LMFDB - Embedded newform 6728.2.a.bf.1.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6728,2,Mod(1,6728)] 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("6728.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6728 = 2^{3} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6728.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,0,4,0,8,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.7233504799$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	6728.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.463807 q^{3} -1.33311 q^{5} -3.08911 q^{7} -2.78488 q^{9} -3.64662 q^{11} -6.91231 q^{13} -0.618307 q^{15} -3.49859 q^{17} -0.994095 q^{19} -1.43275 q^{21} +9.07726 q^{23} -3.22281 q^{25} -2.68307 q^{27} -7.30102 q^{31} -1.69133 q^{33} +4.11814 q^{35} -8.40359 q^{37} -3.20597 q^{39} +1.91431 q^{41} -2.04590 q^{43} +3.71257 q^{45} +6.23503 q^{47} +2.54262 q^{49} -1.62267 q^{51} -4.09383 q^{53} +4.86136 q^{55} -0.461068 q^{57} -4.78766 q^{59} -5.38595 q^{61} +8.60282 q^{63} +9.21489 q^{65} -3.04962 q^{67} +4.21009 q^{69} +5.62301 q^{71} -10.9687 q^{73} -1.49476 q^{75} +11.2648 q^{77} -15.0692 q^{79} +7.11022 q^{81} -3.07444 q^{83} +4.66402 q^{85} +10.3737 q^{89} +21.3529 q^{91} -3.38626 q^{93} +1.32524 q^{95} +10.5611 q^{97} +10.1554 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{3} + 8 q^{5} - 2 q^{7} + 32 q^{9} + 6 q^{11} + 2 q^{13} + 32 q^{15} + 20 q^{17} + 44 q^{19} + 4 q^{21} + 2 q^{23} + 32 q^{25} + 34 q^{27} + 10 q^{31} - 12 q^{33} - 32 q^{35} + 40 q^{37} + 18 q^{39}+ \cdots + 60 q^{99}+O(q^{100}) $ 24 * q + 4 * q^3 + 8 * q^5 - 2 * q^7 + 32 * q^9 + 6 * q^11 + 2 * q^13 + 32 * q^15 + 20 * q^17 + 44 * q^19 + 4 * q^21 + 2 * q^23 + 32 * q^25 + 34 * q^27 + 10 * q^31 - 12 * q^33 - 32 * q^35 + 40 * q^37 + 18 * q^39 + 8 * q^41 + 30 * q^43 + 16 * q^45 + 34 * q^47 + 26 * q^49 - 28 * q^51 + 18 * q^53 + 50 * q^55 + 6 * q^57 - 64 * q^59 + 6 * q^61 - 28 * q^63 + 14 * q^65 - 40 * q^67 + 38 * q^69 + 52 * q^73 + 66 * q^75 + 90 * q^77 + 4 * q^79 + 40 * q^81 - 44 * q^83 + 40 * q^85 - 40 * q^89 - 28 * q^91 - 10 * q^93 + 84 * q^97 + 60 * q^99

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

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

$2$	0	0
$2$	0	0
$3$	0.463807	0.267779	0.133889	−	0.990996i	$-0.457253\pi$
$3$	0.463807	0.267779	0.133889	+	0.990996i	$0.457253\pi$
$4$	0	0
$5$	−1.33311	−0.596187	−0.298093	−	0.954537i	$-0.596351\pi$
$5$	−1.33311	−0.596187	−0.298093	+	0.954537i	$0.596351\pi$
$6$	0	0
$7$	−3.08911	−1.16757	−0.583787	−	0.811907i	$-0.698430\pi$
$7$	−3.08911	−1.16757	−0.583787	+	0.811907i	$0.698430\pi$
$8$	0	0
$9$	−2.78488	−0.928294
$10$	0	0
$11$	−3.64662	−1.09950	−0.549749	−	0.835330i	$-0.685277\pi$
$11$	−3.64662	−1.09950	−0.549749	+	0.835330i	$0.685277\pi$
$12$	0	0
$13$	−6.91231	−1.91713	−0.958564	−	0.284876i	$-0.908048\pi$
$13$	−6.91231	−1.91713	−0.958564	+	0.284876i	$0.908048\pi$
$14$	0	0
$15$	−0.618307	−0.159646
$16$	0	0
$17$	−3.49859	−0.848532	−0.424266	−	0.905538i	$-0.639468\pi$
$17$	−3.49859	−0.848532	−0.424266	+	0.905538i	$0.639468\pi$
$18$	0	0
$19$	−0.994095	−0.228061	−0.114031	−	0.993477i	$-0.536376\pi$
$19$	−0.994095	−0.228061	−0.114031	+	0.993477i	$0.536376\pi$
$20$	0	0
$21$	−1.43275	−0.312652
$22$	0	0
$23$	9.07726	1.89274	0.946369	−	0.323087i	$-0.104720\pi$
$23$	9.07726	1.89274	0.946369	+	0.323087i	$0.104720\pi$
$24$	0	0
$25$	−3.22281	−0.644561
$26$	0	0
$27$	−2.68307	−0.516357
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−7.30102	−1.31130	−0.655651	−	0.755064i	$-0.727606\pi$
$31$	−7.30102	−1.31130	−0.655651	+	0.755064i	$0.727606\pi$
$32$	0	0
$33$	−1.69133	−0.294422
$34$	0	0
$35$	4.11814	0.696092
$36$	0	0
$37$	−8.40359	−1.38154	−0.690771	−	0.723074i	$-0.742729\pi$
$37$	−8.40359	−1.38154	−0.690771	+	0.723074i	$0.742729\pi$
$38$	0	0
$39$	−3.20597	−0.513367
$40$	0	0
$41$	1.91431	0.298964	0.149482	−	0.988764i	$-0.452239\pi$
$41$	1.91431	0.298964	0.149482	+	0.988764i	$0.452239\pi$
$42$	0	0
$43$	−2.04590	−0.311996	−0.155998	−	0.987757i	$-0.549859\pi$
$43$	−2.04590	−0.311996	−0.155998	+	0.987757i	$0.549859\pi$
$44$	0	0
$45$	3.71257	0.553437
$46$	0	0
$47$	6.23503	0.909473	0.454736	−	0.890626i	$-0.349733\pi$
$47$	6.23503	0.909473	0.454736	+	0.890626i	$0.349733\pi$
$48$	0	0
$49$	2.54262	0.363231
$50$	0	0
$51$	−1.62267	−0.227219
$52$	0	0
$53$	−4.09383	−0.562331	−0.281165	−	0.959659i	$-0.590721\pi$
$53$	−4.09383	−0.562331	−0.281165	+	0.959659i	$0.590721\pi$
$54$	0	0
$55$	4.86136	0.655506
$56$	0	0
$57$	−0.461068	−0.0610700
$58$	0	0
$59$	−4.78766	−0.623300	−0.311650	−	0.950197i	$-0.600882\pi$
$59$	−4.78766	−0.623300	−0.311650	+	0.950197i	$0.600882\pi$
$60$	0	0
$61$	−5.38595	−0.689600	−0.344800	−	0.938676i	$-0.612053\pi$
$61$	−5.38595	−0.689600	−0.344800	+	0.938676i	$0.612053\pi$
$62$	0	0
$63$	8.60282	1.08385
$64$	0	0
$65$	9.21489	1.14297
$66$	0	0
$67$	−3.04962	−0.372571	−0.186285	−	0.982496i	$-0.559645\pi$
$67$	−3.04962	−0.372571	−0.186285	+	0.982496i	$0.559645\pi$
$68$	0	0
$69$	4.21009	0.506836
$70$	0	0
$71$	5.62301	0.667329	0.333665	−	0.942692i	$-0.391715\pi$
$71$	5.62301	0.667329	0.333665	+	0.942692i	$0.391715\pi$
$72$	0	0
$73$	−10.9687	−1.28379	−0.641893	−	0.766795i	$-0.721850\pi$
$73$	−10.9687	−1.28379	−0.641893	+	0.766795i	$0.721850\pi$
$74$	0	0
$75$	−1.49476	−0.172600
$76$	0	0
$77$	11.2648	1.28375
$78$	0	0
$79$	−15.0692	−1.69542	−0.847710	−	0.530460i	$-0.822019\pi$
$79$	−15.0692	−1.69542	−0.847710	+	0.530460i	$0.822019\pi$
$80$	0	0
$81$	7.11022	0.790025
$82$	0	0
$83$	−3.07444	−0.337464	−0.168732	−	0.985662i	$-0.553967\pi$
$83$	−3.07444	−0.337464	−0.168732	+	0.985662i	$0.553967\pi$
$84$	0	0
$85$	4.66402	0.505884
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.3737	1.09961	0.549804	−	0.835294i	$-0.314702\pi$
$89$	10.3737	1.09961	0.549804	+	0.835294i	$0.314702\pi$
$90$	0	0
$91$	21.3529	2.23839
$92$	0	0
$93$	−3.38626	−0.351139
$94$	0	0
$95$	1.32524	0.135967
$96$	0	0
$97$	10.5611	1.07231	0.536156	−	0.844119i	$-0.319876\pi$
$97$	10.5611	1.07231	0.536156	+	0.844119i	$0.319876\pi$
$98$	0	0
$99$	10.1554	1.02066

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6728.2.a.bf.1.14		24
29.2	odd	28		232.2.q.a.33.5	✓	48
29.15	odd	28		232.2.q.a.225.5	yes	48
29.28	even	2		6728.2.a.be.1.11		24
116.15	even	28		464.2.y.e.225.4		48
116.31	even	28		464.2.y.e.33.4		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.q.a.33.5	✓	48	29.2	odd	28
232.2.q.a.225.5	yes	48	29.15	odd	28
464.2.y.e.33.4		48	116.31	even	28
464.2.y.e.225.4		48	116.15	even	28
6728.2.a.be.1.11		24	29.28	even	2
6728.2.a.bf.1.14		24	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6728 = 2^{3} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6728.a (trivial)

\(f(q)\)	\(=\)	\(q+0.463807 q^{3} -1.33311 q^{5} -3.08911 q^{7} -2.78488 q^{9} -3.64662 q^{11} -6.91231 q^{13} -0.618307 q^{15} -3.49859 q^{17} -0.994095 q^{19} -1.43275 q^{21} +9.07726 q^{23} -3.22281 q^{25} -2.68307 q^{27} -7.30102 q^{31} -1.69133 q^{33} +4.11814 q^{35} -8.40359 q^{37} -3.20597 q^{39} +1.91431 q^{41} -2.04590 q^{43} +3.71257 q^{45} +6.23503 q^{47} +2.54262 q^{49} -1.62267 q^{51} -4.09383 q^{53} +4.86136 q^{55} -0.461068 q^{57} -4.78766 q^{59} -5.38595 q^{61} +8.60282 q^{63} +9.21489 q^{65} -3.04962 q^{67} +4.21009 q^{69} +5.62301 q^{71} -10.9687 q^{73} -1.49476 q^{75} +11.2648 q^{77} -15.0692 q^{79} +7.11022 q^{81} -3.07444 q^{83} +4.66402 q^{85} +10.3737 q^{89} +21.3529 q^{91} -3.38626 q^{93} +1.32524 q^{95} +10.5611 q^{97} +10.1554 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{3} + 8 q^{5} - 2 q^{7} + 32 q^{9} + 6 q^{11} + 2 q^{13} + 32 q^{15} + 20 q^{17} + 44 q^{19} + 4 q^{21} + 2 q^{23} + 32 q^{25} + 34 q^{27} + 10 q^{31} - 12 q^{33} - 32 q^{35} + 40 q^{37} + 18 q^{39}+ \cdots + 60 q^{99}+O(q^{100}) \) 24 * q + 4 * q^3 + 8 * q^5 - 2 * q^7 + 32 * q^9 + 6 * q^11 + 2 * q^13 + 32 * q^15 + 20 * q^17 + 44 * q^19 + 4 * q^21 + 2 * q^23 + 32 * q^25 + 34 * q^27 + 10 * q^31 - 12 * q^33 - 32 * q^35 + 40 * q^37 + 18 * q^39 + 8 * q^41 + 30 * q^43 + 16 * q^45 + 34 * q^47 + 26 * q^49 - 28 * q^51 + 18 * q^53 + 50 * q^55 + 6 * q^57 - 64 * q^59 + 6 * q^61 - 28 * q^63 + 14 * q^65 - 40 * q^67 + 38 * q^69 + 52 * q^73 + 66 * q^75 + 90 * q^77 + 4 * q^79 + 40 * q^81 - 44 * q^83 + 40 * q^85 - 40 * q^89 - 28 * q^91 - 10 * q^93 + 84 * q^97 + 60 * q^99

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