LMFDB - Newform orbit 2352.4.a.ba

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,-2,0,0,0,9,0,8] 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:	$1$
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} - 2 q^{5} + 9 q^{9} + 8 q^{11} + 42 q^{13} - 6 q^{15} + 2 q^{17} - 124 q^{19} - 76 q^{23} - 121 q^{25} + 27 q^{27} + 254 q^{29} - 72 q^{31} + 24 q^{33} + 398 q^{37} + 126 q^{39} - 462 q^{41}+ \cdots + 72 q^{99}+O(q^{100})

q + 3 * q^3 - 2 * q^5 + 9 * q^9 + 8 * q^11 + 42 * q^13 - 6 * q^15 + 2 * q^17 - 124 * q^19 - 76 * q^23 - 121 * q^25 + 27 * q^27 + 254 * q^29 - 72 * q^31 + 24 * q^33 + 398 * q^37 + 126 * q^39 - 462 * q^41 - 212 * q^43 - 18 * q^45 - 264 * q^47 + 6 * q^51 - 162 * q^53 - 16 * q^55 - 372 * q^57 - 772 * q^59 - 30 * q^61 - 84 * q^65 + 764 * q^67 - 228 * q^69 + 236 * q^71 - 418 * q^73 - 363 * q^75 - 552 * q^79 + 81 * q^81 + 1036 * q^83 - 4 * q^85 + 762 * q^87 - 30 * q^89 - 216 * q^93 + 248 * q^95 + 1190 * q^97 + 72 * 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

−2.00000

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.ba		1
4.b	odd	2	1		294.4.a.h		1
7.b	odd	2	1		336.4.a.d		1
12.b	even	2	1		882.4.a.d		1
21.c	even	2	1		1008.4.a.j		1
28.d	even	2	1		42.4.a.b	✓	1
28.f	even	6	2		294.4.e.a		2
28.g	odd	6	2		294.4.e.d		2
56.e	even	2	1		1344.4.a.f		1
56.h	odd	2	1		1344.4.a.t		1
84.h	odd	2	1		126.4.a.c		1
84.j	odd	6	2		882.4.g.s		2
84.n	even	6	2		882.4.g.r		2
140.c	even	2	1		1050.4.a.d		1
140.j	odd	4	2		1050.4.g.n		2

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

T5 + 2

T_{11} - 8

T11 - 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 3$ T - 3
$5$	$T + 2$ T + 2
$7$	$T$ T
$11$	$T - 8$ T - 8
$13$	$T - 42$ T - 42
$17$	$T - 2$ T - 2
$19$	$T + 124$ T + 124
$23$	$T + 76$ T + 76
$29$	$T - 254$ T - 254
$31$	$T + 72$ T + 72
$37$	$T - 398$ T - 398
$41$	$T + 462$ T + 462
$43$	$T + 212$ T + 212
$47$	$T + 264$ T + 264
$53$	$T + 162$ T + 162
$59$	$T + 772$ T + 772
$61$	$T + 30$ T + 30
$67$	$T - 764$ T - 764
$71$	$T - 236$ T - 236
$73$	$T + 418$ T + 418
$79$	$T + 552$ T + 552
$83$	$T - 1036$ T - 1036
$89$	$T + 30$ T + 30
$97$	$T - 1190$ T - 1190
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