LMFDB - Newform orbit 315.4.a.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(1,315)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$315 = 3^{2} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$18.5856016518$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 35)
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} - 7 q^{4} + 5 q^{5} + 7 q^{7} + 15 q^{8} - 5 q^{10} - 12 q^{11} - 78 q^{13} - 7 q^{14} + 41 q^{16} + 94 q^{17} + 40 q^{19} - 35 q^{20} + 12 q^{22} - 32 q^{23} + 25 q^{25} + 78 q^{26} - 49 q^{28}+ \cdots - 49 q^{98}+O(q^{100})

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

−7.00000

5.00000

7.00000

15.0000

−5.00000

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$5$	$-1$
$7$	$-1$

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	315.4.a.c		1
3.b	odd	2	1		35.4.a.a	✓	1
5.b	even	2	1		1575.4.a.g		1
7.b	odd	2	1		2205.4.a.i		1
12.b	even	2	1		560.4.a.p		1
15.d	odd	2	1		175.4.a.a		1
15.e	even	4	2		175.4.b.a		2
21.c	even	2	1		245.4.a.d		1
21.g	even	6	2		245.4.e.b		2
21.h	odd	6	2		245.4.e.e		2
24.f	even	2	1		2240.4.a.b		1
24.h	odd	2	1		2240.4.a.bk		1
105.g	even	2	1		1225.4.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.4.a.a	✓	1	3.b	odd	2	1
175.4.a.a		1	15.d	odd	2	1
175.4.b.a		2	15.e	even	4	2
245.4.a.d		1	21.c	even	2	1
245.4.e.b		2	21.g	even	6	2
245.4.e.e		2	21.h	odd	6	2
315.4.a.c		1	1.a	even	1	1	trivial
560.4.a.p		1	12.b	even	2	1
1225.4.a.e		1	105.g	even	2	1
1575.4.a.g		1	5.b	even	2	1
2205.4.a.i		1	7.b	odd	2	1
2240.4.a.b		1	24.f	even	2	1
2240.4.a.bk		1	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2} + 1$ T2 + 1 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(315))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 1$ T + 1
$3$	$T$ T
$5$	$T - 5$ T - 5
$7$	$T - 7$ T - 7
$11$	$T + 12$ T + 12
$13$	$T + 78$ T + 78
$17$	$T - 94$ T - 94
$19$	$T - 40$ T - 40
$23$	$T + 32$ T + 32
$29$	$T - 50$ T - 50
$31$	$T + 248$ T + 248
$37$	$T + 434$ T + 434
$41$	$T + 402$ T + 402
$43$	$T + 68$ T + 68
$47$	$T + 536$ T + 536
$53$	$T + 22$ T + 22
$59$	$T - 560$ T - 560
$61$	$T + 278$ T + 278
$67$	$T + 164$ T + 164
$71$	$T + 672$ T + 672
$73$	$T - 82$ T - 82
$79$	$T + 1000$ T + 1000
$83$	$T - 448$ T - 448
$89$	$T - 870$ T - 870
$97$	$T - 1026$ T - 1026
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