LMFDB - Newform orbit 352.2.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [352,2,Mod(1,352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("352.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$352 = 2^{5} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	352.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.81073415115$
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 + 3 q^{3} + q^{5} + 6 q^{9} - q^{11} - 6 q^{13} + 3 q^{15} - 4 q^{17} + 6 q^{19} + 3 q^{23} - 4 q^{25} + 9 q^{27} - 4 q^{29} - 9 q^{31} - 3 q^{33} + 7 q^{37} - 18 q^{39} - 2 q^{41} + 6 q^{43} + 6 q^{45}+ \cdots - 6 q^{99}+O(q^{100})

q + 3 * q^3 + q^5 + 6 * q^9 - q^11 - 6 * q^13 + 3 * q^15 - 4 * q^17 + 6 * q^19 + 3 * q^23 - 4 * q^25 + 9 * q^27 - 4 * q^29 - 9 * q^31 - 3 * q^33 + 7 * q^37 - 18 * q^39 - 2 * q^41 + 6 * q^43 + 6 * q^45 + 12 * q^47 - 7 * q^49 - 12 * q^51 + 2 * q^53 - q^55 + 18 * q^57 + 9 * q^59 + 8 * q^61 - 6 * q^65 - 15 * q^67 + 9 * q^69 - 3 * q^71 - 6 * q^73 - 12 * q^75 - 6 * q^79 + 9 * q^81 - 6 * q^83 - 4 * q^85 - 12 * q^87 - 5 * q^89 - 27 * q^93 + 6 * q^95 - 3 * q^97 - 6 * 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

3.00000

1.00000

6.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$11$	$+1$

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	352.2.a.f	yes	1
3.b	odd	2	1		3168.2.a.i		1
4.b	odd	2	1		352.2.a.a	✓	1
5.b	even	2	1		8800.2.a.a		1
8.b	even	2	1		704.2.a.a		1
8.d	odd	2	1		704.2.a.k		1
11.b	odd	2	1		3872.2.a.m		1
12.b	even	2	1		3168.2.a.h		1
16.e	even	4	2		2816.2.c.h		2
16.f	odd	4	2		2816.2.c.g		2
20.d	odd	2	1		8800.2.a.bb		1
24.f	even	2	1		6336.2.a.bt		1
24.h	odd	2	1		6336.2.a.bs		1
44.c	even	2	1		3872.2.a.a		1
88.b	odd	2	1		7744.2.a.a		1
88.g	even	2	1		7744.2.a.bj		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
352.2.a.a	✓	1	4.b	odd	2	1
352.2.a.f	yes	1	1.a	even	1	1	trivial
704.2.a.a		1	8.b	even	2	1
704.2.a.k		1	8.d	odd	2	1
2816.2.c.g		2	16.f	odd	4	2
2816.2.c.h		2	16.e	even	4	2
3168.2.a.h		1	12.b	even	2	1
3168.2.a.i		1	3.b	odd	2	1
3872.2.a.a		1	44.c	even	2	1
3872.2.a.m		1	11.b	odd	2	1
6336.2.a.bs		1	24.h	odd	2	1
6336.2.a.bt		1	24.f	even	2	1
7744.2.a.a		1	88.b	odd	2	1
7744.2.a.bj		1	88.g	even	2	1
8800.2.a.a		1	5.b	even	2	1
8800.2.a.bb		1	20.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 $S_{2}^{\mathrm{new}}(\Gamma_0(352))$ :

T_{3} - 3

T3 - 3

T_{5} - 1

T5 - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 3$ T - 3
$5$	$T - 1$ T - 1
$7$	$T$ T
$11$	$T + 1$ T + 1
$13$	$T + 6$ T + 6
$17$	$T + 4$ T + 4
$19$	$T - 6$ T - 6
$23$	$T - 3$ T - 3
$29$	$T + 4$ T + 4
$31$	$T + 9$ T + 9
$37$	$T - 7$ T - 7
$41$	$T + 2$ T + 2
$43$	$T - 6$ T - 6
$47$	$T - 12$ T - 12
$53$	$T - 2$ T - 2
$59$	$T - 9$ T - 9
$61$	$T - 8$ T - 8
$67$	$T + 15$ T + 15
$71$	$T + 3$ T + 3
$73$	$T + 6$ T + 6
$79$	$T + 6$ T + 6
$83$	$T + 6$ T + 6
$89$	$T + 5$ T + 5
$97$	$T + 3$ T + 3
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