Properties

Label 1152.4.a.g
Level $1152$
Weight $4$
Character orbit 1152.a
Self dual yes
Analytic conductor $67.970$
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] = [1152,4,Mod(1,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, 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("1152.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.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: \(67.9702003266\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 384)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{5} - 10 q^{7} - 4 q^{11} - 26 q^{13} - 14 q^{17} - 8 q^{19} + 148 q^{23} - 109 q^{25} + 72 q^{29} - 18 q^{31} - 40 q^{35} - 262 q^{37} + 378 q^{41} + 432 q^{43} + 148 q^{47} - 243 q^{49} + 360 q^{53} - 16 q^{55} - 428 q^{59} + 442 q^{61} - 104 q^{65} + 692 q^{67} + 540 q^{71} - 1018 q^{73} + 40 q^{77} - 386 q^{79} + 108 q^{83} - 56 q^{85} + 382 q^{89} + 260 q^{91} - 32 q^{95} + 298 q^{97}+O(q^{100}) \) 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 4.00000 0 −10.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1152.4.a.g 1
3.b odd 2 1 384.4.a.f yes 1
4.b odd 2 1 1152.4.a.h 1
8.b even 2 1 1152.4.a.e 1
8.d odd 2 1 1152.4.a.f 1
12.b even 2 1 384.4.a.b 1
24.f even 2 1 384.4.a.g yes 1
24.h odd 2 1 384.4.a.c yes 1
48.i odd 4 2 768.4.d.k 2
48.k even 4 2 768.4.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.a.b 1 12.b even 2 1
384.4.a.c yes 1 24.h odd 2 1
384.4.a.f yes 1 3.b odd 2 1
384.4.a.g yes 1 24.f even 2 1
768.4.d.f 2 48.k even 4 2
768.4.d.k 2 48.i odd 4 2
1152.4.a.e 1 8.b even 2 1
1152.4.a.f 1 8.d odd 2 1
1152.4.a.g 1 1.a even 1 1 trivial
1152.4.a.h 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1152))\):

\( T_{5} - 4 \) Copy content Toggle raw display
\( T_{7} + 10 \) Copy content Toggle raw display
\( T_{13} + 26 \) 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
$5$ \( T - 4 \) Copy content Toggle raw display
$7$ \( T + 10 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T + 26 \) Copy content Toggle raw display
$17$ \( T + 14 \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T - 148 \) Copy content Toggle raw display
$29$ \( T - 72 \) Copy content Toggle raw display
$31$ \( T + 18 \) Copy content Toggle raw display
$37$ \( T + 262 \) Copy content Toggle raw display
$41$ \( T - 378 \) Copy content Toggle raw display
$43$ \( T - 432 \) Copy content Toggle raw display
$47$ \( T - 148 \) Copy content Toggle raw display
$53$ \( T - 360 \) Copy content Toggle raw display
$59$ \( T + 428 \) Copy content Toggle raw display
$61$ \( T - 442 \) Copy content Toggle raw display
$67$ \( T - 692 \) Copy content Toggle raw display
$71$ \( T - 540 \) Copy content Toggle raw display
$73$ \( T + 1018 \) Copy content Toggle raw display
$79$ \( T + 386 \) Copy content Toggle raw display
$83$ \( T - 108 \) Copy content Toggle raw display
$89$ \( T - 382 \) Copy content Toggle raw display
$97$ \( T - 298 \) Copy content Toggle raw display
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