Properties

Label 288.2.a.d
Level $288$
Weight $2$
Character orbit 288.a
Self dual yes
Analytic conductor $2.300$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,2,Mod(1,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("288.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.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: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{5} + 6 q^{13} - 2 q^{17} - q^{25} + 10 q^{29} - 2 q^{37} - 10 q^{41} - 7 q^{49} - 14 q^{53} - 10 q^{61} + 12 q^{65} - 6 q^{73} - 4 q^{85} - 10 q^{89} + 18 q^{97}+O(q^{100}) \) 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 2.00000 0 0 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.a.d 1
3.b odd 2 1 32.2.a.a 1
4.b odd 2 1 CM 288.2.a.d 1
5.b even 2 1 7200.2.a.v 1
5.c odd 4 2 7200.2.f.m 2
8.b even 2 1 576.2.a.c 1
8.d odd 2 1 576.2.a.c 1
9.c even 3 2 2592.2.i.e 2
9.d odd 6 2 2592.2.i.t 2
12.b even 2 1 32.2.a.a 1
15.d odd 2 1 800.2.a.d 1
15.e even 4 2 800.2.c.e 2
16.e even 4 2 2304.2.d.j 2
16.f odd 4 2 2304.2.d.j 2
20.d odd 2 1 7200.2.a.v 1
20.e even 4 2 7200.2.f.m 2
21.c even 2 1 1568.2.a.e 1
21.g even 6 2 1568.2.i.f 2
21.h odd 6 2 1568.2.i.g 2
24.f even 2 1 64.2.a.a 1
24.h odd 2 1 64.2.a.a 1
33.d even 2 1 3872.2.a.f 1
36.f odd 6 2 2592.2.i.e 2
36.h even 6 2 2592.2.i.t 2
39.d odd 2 1 5408.2.a.g 1
48.i odd 4 2 256.2.b.b 2
48.k even 4 2 256.2.b.b 2
51.c odd 2 1 9248.2.a.f 1
60.h even 2 1 800.2.a.d 1
60.l odd 4 2 800.2.c.e 2
84.h odd 2 1 1568.2.a.e 1
84.j odd 6 2 1568.2.i.f 2
84.n even 6 2 1568.2.i.g 2
96.o even 8 4 1024.2.e.j 4
96.p odd 8 4 1024.2.e.j 4
120.i odd 2 1 1600.2.a.n 1
120.m even 2 1 1600.2.a.n 1
120.q odd 4 2 1600.2.c.l 2
120.w even 4 2 1600.2.c.l 2
132.d odd 2 1 3872.2.a.f 1
156.h even 2 1 5408.2.a.g 1
168.e odd 2 1 3136.2.a.m 1
168.i even 2 1 3136.2.a.m 1
204.h even 2 1 9248.2.a.f 1
264.m even 2 1 7744.2.a.v 1
264.p odd 2 1 7744.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 3.b odd 2 1
32.2.a.a 1 12.b even 2 1
64.2.a.a 1 24.f even 2 1
64.2.a.a 1 24.h odd 2 1
256.2.b.b 2 48.i odd 4 2
256.2.b.b 2 48.k even 4 2
288.2.a.d 1 1.a even 1 1 trivial
288.2.a.d 1 4.b odd 2 1 CM
576.2.a.c 1 8.b even 2 1
576.2.a.c 1 8.d odd 2 1
800.2.a.d 1 15.d odd 2 1
800.2.a.d 1 60.h even 2 1
800.2.c.e 2 15.e even 4 2
800.2.c.e 2 60.l odd 4 2
1024.2.e.j 4 96.o even 8 4
1024.2.e.j 4 96.p odd 8 4
1568.2.a.e 1 21.c even 2 1
1568.2.a.e 1 84.h odd 2 1
1568.2.i.f 2 21.g even 6 2
1568.2.i.f 2 84.j odd 6 2
1568.2.i.g 2 21.h odd 6 2
1568.2.i.g 2 84.n even 6 2
1600.2.a.n 1 120.i odd 2 1
1600.2.a.n 1 120.m even 2 1
1600.2.c.l 2 120.q odd 4 2
1600.2.c.l 2 120.w even 4 2
2304.2.d.j 2 16.e even 4 2
2304.2.d.j 2 16.f odd 4 2
2592.2.i.e 2 9.c even 3 2
2592.2.i.e 2 36.f odd 6 2
2592.2.i.t 2 9.d odd 6 2
2592.2.i.t 2 36.h even 6 2
3136.2.a.m 1 168.e odd 2 1
3136.2.a.m 1 168.i even 2 1
3872.2.a.f 1 33.d even 2 1
3872.2.a.f 1 132.d odd 2 1
5408.2.a.g 1 39.d odd 2 1
5408.2.a.g 1 156.h even 2 1
7200.2.a.v 1 5.b even 2 1
7200.2.a.v 1 20.d odd 2 1
7200.2.f.m 2 5.c odd 4 2
7200.2.f.m 2 20.e even 4 2
7744.2.a.v 1 264.m even 2 1
7744.2.a.v 1 264.p odd 2 1
9248.2.a.f 1 51.c odd 2 1
9248.2.a.f 1 204.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(288))\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} \) 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
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T \) 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
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 10 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 14 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 10 \) Copy content Toggle raw display
$97$ \( T - 18 \) Copy content Toggle raw display
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