LMFDB - Embedded newform 5408.2.a.bs.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5408,2,Mod(1,5408)] 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("5408.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5408 = 2^{5} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5408.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,7,0,6,0,0,0,10,0,-1,0,0,0,18,0,-7,0,-5,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.1830974131$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 14x^{7} + 23x^{6} + 63x^{5} - 85x^{4} - 99x^{3} + 98x^{2} + 35x - 7 $ x^9 - 2x^8 - 14x^7 + 23x^6 + 63x^5 - 85x^4 - 99x^3 + 98x^2 + 35x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.89279$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.892791 q^{3} -2.29689 q^{5} -0.141905 q^{7} -2.20292 q^{9} +0.287920 q^{11} +2.05065 q^{15} -2.71023 q^{17} -7.96689 q^{19} +0.126691 q^{21} -4.74263 q^{23} +0.275716 q^{25} +4.64512 q^{27} +1.65930 q^{29} -4.01133 q^{31} -0.257052 q^{33} +0.325940 q^{35} -1.62801 q^{37} +3.90907 q^{41} +5.08799 q^{43} +5.05988 q^{45} -4.97220 q^{47} -6.97986 q^{49} +2.41966 q^{51} -8.08150 q^{53} -0.661321 q^{55} +7.11277 q^{57} +8.10057 q^{59} +7.78175 q^{61} +0.312605 q^{63} +7.26322 q^{67} +4.23418 q^{69} -12.5732 q^{71} -11.8471 q^{73} -0.246157 q^{75} -0.0408572 q^{77} +8.82927 q^{79} +2.46165 q^{81} -11.1348 q^{83} +6.22510 q^{85} -1.48141 q^{87} +1.31676 q^{89} +3.58128 q^{93} +18.2991 q^{95} -8.17585 q^{97} -0.634265 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 7 q^{3} + 6 q^{5} + 10 q^{9} - q^{11} + 18 q^{15} - 7 q^{17} - 5 q^{19} + 12 q^{21} + 12 q^{23} + 11 q^{25} + 34 q^{27} - 8 q^{29} + 20 q^{31} - 12 q^{33} + 6 q^{35} + 6 q^{37} - 31 q^{41} + 33 q^{43}+ \cdots - 47 q^{99}+O(q^{100}) $ 9 * q + 7 * q^3 + 6 * q^5 + 10 * q^9 - q^11 + 18 * q^15 - 7 * q^17 - 5 * q^19 + 12 * q^21 + 12 * q^23 + 11 * q^25 + 34 * q^27 - 8 * q^29 + 20 * q^31 - 12 * q^33 + 6 * q^35 + 6 * q^37 - 31 * q^41 + 33 * q^43 + 22 * q^45 - 14 * q^47 + 3 * q^49 + 12 * q^51 + 14 * q^53 + 38 * q^55 - 10 * q^57 + 9 * q^59 + 8 * q^61 + 20 * q^63 + 3 * q^67 + 26 * q^69 - 26 * q^71 - 5 * q^73 + 25 * q^75 - 6 * q^77 + 44 * q^79 + 45 * q^81 + 17 * q^83 + 6 * q^85 + 24 * q^87 - 41 * q^89 + 26 * q^93 + 4 * q^95 - 15 * q^97 - 47 * 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
$3$	−0.892791	−0.515453	−0.257727	−	0.966218i	$-0.582973\pi$
$3$	−0.892791	−0.515453	−0.257727	+	0.966218i	$0.582973\pi$
$4$	0	0
$5$	−2.29689	−1.02720	−0.513601	−	0.858029i	$-0.671689\pi$
$5$	−2.29689	−1.02720	−0.513601	+	0.858029i	$0.671689\pi$
$6$	0	0
$7$	−0.141905	−0.0536350	−0.0268175	−	0.999640i	$-0.508537\pi$
$7$	−0.141905	−0.0536350	−0.0268175	+	0.999640i	$0.508537\pi$
$8$	0	0
$9$	−2.20292	−0.734308
$10$	0	0
$11$	0.287920	0.0868111	0.0434055	−	0.999058i	$-0.486179\pi$
$11$	0.287920	0.0868111	0.0434055	+	0.999058i	$0.486179\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.05065	0.529474
$16$	0	0
$17$	−2.71023	−0.657326	−0.328663	−	0.944447i	$-0.606598\pi$
$17$	−2.71023	−0.657326	−0.328663	+	0.944447i	$0.606598\pi$
$18$	0	0
$19$	−7.96689	−1.82773	−0.913865	−	0.406018i	$-0.866917\pi$
$19$	−7.96689	−1.82773	−0.913865	+	0.406018i	$0.866917\pi$
$20$	0	0
$21$	0.126691	0.0276463
$22$	0	0
$23$	−4.74263	−0.988907	−0.494454	−	0.869204i	$-0.664632\pi$
$23$	−4.74263	−0.988907	−0.494454	+	0.869204i	$0.664632\pi$
$24$	0	0
$25$	0.275716	0.0551433
$26$	0	0
$27$	4.64512	0.893955
$28$	0	0
$29$	1.65930	0.308124	0.154062	−	0.988061i	$-0.450764\pi$
$29$	1.65930	0.308124	0.154062	+	0.988061i	$0.450764\pi$
$30$	0	0
$31$	−4.01133	−0.720456	−0.360228	−	0.932864i	$-0.617301\pi$
$31$	−4.01133	−0.720456	−0.360228	+	0.932864i	$0.617301\pi$
$32$	0	0
$33$	−0.257052	−0.0447470
$34$	0	0
$35$	0.325940	0.0550939
$36$	0	0
$37$	−1.62801	−0.267643	−0.133821	−	0.991005i	$-0.542725\pi$
$37$	−1.62801	−0.267643	−0.133821	+	0.991005i	$0.542725\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.90907	0.610494	0.305247	−	0.952273i	$-0.401261\pi$
$41$	3.90907	0.610494	0.305247	+	0.952273i	$0.401261\pi$
$42$	0	0
$43$	5.08799	0.775912	0.387956	−	0.921678i	$-0.373181\pi$
$43$	5.08799	0.775912	0.387956	+	0.921678i	$0.373181\pi$
$44$	0	0
$45$	5.05988	0.754282
$46$	0	0
$47$	−4.97220	−0.725269	−0.362635	−	0.931931i	$-0.618123\pi$
$47$	−4.97220	−0.725269	−0.362635	+	0.931931i	$0.618123\pi$
$48$	0	0
$49$	−6.97986	−0.997123
$50$	0	0
$51$	2.41966	0.338821
$52$	0	0
$53$	−8.08150	−1.11008	−0.555040	−	0.831824i	$-0.687297\pi$
$53$	−8.08150	−1.11008	−0.555040	+	0.831824i	$0.687297\pi$
$54$	0	0
$55$	−0.661321	−0.0891725
$56$	0	0
$57$	7.11277	0.942109
$58$	0	0
$59$	8.10057	1.05460	0.527302	−	0.849678i	$-0.323204\pi$
$59$	8.10057	1.05460	0.527302	+	0.849678i	$0.323204\pi$
$60$	0	0
$61$	7.78175	0.996350	0.498175	−	0.867076i	$-0.334004\pi$
$61$	7.78175	0.996350	0.498175	+	0.867076i	$0.334004\pi$
$62$	0	0
$63$	0.312605	0.0393846
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.26322	0.887343	0.443672	−	0.896189i	$-0.353676\pi$
$67$	7.26322	0.887343	0.443672	+	0.896189i	$0.353676\pi$
$68$	0	0
$69$	4.23418	0.509735
$70$	0	0
$71$	−12.5732	−1.49216	−0.746082	−	0.665854i	$-0.768067\pi$
$71$	−12.5732	−1.49216	−0.746082	+	0.665854i	$0.768067\pi$
$72$	0	0
$73$	−11.8471	−1.38660	−0.693300	−	0.720649i	$-0.743844\pi$
$73$	−11.8471	−1.38660	−0.693300	+	0.720649i	$0.743844\pi$
$74$	0	0
$75$	−0.246157	−0.0284238
$76$	0	0
$77$	−0.0408572	−0.00465611
$78$	0	0
$79$	8.82927	0.993370	0.496685	−	0.867931i	$-0.334550\pi$
$79$	8.82927	0.993370	0.496685	+	0.867931i	$0.334550\pi$
$80$	0	0
$81$	2.46165	0.273516
$82$	0	0
$83$	−11.1348	−1.22221	−0.611104	−	0.791550i	$-0.709274\pi$
$83$	−11.1348	−1.22221	−0.611104	+	0.791550i	$0.709274\pi$
$84$	0	0
$85$	6.22510	0.675207
$86$	0	0
$87$	−1.48141	−0.158824
$88$	0	0
$89$	1.31676	0.139576	0.0697882	−	0.997562i	$-0.477768\pi$
$89$	1.31676	0.139576	0.0697882	+	0.997562i	$0.477768\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.58128	0.371361
$94$	0	0
$95$	18.2991	1.87745
$96$	0	0
$97$	−8.17585	−0.830131	−0.415066	−	0.909791i	$-0.636241\pi$
$97$	−8.17585	−0.830131	−0.415066	+	0.909791i	$0.636241\pi$
$98$	0	0
$99$	−0.634265	−0.0637461

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	5408.2.a.bs.1.3	yes	9
4.3	odd	2		5408.2.a.bq.1.7	yes	9
13.12	even	2		5408.2.a.br.1.3	yes	9
52.51	odd	2		5408.2.a.bp.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5408.2.a.bp.1.7	✓	9	52.51	odd	2
5408.2.a.bq.1.7	yes	9	4.3	odd	2
5408.2.a.br.1.3	yes	9	13.12	even	2
5408.2.a.bs.1.3	yes	9	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

\(f(q)\)	\(=\)	\(q-0.892791 q^{3} -2.29689 q^{5} -0.141905 q^{7} -2.20292 q^{9} +0.287920 q^{11} +2.05065 q^{15} -2.71023 q^{17} -7.96689 q^{19} +0.126691 q^{21} -4.74263 q^{23} +0.275716 q^{25} +4.64512 q^{27} +1.65930 q^{29} -4.01133 q^{31} -0.257052 q^{33} +0.325940 q^{35} -1.62801 q^{37} +3.90907 q^{41} +5.08799 q^{43} +5.05988 q^{45} -4.97220 q^{47} -6.97986 q^{49} +2.41966 q^{51} -8.08150 q^{53} -0.661321 q^{55} +7.11277 q^{57} +8.10057 q^{59} +7.78175 q^{61} +0.312605 q^{63} +7.26322 q^{67} +4.23418 q^{69} -12.5732 q^{71} -11.8471 q^{73} -0.246157 q^{75} -0.0408572 q^{77} +8.82927 q^{79} +2.46165 q^{81} -11.1348 q^{83} +6.22510 q^{85} -1.48141 q^{87} +1.31676 q^{89} +3.58128 q^{93} +18.2991 q^{95} -8.17585 q^{97} -0.634265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 7 q^{3} + 6 q^{5} + 10 q^{9} - q^{11} + 18 q^{15} - 7 q^{17} - 5 q^{19} + 12 q^{21} + 12 q^{23} + 11 q^{25} + 34 q^{27} - 8 q^{29} + 20 q^{31} - 12 q^{33} + 6 q^{35} + 6 q^{37} - 31 q^{41} + 33 q^{43}+ \cdots - 47 q^{99}+O(q^{100}) \) 9 * q + 7 * q^3 + 6 * q^5 + 10 * q^9 - q^11 + 18 * q^15 - 7 * q^17 - 5 * q^19 + 12 * q^21 + 12 * q^23 + 11 * q^25 + 34 * q^27 - 8 * q^29 + 20 * q^31 - 12 * q^33 + 6 * q^35 + 6 * q^37 - 31 * q^41 + 33 * q^43 + 22 * q^45 - 14 * q^47 + 3 * q^49 + 12 * q^51 + 14 * q^53 + 38 * q^55 - 10 * q^57 + 9 * q^59 + 8 * q^61 + 20 * q^63 + 3 * q^67 + 26 * q^69 - 26 * q^71 - 5 * q^73 + 25 * q^75 - 6 * q^77 + 44 * q^79 + 45 * q^81 + 17 * q^83 + 6 * q^85 + 24 * q^87 - 41 * q^89 + 26 * q^93 + 4 * q^95 - 15 * q^97 - 47 * q^99

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