9.00000 q^{55} -7.00000 q^{57} -4.00000 q^{59} +10.0000 q^{61} -1.00000 q^{63} -3.00000 q^{65} +8.00000 q^{67} +1.00000 q^{69} -8.00000 q^{71} -2.00000 q^{73} -4.00000 q^{75} -3.00000 q^{77} -16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +3.00000 q^{85} +10.0000 q^{87} -8.00000 q^{89} -1.00000 q^{91} +4.00000 q^{93} -21.0000 q^{95} -4.00000 q^{97} +3.00000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−9.00000	−1.21356
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	10.0000	1.07211
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−21.0000	−2.15455
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−3.00000	−0.295599	−0.147799	−	0.989017i	$-0.547219\pi$
$103$	−3.00000	−0.295599	−0.147799	+	0.989017i	$0.547219\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	19.0000	1.78737	0.893685	−	0.448695i	$-0.148111\pi$
$113$	19.0000	1.78737	0.893685	+	0.448695i	$0.148111\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−3.00000	−0.270501
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−1.00000	−0.0873704	−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000	−0.0873704	−0.0436852	+	0.999045i	$0.513910\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	0	0
$135$	3.00000	0.258199
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	3.00000	0.250873
$144$	0	0
$145$	30.0000	2.49136
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	24.0000	1.95309	0.976546	−	0.215308i	$-0.0690756\pi$
$151$	24.0000	1.95309	0.976546	+	0.215308i	$0.0690756\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	0	0
$165$	9.00000	0.700649
$166$	0	0
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	7.00000	0.535303
$172$	0	0
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	30.0000	2.20564
$186$	0	0
$187$	−3.00000	−0.219382
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−2.00000	−0.144715	−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000	−0.144715	−0.0723575	+	0.997379i	$0.523052\pi$
$192$	0	0
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	0	0
$195$	3.00000	0.214834
$196$	0	0
$197$	11.0000	0.783718	0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000	0.783718	0.391859	+	0.920025i	$0.371832\pi$
$198$	0	0
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	10.0000	0.701862
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	21.0000	1.45260
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	−33.0000	−2.25058
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	23.0000	1.52656	0.763282	−	0.646066i	$-0.223587\pi$
$227$	23.0000	1.52656	0.763282	+	0.646066i	$0.223587\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	0	0
$237$	16.0000	1.03931
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	7.00000	0.445399
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	−10.0000	−0.618984
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	8.00000	0.489592
$268$	0	0
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	0	0
$285$	21.0000	1.24393
$286$	0	0
$287$	−3.00000	−0.177084
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	4.00000	0.234484
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	−11.0000	−0.634029
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−30.0000	−1.71780
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	3.00000	0.170664
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	−30.0000	−1.67968
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	−7.00000	−0.389490
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−24.0000	−1.31126
$336$	0	0
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	0	0
$339$	−19.0000	−1.03194
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.00000	−0.161515
$346$	0	0
$347$	36.0000	1.93258	0.966291	−	0.257454i	$-0.0828835\pi$
$347$	36.0000	1.93258	0.966291	+	0.257454i	$0.0828835\pi$
$348$	0	0
$349$	35.0000	1.87351	0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000	1.87351	0.936754	+	0.349990i	$0.113815\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	28.0000	1.49029	0.745145	−	0.666903i	$-0.232380\pi$
$353$	28.0000	1.49029	0.745145	+	0.666903i	$0.232380\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	0	0
$369$	3.00000	0.156174
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	−10.0000	−0.515026
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−1.00000	−0.0512316
$382$	0	0
$383$	−2.00000	−0.102195	−0.0510976	−	0.998694i	$-0.516272\pi$
$383$	−2.00000	−0.102195	−0.0510976	+	0.998694i	$0.516272\pi$
$384$	0	0
$385$	9.00000	0.458682
$386$	0	0
$387$	11.0000	0.559161
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	1.00000	0.0505722
$392$	0	0
$393$	1.00000	0.0504433
$394$	0	0
$395$	48.0000	2.41514
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	0	0
$399$	7.00000	0.350438
$400$	0	0
$401$	−39.0000	−1.94757	−0.973784	−	0.227477i	$-0.926952\pi$
$401$	−39.0000	−1.94757	−0.973784	+	0.227477i	$0.926952\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	−30.0000	−1.48704
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	18.0000	0.883585
$416$	0	0
$417$	2.00000	0.0979404
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	27.0000	1.31590	0.657950	−	0.753062i	$-0.271424\pi$
$421$	27.0000	1.31590	0.657950	+	0.753062i	$0.271424\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	−3.00000	−0.144841
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	9.00000	0.432512	0.216256	−	0.976337i	$-0.430615\pi$
$433$	9.00000	0.432512	0.216256	+	0.976337i	$0.430615\pi$
$434$	0	0
$435$	−30.0000	−1.43839
$436$	0	0
$437$	−7.00000	−0.334855
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	0	0
$453$	−24.0000	−1.12762
$454$	0	0
$455$	3.00000	0.140642
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	−4.00000	−0.186299	−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000	−0.186299	−0.0931493	+	0.995652i	$0.529693\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−17.0000	−0.783319
$472$	0	0
$473$	33.0000	1.51734
$474$	0	0
$475$	28.0000	1.28473
$476$	0	0
$477$	−4.00000	−0.183147
$478$	0	0
$479$	−35.0000	−1.59919	−0.799595	−	0.600539i	$-0.794953\pi$
$479$	−35.0000	−1.59919	−0.799595	+	0.600539i	$0.794953\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	12.0000	0.544892
$486$	0	0
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	−26.0000	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$492$	0	0
$493$	10.0000	0.450377
$494$	0	0
$495$	−9.00000	−0.404520
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	−21.0000	−0.938211
$502$	0	0
$503$	35.0000	1.56057	0.780286	−	0.625422i	$-0.215073\pi$
$503$	35.0000	1.56057	0.780286	+	0.625422i	$0.215073\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−32.0000	−1.41838	−0.709188	−	0.705020i	$-0.750938\pi$
$509$	−32.0000	−1.41838	−0.709188	+	0.705020i	$0.750938\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	9.00000	0.396587
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	−1.00000	−0.0438951
$520$	0	0
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	3.00000	0.129944
$534$	0	0
$535$	27.0000	1.16731
$536$	0	0
$537$	2.00000	0.0863064
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	40.0000	1.71973	0.859867	−	0.510518i	$-0.170546\pi$
$541$	40.0000	1.71973	0.859867	+	0.510518i	$0.170546\pi$
$542$	0	0
$543$	−18.0000	−0.772454
$544$	0	0
$545$	18.0000	0.771035
$546$	0	0
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−70.0000	−2.98210
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	−30.0000	−1.27343
$556$	0	0
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	11.0000	0.465250
$560$	0	0
$561$	3.00000	0.126660
$562$	0	0
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	−57.0000	−2.39801
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−40.0000	−1.67689	−0.838444	−	0.544988i	$-0.816534\pi$
$569$	−40.0000	−1.67689	−0.838444	+	0.544988i	$0.816534\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	2.00000	0.0835512
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	0	0
$579$	−8.00000	−0.332469
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	−12.0000	−0.496989
$584$	0	0
$585$	−3.00000	−0.124035
$586$	0	0
$587$	−38.0000	−1.56843	−0.784214	−	0.620491i	$-0.786934\pi$
$587$	−38.0000	−1.56843	−0.784214	+	0.620491i	$0.786934\pi$
$588$	0	0
$589$	−28.0000	−1.15372
$590$	0	0
$591$	−11.0000	−0.452480
$592$	0	0
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	−6.00000	−0.245564
$598$	0	0
$599$	38.0000	1.55264	0.776319	−	0.630340i	$-0.217085\pi$
$599$	38.0000	1.55264	0.776319	+	0.630340i	$0.217085\pi$
$600$	0	0
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	6.00000	0.243935
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	−10.0000	−0.405220
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	0	0
$615$	9.00000	0.362915
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	8.00000	0.320513
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−21.0000	−0.838659
$628$	0	0
$629$	10.0000	0.398726
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	−3.00000	−0.119051
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	33.0000	1.29937
$646$	0	0
$647$	46.0000	1.80845	0.904223	−	0.427060i	$-0.140451\pi$
$647$	46.0000	1.80845	0.904223	+	0.427060i	$0.140451\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	31.0000	1.21312	0.606562	−	0.795036i	$-0.292548\pi$
$653$	31.0000	1.21312	0.606562	+	0.795036i	$0.292548\pi$
$654$	0	0
$655$	3.00000	0.117220
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	0	0
$663$	1.00000	0.0388368
$664$	0	0
$665$	21.0000	0.814345
$666$	0	0
$667$	10.0000	0.387202
$668$	0	0
$669$	1.00000	0.0386622
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−9.00000	−0.345898	−0.172949	−	0.984931i	$-0.555330\pi$
$677$	−9.00000	−0.345898	−0.172949	+	0.984931i	$0.555330\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	−23.0000	−0.881362
$682$	0	0
$683$	5.00000	0.191320	0.0956598	−	0.995414i	$-0.469504\pi$
$683$	5.00000	0.191320	0.0956598	+	0.995414i	$0.469504\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	0	0
$687$	−26.0000	−0.991962
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	6.00000	0.227593
$696$	0	0
$697$	−3.00000	−0.113633
$698$	0	0
$699$	3.00000	0.113470
$700$	0	0
$701$	28.0000	1.05755	0.528773	−	0.848763i	$-0.322652\pi$
$701$	28.0000	1.05755	0.528773	+	0.848763i	$0.322652\pi$
$702$	0	0
$703$	−70.0000	−2.64010
$704$	0	0
$705$	24.0000	0.903892
$706$	0	0
$707$	−6.00000	−0.225653
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	4.00000	0.149801
$714$	0	0
$715$	−9.00000	−0.336581
$716$	0	0
$717$	−6.00000	−0.224074
$718$	0	0
$719$	43.0000	1.60363	0.801815	−	0.597573i	$-0.203868\pi$
$719$	43.0000	1.60363	0.801815	+	0.597573i	$0.203868\pi$
$720$	0	0
$721$	3.00000	0.111726
$722$	0	0
$723$	20.0000	0.743808
$724$	0	0
$725$	−40.0000	−1.48556
$726$	0	0
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.0000	−0.406850
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	−35.0000	−1.28750	−0.643748	−	0.765238i	$-0.722621\pi$
$739$	−35.0000	−1.28750	−0.643748	+	0.765238i	$0.722621\pi$
$740$	0	0
$741$	−7.00000	−0.257151
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−6.00000	−0.219529
$748$	0	0
$749$	9.00000	0.328853
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	−10.0000	−0.364420
$754$	0	0
$755$	−72.0000	−2.62035
$756$	0	0
$757$	−47.0000	−1.70824	−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000	−1.70824	−0.854122	+	0.520073i	$0.825905\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−49.0000	−1.76699	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−49.0000	−1.76699	−0.883493	+	0.468445i	$0.844814\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	−10.0000	−0.358748
$778$	0	0
$779$	21.0000	0.752403
$780$	0	0
$781$	−24.0000	−0.858788
$782$	0	0
$783$	10.0000	0.357371
$784$	0	0
$785$	−51.0000	−1.82027
$786$	0	0
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	−19.0000	−0.675562
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	−12.0000	−0.425596
$796$	0	0
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	−3.00000	−0.105736
$806$	0	0
$807$	−15.0000	−0.528025
$808$	0	0
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	0	0
$813$	13.0000	0.455930
$814$	0	0
$815$	−42.0000	−1.47120
$816$	0	0
$817$	77.0000	2.69389
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−1.00000	−0.0349002	−0.0174501	−	0.999848i	$-0.505555\pi$
$821$	−1.00000	−0.0349002	−0.0174501	+	0.999848i	$0.505555\pi$
$822$	0	0
$823$	46.0000	1.60346	0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000	1.60346	0.801730	+	0.597687i	$0.203913\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	33.0000	1.14752	0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000	1.14752	0.573761	+	0.819023i	$0.305484\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−63.0000	−2.18020
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	23.0000	0.794048	0.397024	−	0.917808i	$-0.370043\pi$
$839$	23.0000	0.794048	0.397024	+	0.917808i	$0.370043\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	16.0000	0.551069
$844$	0	0
$845$	36.0000	1.23844
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	10.0000	0.342796
$852$	0	0
$853$	−32.0000	−1.09566	−0.547830	−	0.836590i	$-0.684546\pi$
$853$	−32.0000	−1.09566	−0.547830	+	0.836590i	$0.684546\pi$
$854$	0	0
$855$	−21.0000	−0.718185
$856$	0	0
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	48.0000	1.63774	0.818869	−	0.573980i	$-0.194601\pi$
$859$	48.0000	1.63774	0.818869	+	0.573980i	$0.194601\pi$
$860$	0	0
$861$	3.00000	0.102240
$862$	0	0
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	0	0
$865$	−3.00000	−0.102003
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−48.0000	−1.62829
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	−7.00000	−0.235569	−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−7.00000	−0.235569	−0.117784	+	0.993039i	$0.537579\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	−45.0000	−1.51095	−0.755476	−	0.655176i	$-0.772594\pi$
$887$	−45.0000	−1.51095	−0.755476	+	0.655176i	$0.772594\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	56.0000	1.87397
$894$	0	0
$895$	6.00000	0.200558
$896$	0	0
$897$	1.00000	0.0333890
$898$	0	0
$899$	40.0000	1.33407
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	11.0000	0.366057
$904$	0	0
$905$	−54.0000	−1.79502
$906$	0	0
$907$	38.0000	1.26177	0.630885	−	0.775877i	$-0.282692\pi$
$907$	38.0000	1.26177	0.630885	+	0.775877i	$0.282692\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−33.0000	−1.09334	−0.546669	−	0.837349i	$-0.684105\pi$
$911$	−33.0000	−1.09334	−0.546669	+	0.837349i	$0.684105\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	0	0
$915$	30.0000	0.991769
$916$	0	0
$917$	1.00000	0.0330229
$918$	0	0
$919$	25.0000	0.824674	0.412337	−	0.911031i	$-0.364713\pi$
$919$	25.0000	0.824674	0.412337	+	0.911031i	$0.364713\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−40.0000	−1.31519
$926$	0	0
$927$	−3.00000	−0.0985329
$928$	0	0
$929$	57.0000	1.87011	0.935055	−	0.354504i	$-0.115350\pi$
$929$	57.0000	1.87011	0.935055	+	0.354504i	$0.115350\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	0	0
$933$	32.0000	1.04763
$934$	0	0
$935$	9.00000	0.294331
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	−3.00000	−0.0976934
$944$	0	0
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	14.0000	0.453981
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	6.00000	0.194155
$956$	0	0
$957$	30.0000	0.969762
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−9.00000	−0.290021
$964$	0	0
$965$	−24.0000	−0.772587
$966$	0	0
$967$	23.0000	0.739630	0.369815	−	0.929105i	$-0.379421\pi$
$967$	23.0000	0.739630	0.369815	+	0.929105i	$0.379421\pi$
$968$	0	0
$969$	7.00000	0.224872
$970$	0	0
$971$	−46.0000	−1.47621	−0.738105	−	0.674686i	$-0.764279\pi$
$971$	−46.0000	−1.47621	−0.738105	+	0.674686i	$0.764279\pi$
$972$	0	0
$973$	2.00000	0.0641171
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	43.0000	1.37149	0.685744	−	0.727843i	$-0.259477\pi$
$983$	43.0000	1.37149	0.685744	+	0.727843i	$0.259477\pi$
$984$	0	0
$985$	−33.0000	−1.05147
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	−11.0000	−0.349780
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−13.0000	−0.412543
$994$	0	0
$995$	−18.0000	−0.570638
$996$	0	0
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5712.2.a.c.1.1		1
4.3	odd	2		357.2.a.a.1.1	✓	1
12.11	even	2		1071.2.a.d.1.1		1
20.19	odd	2		8925.2.a.z.1.1		1
28.27	even	2		2499.2.a.a.1.1		1
68.67	odd	2		6069.2.a.a.1.1		1
84.83	odd	2		7497.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.a.a.1.1	✓	1	4.3	odd	2
1071.2.a.d.1.1		1	12.11	even	2
2499.2.a.a.1.1		1	28.27	even	2
5712.2.a.c.1.1		1	1.1	even	1	trivial
6069.2.a.a.1.1		1	68.67	odd	2
7497.2.a.o.1.1		1	84.83	odd	2
8925.2.a.z.1.1		1	20.19	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:	\( N \)	\(=\)	\( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5712.a (trivial)

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