Properties

Label 9600.2.a.bb
Level $9600$
Weight $2$
Character orbit 9600.a
Self dual yes
Analytic conductor $76.656$
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] = [9600,2,Mod(1,9600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9600 = 2^{7} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9600.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: \(76.6563859404\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1920)
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)
 
\(f(q)\) \(=\) \( q - q^{3} + 4 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + 4 q^{7} + q^{9} + 6 q^{13} - 2 q^{17} - 6 q^{19} - 4 q^{21} + 6 q^{23} - q^{27} - 8 q^{29} + 8 q^{31} + 10 q^{37} - 6 q^{39} - 6 q^{41} - 4 q^{43} + 2 q^{47} + 9 q^{49} + 2 q^{51} - 6 q^{53} + 6 q^{57} + 12 q^{59} + 14 q^{61} + 4 q^{63} - 4 q^{67} - 6 q^{69} + 8 q^{71} - 4 q^{73} + 12 q^{79} + q^{81} - 8 q^{83} + 8 q^{87} + 6 q^{89} + 24 q^{91} - 8 q^{93}+O(q^{100}) \) 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 −1.00000 0 0 0 4.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)

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 9600.2.a.bb 1
4.b odd 2 1 9600.2.a.bd 1
5.b even 2 1 9600.2.a.bc 1
5.c odd 4 2 1920.2.f.k yes 2
8.b even 2 1 9600.2.a.ca 1
8.d odd 2 1 9600.2.a.c 1
20.d odd 2 1 9600.2.a.ba 1
20.e even 4 2 1920.2.f.j yes 2
40.e odd 2 1 9600.2.a.cb 1
40.f even 2 1 9600.2.a.d 1
40.i odd 4 2 1920.2.f.c yes 2
40.k even 4 2 1920.2.f.b 2
80.i odd 4 2 3840.2.d.z 2
80.j even 4 2 3840.2.d.w 2
80.s even 4 2 3840.2.d.i 2
80.t odd 4 2 3840.2.d.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1920.2.f.b 2 40.k even 4 2
1920.2.f.c yes 2 40.i odd 4 2
1920.2.f.j yes 2 20.e even 4 2
1920.2.f.k yes 2 5.c odd 4 2
3840.2.d.h 2 80.t odd 4 2
3840.2.d.i 2 80.s even 4 2
3840.2.d.w 2 80.j even 4 2
3840.2.d.z 2 80.i odd 4 2
9600.2.a.c 1 8.d odd 2 1
9600.2.a.d 1 40.f even 2 1
9600.2.a.ba 1 20.d odd 2 1
9600.2.a.bb 1 1.a even 1 1 trivial
9600.2.a.bc 1 5.b even 2 1
9600.2.a.bd 1 4.b odd 2 1
9600.2.a.ca 1 8.b even 2 1
9600.2.a.cb 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9600))\):

\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{29} + 8 \) Copy content Toggle raw display
\( T_{31} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 6 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 8 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 2 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 14 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T + 4 \) Copy content Toggle raw display
$79$ \( T - 12 \) Copy content Toggle raw display
$83$ \( T + 8 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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