LMFDB - Embedded newform 5408.2.a.bp.1.6

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:	$1$
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			1.6
Root			$1.21063$ 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)

$f(q)$	$=$	$q+0.210627 q^{3} -2.48480 q^{5} +3.64637 q^{7} -2.95564 q^{9} -0.353488 q^{11} -0.523366 q^{15} +1.59796 q^{17} -1.96732 q^{19} +0.768022 q^{21} +7.27088 q^{23} +1.17424 q^{25} -1.25442 q^{27} -7.72133 q^{29} +8.63015 q^{31} -0.0744540 q^{33} -9.06051 q^{35} -9.87328 q^{37} -5.81206 q^{41} -6.16851 q^{43} +7.34417 q^{45} -4.59660 q^{47} +6.29601 q^{49} +0.336573 q^{51} +4.44641 q^{53} +0.878348 q^{55} -0.414370 q^{57} +14.0603 q^{59} -1.42793 q^{61} -10.7773 q^{63} +8.24537 q^{67} +1.53144 q^{69} +6.72644 q^{71} -6.76070 q^{73} +0.247327 q^{75} -1.28895 q^{77} -11.1284 q^{79} +8.60270 q^{81} +5.43005 q^{83} -3.97062 q^{85} -1.62632 q^{87} +13.2450 q^{89} +1.81774 q^{93} +4.88841 q^{95} -12.5220 q^{97} +1.04478 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.210627	0.121605	0.0608027	−	0.998150i	$-0.480634\pi$
$3$	0.210627	0.121605	0.0608027	+	0.998150i	$0.480634\pi$
$4$	0	0
$5$	−2.48480	−1.11124	−0.555619	−	0.831437i	$-0.687519\pi$
$5$	−2.48480	−1.11124	−0.555619	+	0.831437i	$0.687519\pi$
$6$	0	0
$7$	3.64637	1.37820	0.689099	−	0.724667i	$-0.258007\pi$
$7$	3.64637	1.37820	0.689099	+	0.724667i	$0.258007\pi$
$8$	0	0
$9$	−2.95564	−0.985212
$10$	0	0
$11$	−0.353488	−0.106581	−0.0532904	−	0.998579i	$-0.516971\pi$
$11$	−0.353488	−0.106581	−0.0532904	+	0.998579i	$0.516971\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.523366	−0.135132
$16$	0	0
$17$	1.59796	0.387563	0.193781	−	0.981045i	$-0.437925\pi$
$17$	1.59796	0.387563	0.193781	+	0.981045i	$0.437925\pi$
$18$	0	0
$19$	−1.96732	−0.451335	−0.225667	−	0.974204i	$-0.572456\pi$
$19$	−1.96732	−0.451335	−0.225667	+	0.974204i	$0.572456\pi$
$20$	0	0
$21$	0.768022	0.167596
$22$	0	0
$23$	7.27088	1.51608	0.758041	−	0.652207i	$-0.226157\pi$
$23$	7.27088	1.51608	0.758041	+	0.652207i	$0.226157\pi$
$24$	0	0
$25$	1.17424	0.234849
$26$	0	0
$27$	−1.25442	−0.241412
$28$	0	0
$29$	−7.72133	−1.43382	−0.716908	−	0.697168i	$-0.754443\pi$
$29$	−7.72133	−1.43382	−0.716908	+	0.697168i	$0.754443\pi$
$30$	0	0
$31$	8.63015	1.55002	0.775010	−	0.631949i	$-0.217745\pi$
$31$	8.63015	1.55002	0.775010	+	0.631949i	$0.217745\pi$
$32$	0	0
$33$	−0.0744540	−0.0129608
$34$	0	0
$35$	−9.06051	−1.53151
$36$	0	0
$37$	−9.87328	−1.62316	−0.811579	−	0.584243i	$-0.801392\pi$
$37$	−9.87328	−1.62316	−0.811579	+	0.584243i	$0.801392\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.81206	−0.907691	−0.453845	−	0.891080i	$-0.649948\pi$
$41$	−5.81206	−0.907691	−0.453845	+	0.891080i	$0.649948\pi$
$42$	0	0
$43$	−6.16851	−0.940689	−0.470344	−	0.882483i	$-0.655870\pi$
$43$	−6.16851	−0.940689	−0.470344	+	0.882483i	$0.655870\pi$
$44$	0	0
$45$	7.34417	1.09480
$46$	0	0
$47$	−4.59660	−0.670482	−0.335241	−	0.942132i	$-0.608818\pi$
$47$	−4.59660	−0.670482	−0.335241	+	0.942132i	$0.608818\pi$
$48$	0	0
$49$	6.29601	0.899429
$50$	0	0
$51$	0.336573	0.0471297
$52$	0	0
$53$	4.44641	0.610762	0.305381	−	0.952230i	$-0.401216\pi$
$53$	4.44641	0.610762	0.305381	+	0.952230i	$0.401216\pi$
$54$	0	0
$55$	0.878348	0.118436
$56$	0	0
$57$	−0.414370	−0.0548847
$58$	0	0
$59$	14.0603	1.83050	0.915248	−	0.402892i	$-0.131995\pi$
$59$	14.0603	1.83050	0.915248	+	0.402892i	$0.131995\pi$
$60$	0	0
$61$	−1.42793	−0.182828	−0.0914140	−	0.995813i	$-0.529139\pi$
$61$	−1.42793	−0.182828	−0.0914140	+	0.995813i	$0.529139\pi$
$62$	0	0
$63$	−10.7773	−1.35782
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.24537	1.00733	0.503666	−	0.863898i	$-0.331984\pi$
$67$	8.24537	1.00733	0.503666	+	0.863898i	$0.331984\pi$
$68$	0	0
$69$	1.53144	0.184364
$70$	0	0
$71$	6.72644	0.798281	0.399141	−	0.916890i	$-0.369309\pi$
$71$	6.72644	0.798281	0.399141	+	0.916890i	$0.369309\pi$
$72$	0	0
$73$	−6.76070	−0.791280	−0.395640	−	0.918406i	$-0.629477\pi$
$73$	−6.76070	−0.791280	−0.395640	+	0.918406i	$0.629477\pi$
$74$	0	0
$75$	0.247327	0.0285589
$76$	0	0
$77$	−1.28895	−0.146889
$78$	0	0
$79$	−11.1284	−1.25204	−0.626022	−	0.779805i	$-0.715318\pi$
$79$	−11.1284	−1.25204	−0.626022	+	0.779805i	$0.715318\pi$
$80$	0	0
$81$	8.60270	0.955855
$82$	0	0
$83$	5.43005	0.596025	0.298013	−	0.954562i	$-0.403676\pi$
$83$	5.43005	0.596025	0.298013	+	0.954562i	$0.403676\pi$
$84$	0	0
$85$	−3.97062	−0.430674
$86$	0	0
$87$	−1.62632	−0.174360
$88$	0	0
$89$	13.2450	1.40397	0.701983	−	0.712193i	$-0.252298\pi$
$89$	13.2450	1.40397	0.701983	+	0.712193i	$0.252298\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.81774	0.188491
$94$	0	0
$95$	4.88841	0.501540
$96$	0	0
$97$	−12.5220	−1.27142	−0.635710	−	0.771928i	$-0.719293\pi$
$97$	−12.5220	−1.27142	−0.635710	+	0.771928i	$0.719293\pi$
$98$	0	0
$99$	1.04478	0.105005

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.bp.1.6	✓	9
4.3	odd	2		5408.2.a.br.1.4	yes	9
13.12	even	2		5408.2.a.bq.1.6	yes	9
52.51	odd	2		5408.2.a.bs.1.4	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5408.2.a.bp.1.6	✓	9	1.1	even	1	trivial
5408.2.a.bq.1.6	yes	9	13.12	even	2
5408.2.a.br.1.4	yes	9	4.3	odd	2
5408.2.a.bs.1.4	yes	9	52.51	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+0.210627 q^{3} -2.48480 q^{5} +3.64637 q^{7} -2.95564 q^{9} -0.353488 q^{11} -0.523366 q^{15} +1.59796 q^{17} -1.96732 q^{19} +0.768022 q^{21} +7.27088 q^{23} +1.17424 q^{25} -1.25442 q^{27} -7.72133 q^{29} +8.63015 q^{31} -0.0744540 q^{33} -9.06051 q^{35} -9.87328 q^{37} -5.81206 q^{41} -6.16851 q^{43} +7.34417 q^{45} -4.59660 q^{47} +6.29601 q^{49} +0.336573 q^{51} +4.44641 q^{53} +0.878348 q^{55} -0.414370 q^{57} +14.0603 q^{59} -1.42793 q^{61} -10.7773 q^{63} +8.24537 q^{67} +1.53144 q^{69} +6.72644 q^{71} -6.76070 q^{73} +0.247327 q^{75} -1.28895 q^{77} -11.1284 q^{79} +8.60270 q^{81} +5.43005 q^{83} -3.97062 q^{85} -1.62632 q^{87} +13.2450 q^{89} +1.81774 q^{93} +4.88841 q^{95} -12.5220 q^{97} +1.04478 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 7 q^{3} - 6 q^{5} + 10 q^{9} - q^{11} + 18 q^{15} - 7 q^{17} - 5 q^{19} - 12 q^{21} - 12 q^{23} + 11 q^{25} - 34 q^{27} - 8 q^{29} + 20 q^{31} + 12 q^{33} - 6 q^{35} - 6 q^{37} + 31 q^{41} - 33 q^{43}+ \cdots - 47 q^{99}+O(q^{100}) \) 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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