LMFDB - Newform orbit 2304.2.a.z

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2304,2,Mod(1,2304)] 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("2304.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1152)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{5} + \beta_1 q^{7} + \beta_{3} q^{11} + 7 q^{25} + 3 \beta_{2} q^{29} + \beta_1 q^{31} - 3 \beta_{3} q^{35} + 17 q^{49} - \beta_{2} q^{53} - 4 \beta_1 q^{55} + 2 \beta_{3} q^{59} + 14 q^{73}+ \cdots + 2 q^{97}+O(q^{100}) $ q + b2 * q^5 + b1 * q^7 + b3 * q^11 + 7 * q^25 + 3b2 q^29 + b1 * q^31 - 3b3 q^35 + 17 * q^49 - b2 * q^53 - 4b1 q^55 + 2b3 q^59 + 14 * q^73 - 8b2 q^77 - 3b1 q^79 - b3 * q^83 + 2 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{25} + 68 q^{49} + 56 q^{73} + 8 q^{97}+O(q^{100}) $ 4 * q + 28 * q^25 + 68 * q^49 + 56 * q^73 + 8 * q^97

Basis of coefficient ring in terms of $\nu = \zeta_{24} + \zeta_{24}^{-1}$:

$\beta_{1}$	$=$	$ 2\nu^{3} - 10\nu $ 2v^3 - 10v
$\beta_{2}$	$=$	$ 2\nu^{2} - 4 $ 2*v^2 - 4
$\beta_{3}$	$=$	$ 4\nu^{3} - 12\nu $ 4v^3 - 12v

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{3} - 2\beta_1 ) / 8 $ (b3 - 2*b1) / 8
$\nu^{2}$	$=$	$ ( \beta_{2} + 4 ) / 2 $ (b2 + 4) / 2
$\nu^{3}$	$=$	$ ( 5\beta_{3} - 6\beta_1 ) / 8 $ (5b3 - 6b1) / 8

Display $\beta_i$ in terms of $\nu^j$

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

0.517638
−0.517638
1.93185
−1.93185

−3.46410

−4.89898

1.2

−3.46410

4.89898

1.3

3.46410

−4.89898

1.4

3.46410

4.89898

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by $\Q(\sqrt{-6}) $
3.b	odd	2	1	inner
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
12.b	even	2	1	inner
24.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2304.2.a.z		4
3.b	odd	2	1	inner	2304.2.a.z		4
4.b	odd	2	1	inner	2304.2.a.z		4
8.b	even	2	1	inner	2304.2.a.z		4
8.d	odd	2	1	inner	2304.2.a.z		4
12.b	even	2	1	inner	2304.2.a.z		4
16.e	even	4	2		1152.2.d.g	✓	4
16.f	odd	4	2		1152.2.d.g	✓	4
24.f	even	2	1	inner	2304.2.a.z		4
24.h	odd	2	1	CM	2304.2.a.z		4
48.i	odd	4	2		1152.2.d.g	✓	4
48.k	even	4	2		1152.2.d.g	✓	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1152.2.d.g	✓	4	16.e	even	4	2
1152.2.d.g	✓	4	16.f	odd	4	2
1152.2.d.g	✓	4	48.i	odd	4	2
1152.2.d.g	✓	4	48.k	even	4	2
2304.2.a.z		4	1.a	even	1	1	trivial
2304.2.a.z		4	3.b	odd	2	1	inner
2304.2.a.z		4	4.b	odd	2	1	inner
2304.2.a.z		4	8.b	even	2	1	inner
2304.2.a.z		4	8.d	odd	2	1	inner
2304.2.a.z		4	12.b	even	2	1	inner
2304.2.a.z		4	24.f	even	2	1	inner
2304.2.a.z		4	24.h	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2304))$:

$ T_{5}^{2} - 12 $

T5^2 - 12

$ T_{7}^{2} - 24 $

T7^2 - 24

$ T_{11}^{2} - 32 $

T11^2 - 32

$ T_{13} $

T13

$ T_{19} $

T19

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ (T^{2} - 12)^{2} $ (T^2 - 12)^2
$7$	$ (T^{2} - 24)^{2} $ (T^2 - 24)^2
$11$	$ (T^{2} - 32)^{2} $ (T^2 - 32)^2
$13$	$ T^{4} $ T^4
$17$	$ T^{4} $ T^4
$19$	$ T^{4} $ T^4
$23$	$ T^{4} $ T^4
$29$	$ (T^{2} - 108)^{2} $ (T^2 - 108)^2
$31$	$ (T^{2} - 24)^{2} $ (T^2 - 24)^2
$37$	$ T^{4} $ T^4
$41$	$ T^{4} $ T^4
$43$	$ T^{4} $ T^4
$47$	$ T^{4} $ T^4
$53$	$ (T^{2} - 12)^{2} $ (T^2 - 12)^2
$59$	$ (T^{2} - 128)^{2} $ (T^2 - 128)^2
$61$	$ T^{4} $ T^4
$67$	$ T^{4} $ T^4
$71$	$ T^{4} $ T^4
$73$	$ (T - 14)^{4} $ (T - 14)^4
$79$	$ (T^{2} - 216)^{2} $ (T^2 - 216)^2
$83$	$ (T^{2} - 32)^{2} $ (T^2 - 32)^2
$89$	$ T^{4} $ T^4
$97$	$ (T - 2)^{4} $ (T - 2)^4
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Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{5} + \beta_1 q^{7} + \beta_{3} q^{11} + 7 q^{25} + 3 \beta_{2} q^{29} + \beta_1 q^{31} - 3 \beta_{3} q^{35} + 17 q^{49} - \beta_{2} q^{53} - 4 \beta_1 q^{55} + 2 \beta_{3} q^{59} + 14 q^{73}+ \cdots + 2 q^{97}+O(q^{100}) \) q + b2 * q^5 + b1 * q^7 + b3 * q^11 + 7 * q^25 + 3b2 q^29 + b1 * q^31 - 3b3 q^35 + 17 * q^49 - b2 * q^53 - 4b1 q^55 + 2b3 q^59 + 14 * q^73 - 8b2 q^77 - 3b1 q^79 - b3 * q^83 + 2 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{25} + 68 q^{49} + 56 q^{73} + 8 q^{97}+O(q^{100}) \) 4 * q + 28 * q^25 + 68 * q^49 + 56 * q^73 + 8 * q^97

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