LMFDB - Newform orbit 13.4.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [13,4,Mod(1,13)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	13.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$0.767024830075$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q - 5 q^{2} - 7 q^{3} + 17 q^{4} - 7 q^{5} + 35 q^{6} - 13 q^{7} - 45 q^{8} + 22 q^{9} + 35 q^{10} - 26 q^{11} - 119 q^{12} + 13 q^{13} + 65 q^{14} + 49 q^{15} + 89 q^{16} + 77 q^{17} - 110 q^{18} - 126 q^{19}+ \cdots - 572 q^{99}+O(q^{100})

q - 5 * q^2 - 7 * q^3 + 17 * q^4 - 7 * q^5 + 35 * q^6 - 13 * q^7 - 45 * q^8 + 22 * q^9 + 35 * q^10 - 26 * q^11 - 119 * q^12 + 13 * q^13 + 65 * q^14 + 49 * q^15 + 89 * q^16 + 77 * q^17 - 110 * q^18 - 126 * q^19 - 119 * q^20 + 91 * q^21 + 130 * q^22 - 96 * q^23 + 315 * q^24 - 76 * q^25 - 65 * q^26 + 35 * q^27 - 221 * q^28 - 82 * q^29 - 245 * q^30 + 196 * q^31 - 85 * q^32 + 182 * q^33 - 385 * q^34 + 91 * q^35 + 374 * q^36 - 131 * q^37 + 630 * q^38 - 91 * q^39 + 315 * q^40 + 336 * q^41 - 455 * q^42 - 201 * q^43 - 442 * q^44 - 154 * q^45 + 480 * q^46 - 105 * q^47 - 623 * q^48 - 174 * q^49 + 380 * q^50 - 539 * q^51 + 221 * q^52 - 432 * q^53 - 175 * q^54 + 182 * q^55 + 585 * q^56 + 882 * q^57 + 410 * q^58 - 294 * q^59 + 833 * q^60 - 56 * q^61 - 980 * q^62 - 286 * q^63 - 287 * q^64 - 91 * q^65 - 910 * q^66 + 478 * q^67 + 1309 * q^68 + 672 * q^69 - 455 * q^70 + 9 * q^71 - 990 * q^72 + 98 * q^73 + 655 * q^74 + 532 * q^75 - 2142 * q^76 + 338 * q^77 + 455 * q^78 + 1304 * q^79 - 623 * q^80 - 839 * q^81 - 1680 * q^82 - 308 * q^83 + 1547 * q^84 - 539 * q^85 + 1005 * q^86 + 574 * q^87 + 1170 * q^88 - 1190 * q^89 + 770 * q^90 - 169 * q^91 - 1632 * q^92 - 1372 * q^93 + 525 * q^94 + 882 * q^95 + 595 * q^96 + 70 * q^97 + 870 * q^98 - 572 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

−5.00000

−7.00000

17.0000

−7.00000

35.0000

−13.0000

−45.0000

22.0000

35.0000

Atkin-Lehner signs

$p$	Sign
$13$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	13.4.a.a	✓	1
3.b	odd	2	1		117.4.a.b		1
4.b	odd	2	1		208.4.a.g		1
5.b	even	2	1		325.4.a.d		1
5.c	odd	4	2		325.4.b.b		2
7.b	odd	2	1		637.4.a.a		1
8.b	even	2	1		832.4.a.r		1
8.d	odd	2	1		832.4.a.a		1
11.b	odd	2	1		1573.4.a.a		1
12.b	even	2	1		1872.4.a.k		1
13.b	even	2	1		169.4.a.e		1
13.c	even	3	2		169.4.c.e		2
13.d	odd	4	2		169.4.b.a		2
13.e	even	6	2		169.4.c.a		2
13.f	odd	12	4		169.4.e.e		4
39.d	odd	2	1		1521.4.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.a.a	✓	1	1.a	even	1	1	trivial
117.4.a.b		1	3.b	odd	2	1
169.4.a.e		1	13.b	even	2	1
169.4.b.a		2	13.d	odd	4	2
169.4.c.a		2	13.e	even	6	2
169.4.c.e		2	13.c	even	3	2
169.4.e.e		4	13.f	odd	12	4
208.4.a.g		1	4.b	odd	2	1
325.4.a.d		1	5.b	even	2	1
325.4.b.b		2	5.c	odd	4	2
637.4.a.a		1	7.b	odd	2	1
832.4.a.a		1	8.d	odd	2	1
832.4.a.r		1	8.b	even	2	1
1521.4.a.a		1	39.d	odd	2	1
1573.4.a.a		1	11.b	odd	2	1
1872.4.a.k		1	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2} + 5$ T2 + 5 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(13))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 5$ T + 5
$3$	$T + 7$ T + 7
$5$	$T + 7$ T + 7
$7$	$T + 13$ T + 13
$11$	$T + 26$ T + 26
$13$	$T - 13$ T - 13
$17$	$T - 77$ T - 77
$19$	$T + 126$ T + 126
$23$	$T + 96$ T + 96
$29$	$T + 82$ T + 82
$31$	$T - 196$ T - 196
$37$	$T + 131$ T + 131
$41$	$T - 336$ T - 336
$43$	$T + 201$ T + 201
$47$	$T + 105$ T + 105
$53$	$T + 432$ T + 432
$59$	$T + 294$ T + 294
$61$	$T + 56$ T + 56
$67$	$T - 478$ T - 478
$71$	$T - 9$ T - 9
$73$	$T - 98$ T - 98
$79$	$T - 1304$ T - 1304
$83$	$T + 308$ T + 308
$89$	$T + 1190$ T + 1190
$97$	$T - 70$ T - 70
show more
show less

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

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more