LMFDB - Embedded newform 5712.2.a.bb.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 714)
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)

$f(q)$

$=$

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

$q+1.00000 q^{3} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +3.00000 q^{15} -1.00000 q^{17} -6.00000 q^{19} +1.00000 q^{21} +2.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -1.00000 q^{33} +3.00000 q^{35} +5.00000 q^{37} +1.00000 q^{39} +4.00000 q^{41} +9.00000 q^{43} +3.00000 q^{45} +1.00000 q^{49} -1.00000 q^{51} +11.0000 q^{53} -3.00000 q^{55} -6.00000 q^{57} -4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{63} +3.00000 q^{65} +11.0000 q^{67} +2.00000 q^{69} +12.0000 q^{71} -5.00000 q^{73} +4.00000 q^{75} -1.00000 q^{77} +15.0000 q^{79} +1.00000 q^{81} -1.00000 q^{83} -3.00000 q^{85} -2.00000 q^{87} +9.00000 q^{89} +1.00000 q^{91} -18.0000 q^{95} -9.00000 q^{97} -1.00000 q^{99} +O(q^{100})$

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	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	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
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	9.00000	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	−6.00000	−0.794719
$58$	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
$61$	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
$63$	1.00000	0.125988
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−18.0000	−1.84676
$96$	0	0
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	5.00000	0.474579
$112$	0	0
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	0	0
$129$	9.00000	0.792406
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	3.00000	0.258199
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	11.0000	0.872357
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	15.0000	1.10282
$186$	0	0
$187$	1.00000	0.0731272
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	0	0
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	0	0
$195$	3.00000	0.214834
$196$	0	0
$197$	20.0000	1.42494	0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000	1.42494	0.712470	+	0.701702i	$0.247576\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	11.0000	0.775880
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	27.0000	1.84138
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5.00000	−0.337869
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−17.0000	−1.12339	−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000	−1.12339	−0.561696	+	0.827344i	$0.689851\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	15.0000	0.974355
$238$	0	0
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	−1.00000	−0.0633724
$250$	0	0
$251$	−23.0000	−1.45175	−0.725874	−	0.687828i	$-0.758564\pi$
$251$	−23.0000	−1.45175	−0.725874	+	0.687828i	$0.758564\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	15.0000	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$257$	15.0000	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$258$	0	0
$259$	5.00000	0.310685
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−31.0000	−1.91154	−0.955771	−	0.294112i	$-0.904976\pi$
$263$	−31.0000	−1.91154	−0.955771	+	0.294112i	$0.904976\pi$
$264$	0	0
$265$	33.0000	2.02717
$266$	0	0
$267$	9.00000	0.550791
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	0	0
$285$	−18.0000	−1.06623
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−9.00000	−0.527589
$292$	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
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	9.00000	0.518751
$302$	0	0
$303$	8.00000	0.459588
$304$	0	0
$305$	18.0000	1.03068
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	7.00000	0.398216
$310$	0	0
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	0	0
$313$	−30.0000	−1.69570	−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000	−1.69570	−0.847850	+	0.530236i	$0.822103\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	6.00000	0.333849
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	0	0
$333$	5.00000	0.273998
$334$	0	0
$335$	33.0000	1.80298
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−9.00000	−0.488813
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	−27.0000	−1.44944	−0.724718	−	0.689046i	$-0.758030\pi$
$347$	−27.0000	−1.44944	−0.724718	+	0.689046i	$0.758030\pi$
$348$	0	0
$349$	−13.0000	−0.695874	−0.347937	−	0.937518i	$-0.613118\pi$
$349$	−13.0000	−0.695874	−0.347937	+	0.937518i	$0.613118\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	5.00000	0.266123	0.133062	−	0.991108i	$-0.457519\pi$
$353$	5.00000	0.266123	0.133062	+	0.991108i	$0.457519\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−10.0000	−0.524864
$364$	0	0
$365$	−15.0000	−0.785136
$366$	0	0
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	11.0000	0.571092
$372$	0	0
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	10.0000	0.512316
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	9.00000	0.457496
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	10.0000	0.504433
$394$	0	0
$395$	45.0000	2.26420
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.00000	0.149071
$406$	0	0
$407$	−5.00000	−0.247841
$408$	0	0
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−3.00000	−0.147264
$416$	0	0
$417$	1.00000	0.0489702
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	0	0
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	−12.0000	−0.574038
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	27.0000	1.27992
$446$	0	0
$447$	−9.00000	−0.425685
$448$	0	0
$449$	−1.00000	−0.0471929	−0.0235965	−	0.999722i	$-0.507512\pi$
$449$	−1.00000	−0.0471929	−0.0235965	+	0.999722i	$0.507512\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	3.00000	0.140642
$456$	0	0
$457$	−25.0000	−1.16945	−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000	−1.16945	−0.584725	+	0.811231i	$0.698798\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.0000	1.48078	0.740392	−	0.672176i	$-0.234640\pi$
$467$	32.0000	1.48078	0.740392	+	0.672176i	$0.234640\pi$
$468$	0	0
$469$	11.0000	0.507933
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	−9.00000	−0.413820
$474$	0	0
$475$	−24.0000	−1.10120
$476$	0	0
$477$	11.0000	0.503655
$478$	0	0
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	0	0
$481$	5.00000	0.227980
$482$	0	0
$483$	2.00000	0.0910032
$484$	0	0
$485$	−27.0000	−1.22601
$486$	0	0
$487$	−23.0000	−1.04223	−0.521115	−	0.853487i	$-0.674484\pi$
$487$	−23.0000	−1.04223	−0.521115	+	0.853487i	$0.674484\pi$
$488$	0	0
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	−3.00000	−0.134840
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	0	0
$501$	3.00000	0.134030
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	16.0000	0.709188	0.354594	−	0.935020i	$-0.384619\pi$
$509$	16.0000	0.709188	0.354594	+	0.935020i	$0.384619\pi$
$510$	0	0
$511$	−5.00000	−0.221187
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	21.0000	0.925371
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−16.0000	−0.700973	−0.350486	−	0.936568i	$-0.613984\pi$
$521$	−16.0000	−0.700973	−0.350486	+	0.936568i	$0.613984\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	4.00000	0.173259
$534$	0	0
$535$	24.0000	1.03761
$536$	0	0
$537$	−18.0000	−0.776757
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	15.0000	0.637865
$554$	0	0
$555$	15.0000	0.636715
$556$	0	0
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	9.00000	0.380659
$560$	0	0
$561$	1.00000	0.0422200
$562$	0	0
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	0	0
$565$	−27.0000	−1.13590
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	−15.0000	−0.626634
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	0	0
$579$	−8.00000	−0.332469
$580$	0	0
$581$	−1.00000	−0.0414870
$582$	0	0
$583$	−11.0000	−0.455573
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	23.0000	0.949312	0.474656	−	0.880172i	$-0.342573\pi$
$587$	23.0000	0.949312	0.474656	+	0.880172i	$0.342573\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	20.0000	0.822690
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	11.0000	0.447955
$604$	0	0
$605$	−30.0000	−1.21967
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	43.0000	1.73111	0.865557	−	0.500810i	$-0.166964\pi$
$617$	43.0000	1.73111	0.865557	+	0.500810i	$0.166964\pi$
$618$	0	0
$619$	19.0000	0.763674	0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000	0.763674	0.381837	+	0.924230i	$0.375291\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	9.00000	0.360577
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	6.00000	0.239617
$628$	0	0
$629$	−5.00000	−0.199363
$630$	0	0
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	0	0
$633$	10.0000	0.397464
$634$	0	0
$635$	30.0000	1.19051
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	7.00000	0.276483	0.138242	−	0.990399i	$-0.455855\pi$
$641$	7.00000	0.276483	0.138242	+	0.990399i	$0.455855\pi$
$642$	0	0
$643$	−11.0000	−0.433798	−0.216899	−	0.976194i	$-0.569594\pi$
$643$	−11.0000	−0.433798	−0.216899	+	0.976194i	$0.569594\pi$
$644$	0	0
$645$	27.0000	1.06312
$646$	0	0
$647$	−4.00000	−0.157256	−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000	−0.157256	−0.0786281	+	0.996904i	$0.525054\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	30.0000	1.17220
$656$	0	0
$657$	−5.00000	−0.195069
$658$	0	0
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	−1.00000	−0.0388368
$664$	0	0
$665$	−18.0000	−0.698010
$666$	0	0
$667$	−4.00000	−0.154881
$668$	0	0
$669$	−24.0000	−0.927894
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	−48.0000	−1.85026	−0.925132	−	0.379646i	$-0.876046\pi$
$673$	−48.0000	−1.85026	−0.925132	+	0.379646i	$0.876046\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	0	0
$679$	−9.00000	−0.345388
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−17.0000	−0.648590
$688$	0	0
$689$	11.0000	0.419067
$690$	0	0
$691$	−21.0000	−0.798878	−0.399439	−	0.916760i	$-0.630795\pi$
$691$	−21.0000	−0.798878	−0.399439	+	0.916760i	$0.630795\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	3.00000	0.113796
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	3.00000	0.113470
$700$	0	0
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	−30.0000	−1.13147
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.00000	0.300871
$708$	0	0
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	0	0
$711$	15.0000	0.562544
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−3.00000	−0.112194
$716$	0	0
$717$	15.0000	0.560185
$718$	0	0
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	7.00000	0.260694
$722$	0	0
$723$	22.0000	0.818189
$724$	0	0
$725$	−8.00000	−0.297113
$726$	0	0
$727$	−17.0000	−0.630495	−0.315248	−	0.949009i	$-0.602088\pi$
$727$	−17.0000	−0.630495	−0.315248	+	0.949009i	$0.602088\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.00000	−0.332877
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	−11.0000	−0.405190
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	0	0
$745$	−27.0000	−0.989203
$746$	0	0
$747$	−1.00000	−0.0365881
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	0	0
$753$	−23.0000	−0.838167
$754$	0	0
$755$	24.0000	0.873449
$756$	0	0
$757$	36.0000	1.30844	0.654221	−	0.756303i	$-0.272997\pi$
$757$	36.0000	1.30844	0.654221	+	0.756303i	$0.272997\pi$
$758$	0	0
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	43.0000	1.55875	0.779374	−	0.626559i	$-0.215537\pi$
$761$	43.0000	1.55875	0.779374	+	0.626559i	$0.215537\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	−3.00000	−0.108465
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	15.0000	0.540212
$772$	0	0
$773$	−32.0000	−1.15096	−0.575480	−	0.817816i	$-0.695185\pi$
$773$	−32.0000	−1.15096	−0.575480	+	0.817816i	$0.695185\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.00000	0.179374
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−30.0000	−1.07075
$786$	0	0
$787$	11.0000	0.392108	0.196054	−	0.980593i	$-0.437187\pi$
$787$	11.0000	0.392108	0.196054	+	0.980593i	$0.437187\pi$
$788$	0	0
$789$	−31.0000	−1.10363
$790$	0	0
$791$	−9.00000	−0.320003
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	33.0000	1.17039
$796$	0	0
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	9.00000	0.317999
$802$	0	0
$803$	5.00000	0.176446
$804$	0	0
$805$	6.00000	0.211472
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	29.0000	1.01833	0.509164	−	0.860670i	$-0.329955\pi$
$811$	29.0000	1.01833	0.509164	+	0.860670i	$0.329955\pi$
$812$	0	0
$813$	−15.0000	−0.526073
$814$	0	0
$815$	−30.0000	−1.05085
$816$	0	0
$817$	−54.0000	−1.88922
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	−9.00000	−0.313720	−0.156860	−	0.987621i	$-0.550137\pi$
$823$	−9.00000	−0.313720	−0.156860	+	0.987621i	$0.550137\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	11.0000	0.382507	0.191254	−	0.981541i	$-0.438745\pi$
$827$	11.0000	0.382507	0.191254	+	0.981541i	$0.438745\pi$
$828$	0	0
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	−30.0000	−1.04069
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	9.00000	0.311458
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−11.0000	−0.379762	−0.189881	−	0.981807i	$-0.560810\pi$
$839$	−11.0000	−0.379762	−0.189881	+	0.981807i	$0.560810\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	26.0000	0.895488
$844$	0	0
$845$	−36.0000	−1.23844
$846$	0	0
$847$	−10.0000	−0.343604
$848$	0	0
$849$	21.0000	0.720718
$850$	0	0
$851$	10.0000	0.342796
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	−18.0000	−0.615587
$856$	0	0
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	37.0000	1.25949	0.629747	−	0.776800i	$-0.283158\pi$
$863$	37.0000	1.25949	0.629747	+	0.776800i	$0.283158\pi$
$864$	0	0
$865$	−54.0000	−1.83606
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−15.0000	−0.508840
$870$	0	0
$871$	11.0000	0.372721
$872$	0	0
$873$	−9.00000	−0.304604
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−22.0000	−0.741199	−0.370599	−	0.928793i	$-0.620848\pi$
$881$	−22.0000	−0.741199	−0.370599	+	0.928793i	$0.620848\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	−51.0000	−1.71241	−0.856206	−	0.516634i	$-0.827185\pi$
$887$	−51.0000	−1.71241	−0.856206	+	0.516634i	$0.827185\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−54.0000	−1.80502
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−11.0000	−0.366463
$902$	0	0
$903$	9.00000	0.299501
$904$	0	0
$905$	36.0000	1.19668
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	0	0
$909$	8.00000	0.265343
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	1.00000	0.0330952
$914$	0	0
$915$	18.0000	0.595062
$916$	0	0
$917$	10.0000	0.330229
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	0	0
$923$	12.0000	0.394985
$924$	0	0
$925$	20.0000	0.657596
$926$	0	0
$927$	7.00000	0.229910
$928$	0	0
$929$	28.0000	0.918650	0.459325	−	0.888268i	$-0.348091\pi$
$929$	28.0000	0.918650	0.459325	+	0.888268i	$0.348091\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	3.00000	0.0982156
$934$	0	0
$935$	3.00000	0.0981105
$936$	0	0
$937$	−32.0000	−1.04539	−0.522697	−	0.852518i	$-0.675074\pi$
$937$	−32.0000	−1.04539	−0.522697	+	0.852518i	$0.675074\pi$
$938$	0	0
$939$	−30.0000	−0.979013
$940$	0	0
$941$	49.0000	1.59735	0.798677	−	0.601760i	$-0.205534\pi$
$941$	49.0000	1.59735	0.798677	+	0.601760i	$0.205534\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	−5.00000	−0.162307
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	−45.0000	−1.45617
$956$	0	0
$957$	2.00000	0.0646508
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	8.00000	0.257796
$964$	0	0
$965$	−24.0000	−0.772587
$966$	0	0
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	−53.0000	−1.70085	−0.850425	−	0.526096i	$-0.823655\pi$
$971$	−53.0000	−1.70085	−0.850425	+	0.526096i	$0.823655\pi$
$972$	0	0
$973$	1.00000	0.0320585
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	8.00000	0.255943	0.127971	−	0.991778i	$-0.459153\pi$
$977$	8.00000	0.255943	0.127971	+	0.991778i	$0.459153\pi$
$978$	0	0
$979$	−9.00000	−0.287641
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	60.0000	1.91176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.0000	0.572367
$990$	0	0
$991$	−9.00000	−0.285894	−0.142947	−	0.989730i	$-0.545658\pi$
$991$	−9.00000	−0.285894	−0.142947	+	0.989730i	$0.545658\pi$
$992$	0	0
$993$	−1.00000	−0.0317340
$994$	0	0
$995$	−24.0000	−0.760851
$996$	0	0
$997$	4.00000	0.126681	0.0633406	−	0.997992i	$-0.479825\pi$
$997$	4.00000	0.126681	0.0633406	+	0.997992i	$0.479825\pi$
$998$	0	0
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5712.2.a.bb.1.1		1
4.3	odd	2		714.2.a.g.1.1	✓	1
12.11	even	2		2142.2.a.a.1.1		1
28.27	even	2		4998.2.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.a.g.1.1	✓	1	4.3	odd	2
2142.2.a.a.1.1		1	12.11	even	2
4998.2.a.bj.1.1		1	28.27	even	2
5712.2.a.bb.1.1		1	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:	\( N \)	\(=\)	\( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5712.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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