LMFDB - Embedded newform 52.3.b.a.51.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [52,3,Mod(51,52)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("52.51"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(52, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 52 = 2^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	52.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.41689737467$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			51.1
Character	$\chi$	$=$	52.51

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +12.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +4.00000 q^{11} -13.0000 q^{13} -24.0000 q^{14} +16.0000 q^{16} -18.0000 q^{17} -18.0000 q^{18} -12.0000 q^{19} -8.00000 q^{22} +25.0000 q^{25} +26.0000 q^{26} +48.0000 q^{28} +6.00000 q^{29} -36.0000 q^{31} -32.0000 q^{32} +36.0000 q^{34} +36.0000 q^{36} +24.0000 q^{38} +16.0000 q^{44} -68.0000 q^{47} +95.0000 q^{49} -50.0000 q^{50} -52.0000 q^{52} -102.000 q^{53} -96.0000 q^{56} -12.0000 q^{58} +116.000 q^{59} -86.0000 q^{61} +72.0000 q^{62} +108.000 q^{63} +64.0000 q^{64} -108.000 q^{67} -72.0000 q^{68} +92.0000 q^{71} -72.0000 q^{72} -48.0000 q^{76} +48.0000 q^{77} +81.0000 q^{81} +68.0000 q^{83} -32.0000 q^{88} -156.000 q^{91} +136.000 q^{94} -190.000 q^{98} +36.0000 q^{99} +O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/52\mathbb{Z}\right)^\times$.

$n$	$27$	$41$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−2.00000	−1.00000
$2$	−2.00000	−1.00000
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	4.00000	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	12.0000	1.71429	0.857143	−	0.515079i	$-0.172237\pi$
$7$	12.0000	1.71429	0.857143	+	0.515079i	$0.172237\pi$
$8$	−8.00000	−1.00000
$9$	9.00000	1.00000
$10$	0	0
$11$	4.00000	0.363636	0.181818	−	0.983332i	$-0.441802\pi$
$11$	4.00000	0.363636	0.181818	+	0.983332i	$0.441802\pi$
$12$	0	0
$13$	−13.0000	−1.00000
$13$	−13.0000	−1.00000
$14$	−24.0000	−1.71429
$15$	0	0
$16$	16.0000	1.00000
$17$	−18.0000	−1.05882	−0.529412	−	0.848365i	$-0.677587\pi$
$17$	−18.0000	−1.05882	−0.529412	+	0.848365i	$0.677587\pi$
$18$	−18.0000	−1.00000
$19$	−12.0000	−0.631579	−0.315789	−	0.948829i	$-0.602269\pi$
$19$	−12.0000	−0.631579	−0.315789	+	0.948829i	$0.602269\pi$
$20$	0	0
$21$	0	0
$22$	−8.00000	−0.363636
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	25.0000	1.00000
$26$	26.0000	1.00000
$27$	0	0
$28$	48.0000	1.71429
$29$	6.00000	0.206897	0.103448	−	0.994635i	$-0.467012\pi$
$29$	6.00000	0.206897	0.103448	+	0.994635i	$0.467012\pi$
$30$	0	0
$31$	−36.0000	−1.16129	−0.580645	−	0.814157i	$-0.697200\pi$
$31$	−36.0000	−1.16129	−0.580645	+	0.814157i	$0.697200\pi$
$32$	−32.0000	−1.00000
$33$	0	0
$34$	36.0000	1.05882
$35$	0	0
$36$	36.0000	1.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	24.0000	0.631579
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	16.0000	0.363636
$45$	0	0
$46$	0	0
$47$	−68.0000	−1.44681	−0.723404	−	0.690425i	$-0.757424\pi$
$47$	−68.0000	−1.44681	−0.723404	+	0.690425i	$0.757424\pi$
$48$	0	0
$49$	95.0000	1.93878
$50$	−50.0000	−1.00000
$51$	0	0
$52$	−52.0000	−1.00000
$53$	−102.000	−1.92453	−0.962264	−	0.272117i	$-0.912276\pi$
$53$	−102.000	−1.92453	−0.962264	+	0.272117i	$0.912276\pi$
$54$	0	0
$55$	0	0
$56$	−96.0000	−1.71429
$57$	0	0
$58$	−12.0000	−0.206897
$59$	116.000	1.96610	0.983051	−	0.183333i	$-0.0586887\pi$
$59$	116.000	1.96610	0.983051	+	0.183333i	$0.0586887\pi$
$60$	0	0
$61$	−86.0000	−1.40984	−0.704918	−	0.709289i	$-0.749016\pi$
$61$	−86.0000	−1.40984	−0.704918	+	0.709289i	$0.749016\pi$
$62$	72.0000	1.16129
$63$	108.000	1.71429
$64$	64.0000	1.00000
$65$	0	0
$66$	0	0
$67$	−108.000	−1.61194	−0.805970	−	0.591956i	$-0.798356\pi$
$67$	−108.000	−1.61194	−0.805970	+	0.591956i	$0.798356\pi$
$68$	−72.0000	−1.05882
$69$	0	0
$70$	0	0
$71$	92.0000	1.29577	0.647887	−	0.761736i	$-0.275653\pi$
$71$	92.0000	1.29577	0.647887	+	0.761736i	$0.275653\pi$
$72$	−72.0000	−1.00000
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	−48.0000	−0.631579
$77$	48.0000	0.623377
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	68.0000	0.819277	0.409639	−	0.912248i	$-0.365655\pi$
$83$	68.0000	0.819277	0.409639	+	0.912248i	$0.365655\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−32.0000	−0.363636
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	−156.000	−1.71429
$92$	0	0
$93$	0	0
$94$	136.000	1.44681
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−190.000	−1.93878
$99$	36.0000	0.363636
$100$	100.000	1.00000
$101$	−6.00000	−0.0594059	−0.0297030	−	0.999559i	$-0.509456\pi$
$101$	−6.00000	−0.0594059	−0.0297030	+	0.999559i	$0.509456\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	104.000	1.00000
$105$	0	0
$106$	204.000	1.92453
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	192.000	1.71429
$113$	174.000	1.53982	0.769912	−	0.638151i	$-0.220300\pi$
$113$	174.000	1.53982	0.769912	+	0.638151i	$0.220300\pi$
$114$	0	0
$115$	0	0
$116$	24.0000	0.206897
$117$	−117.000	−1.00000
$118$	−232.000	−1.96610
$119$	−216.000	−1.81513
$120$	0	0
$121$	−105.000	−0.867769
$122$	172.000	1.40984
$123$	0	0
$124$	−144.000	−1.16129
$125$	0	0
$126$	−216.000	−1.71429
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−128.000	−1.00000
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−144.000	−1.08271
$134$	216.000	1.61194
$135$	0	0
$136$	144.000	1.05882
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	−184.000	−1.29577
$143$	−52.0000	−0.363636
$144$	144.000	1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−276.000	−1.82781	−0.913907	−	0.405923i	$-0.866950\pi$
$151$	−276.000	−1.82781	−0.913907	+	0.405923i	$0.866950\pi$
$152$	96.0000	0.631579
$153$	−162.000	−1.05882
$154$	−96.0000	−0.623377
$155$	0	0
$156$	0	0
$157$	262.000	1.66879	0.834395	−	0.551167i	$-0.185817\pi$
$157$	262.000	1.66879	0.834395	+	0.551167i	$0.185817\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−162.000	−1.00000
$163$	324.000	1.98773	0.993865	−	0.110600i	$-0.0352772\pi$
$163$	324.000	1.98773	0.993865	+	0.110600i	$0.0352772\pi$
$164$	0	0
$165$	0	0
$166$	−136.000	−0.819277
$167$	316.000	1.89222	0.946108	−	0.323852i	$-0.104978\pi$
$167$	316.000	1.89222	0.946108	+	0.323852i	$0.104978\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	−108.000	−0.631579
$172$	0	0
$173$	138.000	0.797688	0.398844	−	0.917019i	$-0.369412\pi$
$173$	138.000	0.797688	0.398844	+	0.917019i	$0.369412\pi$
$174$	0	0
$175$	300.000	1.71429
$176$	64.0000	0.363636
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−106.000	−0.585635	−0.292818	−	0.956168i	$-0.594593\pi$
$181$	−106.000	−0.585635	−0.292818	+	0.956168i	$0.594593\pi$
$182$	312.000	1.71429
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−72.0000	−0.385027
$188$	−272.000	−1.44681
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	380.000	1.93878
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−72.0000	−0.363636
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−200.000	−1.00000
$201$	0	0
$202$	12.0000	0.0594059
$203$	72.0000	0.354680
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−208.000	−1.00000
$209$	−48.0000	−0.229665
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−408.000	−1.92453
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−432.000	−1.99078
$218$	0	0
$219$	0	0
$220$	0	0
$221$	234.000	1.05882
$222$	0	0
$223$	204.000	0.914798	0.457399	−	0.889262i	$-0.348781\pi$
$223$	204.000	0.914798	0.457399	+	0.889262i	$0.348781\pi$
$224$	−384.000	−1.71429
$225$	225.000	1.00000
$226$	−348.000	−1.53982
$227$	−428.000	−1.88546	−0.942731	−	0.333553i	$-0.891752\pi$
$227$	−428.000	−1.88546	−0.942731	+	0.333553i	$0.891752\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−48.0000	−0.206897
$233$	−366.000	−1.57082	−0.785408	−	0.618979i	$-0.787547\pi$
$233$	−366.000	−1.57082	−0.785408	+	0.618979i	$0.787547\pi$
$234$	234.000	1.00000
$235$	0	0
$236$	464.000	1.96610
$237$	0	0
$238$	432.000	1.81513
$239$	−244.000	−1.02092	−0.510460	−	0.859901i	$-0.670525\pi$
$239$	−244.000	−1.02092	−0.510460	+	0.859901i	$0.670525\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	210.000	0.867769
$243$	0	0
$244$	−344.000	−1.40984
$245$	0	0
$246$	0	0
$247$	156.000	0.631579
$248$	288.000	1.16129
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	432.000	1.71429
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	−318.000	−1.23735	−0.618677	−	0.785645i	$-0.712331\pi$
$257$	−318.000	−1.23735	−0.618677	+	0.785645i	$0.712331\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	54.0000	0.206897
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	288.000	1.08271
$267$	0	0
$268$	−432.000	−1.61194
$269$	486.000	1.80669	0.903346	−	0.428913i	$-0.141103\pi$
$269$	486.000	1.80669	0.903346	+	0.428913i	$0.141103\pi$
$270$	0	0
$271$	−516.000	−1.90406	−0.952030	−	0.306006i	$-0.901007\pi$
$271$	−516.000	−1.90406	−0.952030	+	0.306006i	$0.901007\pi$
$272$	−288.000	−1.05882
$273$	0	0
$274$	0	0
$275$	100.000	0.363636
$276$	0	0
$277$	346.000	1.24910	0.624549	−	0.780986i	$-0.285283\pi$
$277$	346.000	1.24910	0.624549	+	0.780986i	$0.285283\pi$
$278$	0	0
$279$	−324.000	−1.16129
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	368.000	1.29577
$285$	0	0
$286$	104.000	0.363636
$287$	0	0
$288$	−288.000	−1.00000
$289$	35.0000	0.121107
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	552.000	1.82781
$303$	0	0
$304$	−192.000	−0.631579
$305$	0	0
$306$	324.000	1.05882
$307$	36.0000	0.117264	0.0586319	−	0.998280i	$-0.481326\pi$
$307$	36.0000	0.117264	0.0586319	+	0.998280i	$0.481326\pi$
$308$	192.000	0.623377
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	158.000	0.504792	0.252396	−	0.967624i	$-0.418781\pi$
$313$	158.000	0.504792	0.252396	+	0.967624i	$0.418781\pi$
$314$	−524.000	−1.66879
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	24.0000	0.0752351
$320$	0	0
$321$	0	0
$322$	0	0
$323$	216.000	0.668731
$324$	324.000	1.00000
$325$	−325.000	−1.00000
$326$	−648.000	−1.98773
$327$	0	0
$328$	0	0
$329$	−816.000	−2.48024
$330$	0	0
$331$	612.000	1.84894	0.924471	−	0.381252i	$-0.124507\pi$
$331$	612.000	1.84894	0.924471	+	0.381252i	$0.124507\pi$
$332$	272.000	0.819277
$333$	0	0
$334$	−632.000	−1.89222
$335$	0	0
$336$	0	0
$337$	622.000	1.84570	0.922849	−	0.385163i	$-0.125855\pi$
$337$	622.000	1.84570	0.922849	+	0.385163i	$0.125855\pi$
$338$	−338.000	−1.00000
$339$	0	0
$340$	0	0
$341$	−144.000	−0.422287
$342$	216.000	0.631579
$343$	552.000	1.60933
$344$	0	0
$345$	0	0
$346$	−276.000	−0.797688
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−600.000	−1.71429
$351$	0	0
$352$	−128.000	−0.363636
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	556.000	1.54875	0.774373	−	0.632729i	$-0.218065\pi$
$359$	556.000	1.54875	0.774373	+	0.632729i	$0.218065\pi$
$360$	0	0
$361$	−217.000	−0.601108
$362$	212.000	0.585635
$363$	0	0
$364$	−624.000	−1.71429
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1224.00	−3.29919
$372$	0	0
$373$	278.000	0.745308	0.372654	−	0.927970i	$-0.378448\pi$
$373$	278.000	0.745308	0.372654	+	0.927970i	$0.378448\pi$
$374$	144.000	0.385027
$375$	0	0
$376$	544.000	1.44681
$377$	−78.0000	−0.206897
$378$	0	0
$379$	516.000	1.36148	0.680739	−	0.732526i	$-0.261659\pi$
$379$	516.000	1.36148	0.680739	+	0.732526i	$0.261659\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−116.000	−0.302872	−0.151436	−	0.988467i	$-0.548390\pi$
$383$	−116.000	−0.302872	−0.151436	+	0.988467i	$0.548390\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−522.000	−1.34190	−0.670951	−	0.741502i	$-0.734114\pi$
$389$	−522.000	−1.34190	−0.670951	+	0.741502i	$0.734114\pi$
$390$	0	0
$391$	0	0
$392$	−760.000	−1.93878
$393$	0	0
$394$	0	0
$395$	0	0
$396$	144.000	0.363636
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	400.000	1.00000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	468.000	1.16129
$404$	−24.0000	−0.0594059
$405$	0	0
$406$	−144.000	−0.354680
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1392.00	3.37046
$414$	0	0
$415$	0	0
$416$	416.000	1.00000
$417$	0	0
$418$	96.0000	0.229665
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	−612.000	−1.44681
$424$	816.000	1.92453
$425$	−450.000	−1.05882
$426$	0	0
$427$	−1032.00	−2.41686
$428$	0	0
$429$	0	0
$430$	0	0
$431$	412.000	0.955916	0.477958	−	0.878383i	$-0.341377\pi$
$431$	412.000	0.955916	0.477958	+	0.878383i	$0.341377\pi$
$432$	0	0
$433$	34.0000	0.0785219	0.0392610	−	0.999229i	$-0.487500\pi$
$433$	34.0000	0.0785219	0.0392610	+	0.999229i	$0.487500\pi$
$434$	864.000	1.99078
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	855.000	1.93878
$442$	−468.000	−1.05882
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	−408.000	−0.914798
$447$	0	0
$448$	768.000	1.71429
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−450.000	−1.00000
$451$	0	0
$452$	696.000	1.53982
$453$	0	0
$454$	856.000	1.88546
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	348.000	0.751620	0.375810	−	0.926697i	$-0.377365\pi$
$463$	348.000	0.751620	0.375810	+	0.926697i	$0.377365\pi$
$464$	96.0000	0.206897
$465$	0	0
$466$	732.000	1.57082
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−468.000	−1.00000
$469$	−1296.00	−2.76333
$470$	0	0
$471$	0	0
$472$	−928.000	−1.96610
$473$	0	0
$474$	0	0
$475$	−300.000	−0.631579
$476$	−864.000	−1.81513
$477$	−918.000	−1.92453
$478$	488.000	1.02092
$479$	−724.000	−1.51148	−0.755741	−	0.654870i	$-0.772723\pi$
$479$	−724.000	−1.51148	−0.755741	+	0.654870i	$0.772723\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−420.000	−0.867769
$485$	0	0
$486$	0	0
$487$	−948.000	−1.94661	−0.973306	−	0.229511i	$-0.926287\pi$
$487$	−948.000	−1.94661	−0.973306	+	0.229511i	$0.926287\pi$
$488$	688.000	1.40984
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	−108.000	−0.219067
$494$	−312.000	−0.631579
$495$	0	0
$496$	−576.000	−1.16129
$497$	1104.00	2.22133
$498$	0	0
$499$	276.000	0.553106	0.276553	−	0.960999i	$-0.410808\pi$
$499$	276.000	0.553106	0.276553	+	0.960999i	$0.410808\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−864.000	−1.71429
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−1.00000
$513$	0	0
$514$	636.000	1.23735
$515$	0	0
$516$	0	0
$517$	−272.000	−0.526112
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−258.000	−0.495202	−0.247601	−	0.968862i	$-0.579642\pi$
$521$	−258.000	−0.495202	−0.247601	+	0.968862i	$0.579642\pi$
$522$	−108.000	−0.206897
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	648.000	1.22960
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	1044.00	1.96610
$532$	−576.000	−1.08271
$533$	0	0
$534$	0	0
$535$	0	0
$536$	864.000	1.61194
$537$	0	0
$538$	−972.000	−1.80669
$539$	380.000	0.705009
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	1032.00	1.90406
$543$	0	0
$544$	576.000	1.05882
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−774.000	−1.40984
$550$	−200.000	−0.363636
$551$	−72.0000	−0.130672
$552$	0	0
$553$	0	0
$554$	−692.000	−1.24910
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	648.000	1.16129
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	972.000	1.71429
$568$	−736.000	−1.29577
$569$	306.000	0.537786	0.268893	−	0.963170i	$-0.413342\pi$
$569$	306.000	0.537786	0.268893	+	0.963170i	$0.413342\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	−208.000	−0.363636
$573$	0	0
$574$	0	0
$575$	0	0
$576$	576.000	1.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−70.0000	−0.121107
$579$	0	0
$580$	0	0
$581$	816.000	1.40448
$582$	0	0
$583$	−408.000	−0.699828
$584$	0	0
$585$	0	0
$586$	0	0
$587$	932.000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$587$	932.000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$588$	0	0
$589$	432.000	0.733447
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	734.000	1.22130	0.610649	−	0.791901i	$-0.290909\pi$
$601$	734.000	1.22130	0.610649	+	0.791901i	$0.290909\pi$
$602$	0	0
$603$	−972.000	−1.61194
$604$	−1104.00	−1.82781
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	384.000	0.631579
$609$	0	0
$610$	0	0
$611$	884.000	1.44681
$612$	−648.000	−1.05882
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−72.0000	−0.117264
$615$	0	0
$616$	−384.000	−0.623377
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−1212.00	−1.95800	−0.978998	−	0.203868i	$-0.934649\pi$
$619$	−1212.00	−1.95800	−0.978998	+	0.203868i	$0.934649\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	625.000	1.00000
$626$	−316.000	−0.504792
$627$	0	0
$628$	1048.00	1.66879
$629$	0	0
$630$	0	0
$631$	12.0000	0.0190174	0.00950872	−	0.999955i	$-0.496973\pi$
$631$	12.0000	0.0190174	0.00950872	+	0.999955i	$0.496973\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1235.00	−1.93878
$638$	−48.0000	−0.0752351
$639$	828.000	1.29577
$640$	0	0
$641$	−1266.00	−1.97504	−0.987520	−	0.157497i	$-0.949658\pi$
$641$	−1266.00	−1.97504	−0.987520	+	0.157497i	$0.949658\pi$
$642$	0	0
$643$	−636.000	−0.989114	−0.494557	−	0.869145i	$-0.664670\pi$
$643$	−636.000	−0.989114	−0.494557	+	0.869145i	$0.664670\pi$
$644$	0	0
$645$	0	0
$646$	−432.000	−0.668731
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−648.000	−1.00000
$649$	464.000	0.714946
$650$	650.000	1.00000
$651$	0	0
$652$	1296.00	1.98773
$653$	−1242.00	−1.90199	−0.950995	−	0.309205i	$-0.899937\pi$
$653$	−1242.00	−1.90199	−0.950995	+	0.309205i	$0.899937\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	1632.00	2.48024
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−1224.00	−1.84894
$663$	0	0
$664$	−544.000	−0.819277
$665$	0	0
$666$	0	0
$667$	0	0
$668$	1264.00	1.89222
$669$	0	0
$670$	0	0
$671$	−344.000	−0.512668
$672$	0	0
$673$	−1202.00	−1.78603	−0.893016	−	0.450024i	$-0.851415\pi$
$673$	−1202.00	−1.78603	−0.893016	+	0.450024i	$0.851415\pi$
$674$	−1244.00	−1.84570
$675$	0	0
$676$	676.000	1.00000
$677$	1146.00	1.69276	0.846381	−	0.532578i	$-0.178777\pi$
$677$	1146.00	1.69276	0.846381	+	0.532578i	$0.178777\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	288.000	0.422287
$683$	−92.0000	−0.134700	−0.0673499	−	0.997729i	$-0.521454\pi$
$683$	−92.0000	−0.134700	−0.0673499	+	0.997729i	$0.521454\pi$
$684$	−432.000	−0.631579
$685$	0	0
$686$	−1104.00	−1.60933
$687$	0	0
$688$	0	0
$689$	1326.00	1.92453
$690$	0	0
$691$	−1356.00	−1.96237	−0.981187	−	0.193061i	$-0.938158\pi$
$691$	−1356.00	−1.96237	−0.981187	+	0.193061i	$0.938158\pi$
$692$	552.000	0.797688
$693$	432.000	0.623377
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	1200.00	1.71429
$701$	−1146.00	−1.63481	−0.817404	−	0.576065i	$-0.804587\pi$
$701$	−1146.00	−1.63481	−0.817404	+	0.576065i	$0.804587\pi$
$702$	0	0
$703$	0	0
$704$	256.000	0.363636
$705$	0	0
$706$	0	0
$707$	−72.0000	−0.101839
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−1112.00	−1.54875
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	434.000	0.601108
$723$	0	0
$724$	−424.000	−0.585635
$725$	150.000	0.206897
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	1248.00	1.71429
$729$	729.000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−432.000	−0.586160
$738$	0	0
$739$	−204.000	−0.276049	−0.138024	−	0.990429i	$-0.544075\pi$
$739$	−204.000	−0.276049	−0.138024	+	0.990429i	$0.544075\pi$
$740$	0	0
$741$	0	0
$742$	2448.00	3.29919
$743$	−1252.00	−1.68506	−0.842530	−	0.538649i	$-0.818935\pi$
$743$	−1252.00	−1.68506	−0.842530	+	0.538649i	$0.818935\pi$
$744$	0	0
$745$	0	0
$746$	−556.000	−0.745308
$747$	612.000	0.819277
$748$	−288.000	−0.385027
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−1088.00	−1.44681
$753$	0	0
$754$	156.000	0.206897
$755$	0	0
$756$	0	0
$757$	−358.000	−0.472919	−0.236460	−	0.971641i	$-0.575987\pi$
$757$	−358.000	−0.472919	−0.236460	+	0.971641i	$0.575987\pi$
$758$	−1032.00	−1.36148
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	232.000	0.302872
$767$	−1508.00	−1.96610
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−900.000	−1.16129
$776$	0	0
$777$	0	0
$778$	1044.00	1.34190
$779$	0	0
$780$	0	0
$781$	368.000	0.471191
$782$	0	0
$783$	0	0
$784$	1520.00	1.93878
$785$	0	0
$786$	0	0
$787$	1572.00	1.99746	0.998729	−	0.0503953i	$-0.0160481\pi$
$787$	1572.00	1.99746	0.998729	+	0.0503953i	$0.0160481\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2088.00	2.63970
$792$	−288.000	−0.363636
$793$	1118.00	1.40984
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1542.00	1.93476	0.967378	−	0.253339i	$-0.0815287\pi$
$797$	1542.00	1.93476	0.967378	+	0.253339i	$0.0815287\pi$
$798$	0	0
$799$	1224.00	1.53191
$800$	−800.000	−1.00000
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−936.000	−1.16129
$807$	0	0
$808$	48.0000	0.0594059
$809$	318.000	0.393078	0.196539	−	0.980496i	$-0.437030\pi$
$809$	318.000	0.393078	0.196539	+	0.980496i	$0.437030\pi$
$810$	0	0
$811$	1524.00	1.87916	0.939581	−	0.342327i	$-0.111215\pi$
$811$	1524.00	1.87916	0.939581	+	0.342327i	$0.111215\pi$
$812$	288.000	0.354680
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−1404.00	−1.71429
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	−2784.00	−3.37046
$827$	1492.00	1.80411	0.902056	−	0.431620i	$-0.142058\pi$
$827$	1492.00	1.80411	0.902056	+	0.431620i	$0.142058\pi$
$828$	0	0
$829$	−214.000	−0.258142	−0.129071	−	0.991635i	$-0.541200\pi$
$829$	−214.000	−0.258142	−0.129071	+	0.991635i	$0.541200\pi$
$830$	0	0
$831$	0	0
$832$	−832.000	−1.00000
$833$	−1710.00	−2.05282
$834$	0	0
$835$	0	0
$836$	−192.000	−0.229665
$837$	0	0
$838$	0	0
$839$	428.000	0.510131	0.255066	−	0.966924i	$-0.417903\pi$
$839$	428.000	0.510131	0.255066	+	0.966924i	$0.417903\pi$
$840$	0	0
$841$	−805.000	−0.957194
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	1224.00	1.44681
$847$	−1260.00	−1.48760
$848$	−1632.00	−1.92453
$849$	0	0
$850$	900.000	1.05882
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	2064.00	2.41686
$855$	0	0
$856$	0	0
$857$	−1614.00	−1.88331	−0.941657	−	0.336574i	$-0.890732\pi$
$857$	−1614.00	−1.88331	−0.941657	+	0.336574i	$0.890732\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	−824.000	−0.955916
$863$	−452.000	−0.523754	−0.261877	−	0.965101i	$-0.584342\pi$
$863$	−452.000	−0.523754	−0.261877	+	0.965101i	$0.584342\pi$
$864$	0	0
$865$	0	0
$866$	−68.0000	−0.0785219
$867$	0	0
$868$	−1728.00	−1.99078
$869$	0	0
$870$	0	0
$871$	1404.00	1.61194
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1566.00	−1.77753	−0.888763	−	0.458367i	$-0.848434\pi$
$881$	−1566.00	−1.77753	−0.888763	+	0.458367i	$0.848434\pi$
$882$	−1710.00	−1.93878
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	936.000	1.05882
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	324.000	0.363636
$892$	816.000	0.914798
$893$	816.000	0.913774
$894$	0	0
$895$	0	0
$896$	−1536.00	−1.71429
$897$	0	0
$898$	0	0
$899$	−216.000	−0.240267
$900$	900.000	1.00000
$901$	1836.00	2.03774
$902$	0	0
$903$	0	0
$904$	−1392.00	−1.53982
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−1712.00	−1.88546
$909$	−54.0000	−0.0594059
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	272.000	0.297919
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1196.00	−1.29577
$924$	0	0
$925$	0	0
$926$	−696.000	−0.751620
$927$	0	0
$928$	−192.000	−0.206897
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−1140.00	−1.22449
$932$	−1464.00	−1.57082
$933$	0	0
$934$	0	0
$935$	0	0
$936$	936.000	1.00000
$937$	1042.00	1.11206	0.556030	−	0.831162i	$-0.312324\pi$
$937$	1042.00	1.11206	0.556030	+	0.831162i	$0.312324\pi$
$938$	2592.00	2.76333
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	1856.00	1.96610
$945$	0	0
$946$	0	0
$947$	212.000	0.223865	0.111932	−	0.993716i	$-0.464296\pi$
$947$	212.000	0.223865	0.111932	+	0.993716i	$0.464296\pi$
$948$	0	0
$949$	0	0
$950$	600.000	0.631579
$951$	0	0
$952$	1728.00	1.81513
$953$	−1422.00	−1.49213	−0.746065	−	0.665873i	$-0.768059\pi$
$953$	−1422.00	−1.49213	−0.746065	+	0.665873i	$0.768059\pi$
$954$	1836.00	1.92453
$955$	0	0
$956$	−976.000	−1.02092
$957$	0	0
$958$	1448.00	1.51148
$959$	0	0
$960$	0	0
$961$	335.000	0.348595
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1212.00	1.25336	0.626680	−	0.779276i	$-0.284413\pi$
$967$	1212.00	1.25336	0.626680	+	0.779276i	$0.284413\pi$
$968$	840.000	0.867769
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	1896.00	1.94661
$975$	0	0
$976$	−1376.00	−1.40984
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1732.00	−1.76195	−0.880977	−	0.473160i	$-0.843113\pi$
$983$	−1732.00	−1.76195	−0.880977	+	0.473160i	$0.843113\pi$
$984$	0	0
$985$	0	0
$986$	216.000	0.219067
$987$	0	0
$988$	624.000	0.631579
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	1152.00	1.16129
$993$	0	0
$994$	−2208.00	−2.22133
$995$	0	0
$996$	0	0
$997$	122.000	0.122367	0.0611836	−	0.998127i	$-0.480512\pi$
$997$	122.000	0.122367	0.0611836	+	0.998127i	$0.480512\pi$
$998$	−552.000	−0.553106
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	52.3.b.a.51.1	✓	1
3.2	odd	2		468.3.e.b.415.1		1
4.3	odd	2		52.3.b.b.51.1	yes	1
8.3	odd	2		832.3.c.a.831.1		1
8.5	even	2		832.3.c.b.831.1		1
12.11	even	2		468.3.e.a.415.1		1
13.12	even	2		52.3.b.b.51.1	yes	1
39.38	odd	2		468.3.e.a.415.1		1
52.51	odd	2	CM	52.3.b.a.51.1	✓	1
104.51	odd	2		832.3.c.b.831.1		1
104.77	even	2		832.3.c.a.831.1		1
156.155	even	2		468.3.e.b.415.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.3.b.a.51.1	✓	1	1.1	even	1	trivial
52.3.b.a.51.1	✓	1	52.51	odd	2	CM
52.3.b.b.51.1	yes	1	4.3	odd	2
52.3.b.b.51.1	yes	1	13.12	even	2
468.3.e.a.415.1		1	12.11	even	2
468.3.e.a.415.1		1	39.38	odd	2
468.3.e.b.415.1		1	3.2	odd	2
468.3.e.b.415.1		1	156.155	even	2
832.3.c.a.831.1		1	8.3	odd	2
832.3.c.a.831.1		1	104.77	even	2
832.3.c.b.831.1		1	8.5	even	2
832.3.c.b.831.1		1	104.51	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	52.b (of order \(2\), degree \(1\), minimal)

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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