LMFDB - Newform orbit 2352.4.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2352,4,Mod(1,2352)] 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("2352.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$2352 = 2^{4} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2352.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,-18,0,0,0,9,0,72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$138.772492334$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
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} - 18 q^{5} + 9 q^{9} + 72 q^{11} + 34 q^{13} + 54 q^{15} - 6 q^{17} + 92 q^{19} + 180 q^{23} + 199 q^{25} - 27 q^{27} - 114 q^{29} + 56 q^{31} - 216 q^{33} - 34 q^{37} - 102 q^{39} - 6 q^{41}+ \cdots + 648 q^{99}+O(q^{100})

q - 3 * q^3 - 18 * q^5 + 9 * q^9 + 72 * q^11 + 34 * q^13 + 54 * q^15 - 6 * q^17 + 92 * q^19 + 180 * q^23 + 199 * q^25 - 27 * q^27 - 114 * q^29 + 56 * q^31 - 216 * q^33 - 34 * q^37 - 102 * q^39 - 6 * q^41 - 164 * q^43 - 162 * q^45 + 168 * q^47 + 18 * q^51 + 654 * q^53 - 1296 * q^55 - 276 * q^57 - 492 * q^59 + 250 * q^61 - 612 * q^65 + 124 * q^67 - 540 * q^69 - 36 * q^71 - 1010 * q^73 - 597 * q^75 - 56 * q^79 + 81 * q^81 + 228 * q^83 + 108 * q^85 + 342 * q^87 - 390 * q^89 - 168 * q^93 - 1656 * q^95 + 70 * q^97 + 648 * 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

−18.0000

9.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$7$	$-1$

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	2352.4.a.a		1
4.b	odd	2	1		294.4.a.i		1
7.b	odd	2	1		336.4.a.l		1
12.b	even	2	1		882.4.a.g		1
21.c	even	2	1		1008.4.a.b		1
28.d	even	2	1		42.4.a.a	✓	1
28.f	even	6	2		294.4.e.c		2
28.g	odd	6	2		294.4.e.b		2
56.e	even	2	1		1344.4.a.o		1
56.h	odd	2	1		1344.4.a.a		1
84.h	odd	2	1		126.4.a.a		1
84.j	odd	6	2		882.4.g.w		2
84.n	even	6	2		882.4.g.o		2
140.c	even	2	1		1050.4.a.g		1
140.j	odd	4	2		1050.4.g.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.a.a	✓	1	28.d	even	2	1
126.4.a.a		1	84.h	odd	2	1
294.4.a.i		1	4.b	odd	2	1
294.4.e.b		2	28.g	odd	6	2
294.4.e.c		2	28.f	even	6	2
336.4.a.l		1	7.b	odd	2	1
882.4.a.g		1	12.b	even	2	1
882.4.g.o		2	84.n	even	6	2
882.4.g.w		2	84.j	odd	6	2
1008.4.a.b		1	21.c	even	2	1
1050.4.a.g		1	140.c	even	2	1
1050.4.g.a		2	140.j	odd	4	2
1344.4.a.a		1	56.h	odd	2	1
1344.4.a.o		1	56.e	even	2	1
2352.4.a.a		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2352))$ :

T_{5} + 18

T5 + 18

T_{11} - 72

T11 - 72

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 3$ T + 3
$5$	$T + 18$ T + 18
$7$	$T$ T
$11$	$T - 72$ T - 72
$13$	$T - 34$ T - 34
$17$	$T + 6$ T + 6
$19$	$T - 92$ T - 92
$23$	$T - 180$ T - 180
$29$	$T + 114$ T + 114
$31$	$T - 56$ T - 56
$37$	$T + 34$ T + 34
$41$	$T + 6$ T + 6
$43$	$T + 164$ T + 164
$47$	$T - 168$ T - 168
$53$	$T - 654$ T - 654
$59$	$T + 492$ T + 492
$61$	$T - 250$ T - 250
$67$	$T - 124$ T - 124
$71$	$T + 36$ T + 36
$73$	$T + 1010$ T + 1010
$79$	$T + 56$ T + 56
$83$	$T - 228$ T - 228
$89$	$T + 390$ T + 390
$97$	$T - 70$ T - 70
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