LMFDB - Newform orbit 702.2.a.k

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [702,2,Mod(1,702)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(702, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("702.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$702 = 2 \cdot 3^{3} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	702.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,-2,0,4] 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:	$5.60549822189$
Analytic rank:	$0$
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 + q^{2} + q^{4} - 2 q^{5} + 4 q^{7} + q^{8} - 2 q^{10} - 2 q^{11} + q^{13} + 4 q^{14} + q^{16} + 4 q^{17} + 7 q^{19} - 2 q^{20} - 2 q^{22} - q^{25} + q^{26} + 4 q^{28} + 9 q^{29} - 10 q^{31} + q^{32}+ \cdots + 9 q^{98}+O(q^{100})

q + q^2 + q^4 - 2 * q^5 + 4 * q^7 + q^8 - 2 * q^10 - 2 * q^11 + q^13 + 4 * q^14 + q^16 + 4 * q^17 + 7 * q^19 - 2 * q^20 - 2 * q^22 - q^25 + q^26 + 4 * q^28 + 9 * q^29 - 10 * q^31 + q^32 + 4 * q^34 - 8 * q^35 + 7 * q^37 + 7 * q^38 - 2 * q^40 - 5 * q^41 - 2 * q^43 - 2 * q^44 + q^47 + 9 * q^49 - q^50 + q^52 - 14 * q^53 + 4 * q^55 + 4 * q^56 + 9 * q^58 + 8 * q^59 + 10 * q^61 - 10 * q^62 + q^64 - 2 * q^65 - 4 * q^67 + 4 * q^68 - 8 * q^70 - q^71 - 4 * q^73 + 7 * q^74 + 7 * q^76 - 8 * q^77 - 7 * q^79 - 2 * q^80 - 5 * q^82 - 8 * q^85 - 2 * q^86 - 2 * q^88 - 17 * q^89 + 4 * q^91 + q^94 - 14 * q^95 - 8 * q^97 + 9 * q^98

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

1.00000

−2.00000

4.00000

1.00000

−2.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$13$	$-1$

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	702.2.a.k	yes	1
3.b	odd	2	1		702.2.a.g	✓	1
4.b	odd	2	1		5616.2.a.f		1
9.c	even	3	2		2106.2.e.k		2
9.d	odd	6	2		2106.2.e.p		2
12.b	even	2	1		5616.2.a.y		1
13.b	even	2	1		9126.2.a.p		1
39.d	odd	2	1		9126.2.a.v		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
702.2.a.g	✓	1	3.b	odd	2	1
702.2.a.k	yes	1	1.a	even	1	1	trivial
2106.2.e.k		2	9.c	even	3	2
2106.2.e.p		2	9.d	odd	6	2
5616.2.a.f		1	4.b	odd	2	1
5616.2.a.y		1	12.b	even	2	1
9126.2.a.p		1	13.b	even	2	1
9126.2.a.v		1	39.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(702))$ :

T_{5} + 2

T5 + 2

T_{7} - 4

T7 - 4

T_{11} + 2

T11 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 1$ T - 1
$3$	$T$ T
$5$	$T + 2$ T + 2
$7$	$T - 4$ T - 4
$11$	$T + 2$ T + 2
$13$	$T - 1$ T - 1
$17$	$T - 4$ T - 4
$19$	$T - 7$ T - 7
$23$	$T$ T
$29$	$T - 9$ T - 9
$31$	$T + 10$ T + 10
$37$	$T - 7$ T - 7
$41$	$T + 5$ T + 5
$43$	$T + 2$ T + 2
$47$	$T - 1$ T - 1
$53$	$T + 14$ T + 14
$59$	$T - 8$ T - 8
$61$	$T - 10$ T - 10
$67$	$T + 4$ T + 4
$71$	$T + 1$ T + 1
$73$	$T + 4$ T + 4
$79$	$T + 7$ T + 7
$83$	$T$ T
$89$	$T + 17$ T + 17
$97$	$T + 8$ T + 8
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