Properties

Label 2448.4.a.f
Level $2448$
Weight $4$
Character orbit 2448.a
Self dual yes
Analytic conductor $144.437$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2448,4,Mod(1,2448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2448.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2448.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: \(144.436675694\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
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)
 
\(f(q)\) \(=\) \( q - 6 q^{5} + 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{5} + 28 q^{7} - 24 q^{11} - 58 q^{13} - 17 q^{17} - 116 q^{19} - 60 q^{23} - 89 q^{25} - 30 q^{29} + 172 q^{31} - 168 q^{35} - 58 q^{37} + 342 q^{41} + 148 q^{43} + 288 q^{47} + 441 q^{49} - 318 q^{53} + 144 q^{55} + 252 q^{59} + 110 q^{61} + 348 q^{65} + 484 q^{67} - 708 q^{71} + 362 q^{73} - 672 q^{77} + 484 q^{79} + 756 q^{83} + 102 q^{85} + 774 q^{89} - 1624 q^{91} + 696 q^{95} - 382 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
0 0 0 −6.00000 0 28.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(17\) \(1\)

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 2448.4.a.f 1
3.b odd 2 1 272.4.a.d 1
4.b odd 2 1 153.4.a.d 1
12.b even 2 1 17.4.a.a 1
24.f even 2 1 1088.4.a.l 1
24.h odd 2 1 1088.4.a.a 1
60.h even 2 1 425.4.a.d 1
60.l odd 4 2 425.4.b.c 2
84.h odd 2 1 833.4.a.a 1
132.d odd 2 1 2057.4.a.d 1
204.h even 2 1 289.4.a.a 1
204.l even 4 2 289.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 12.b even 2 1
153.4.a.d 1 4.b odd 2 1
272.4.a.d 1 3.b odd 2 1
289.4.a.a 1 204.h even 2 1
289.4.b.a 2 204.l even 4 2
425.4.a.d 1 60.h even 2 1
425.4.b.c 2 60.l odd 4 2
833.4.a.a 1 84.h odd 2 1
1088.4.a.a 1 24.h odd 2 1
1088.4.a.l 1 24.f even 2 1
2057.4.a.d 1 132.d odd 2 1
2448.4.a.f 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2448))\):

\( T_{5} + 6 \) Copy content Toggle raw display
\( T_{7} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T - 28 \) Copy content Toggle raw display
$11$ \( T + 24 \) Copy content Toggle raw display
$13$ \( T + 58 \) Copy content Toggle raw display
$17$ \( T + 17 \) Copy content Toggle raw display
$19$ \( T + 116 \) Copy content Toggle raw display
$23$ \( T + 60 \) Copy content Toggle raw display
$29$ \( T + 30 \) Copy content Toggle raw display
$31$ \( T - 172 \) Copy content Toggle raw display
$37$ \( T + 58 \) Copy content Toggle raw display
$41$ \( T - 342 \) Copy content Toggle raw display
$43$ \( T - 148 \) Copy content Toggle raw display
$47$ \( T - 288 \) Copy content Toggle raw display
$53$ \( T + 318 \) Copy content Toggle raw display
$59$ \( T - 252 \) Copy content Toggle raw display
$61$ \( T - 110 \) Copy content Toggle raw display
$67$ \( T - 484 \) Copy content Toggle raw display
$71$ \( T + 708 \) Copy content Toggle raw display
$73$ \( T - 362 \) Copy content Toggle raw display
$79$ \( T - 484 \) Copy content Toggle raw display
$83$ \( T - 756 \) Copy content Toggle raw display
$89$ \( T - 774 \) Copy content Toggle raw display
$97$ \( T + 382 \) Copy content Toggle raw display
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