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Properties

Label 768.2.a.e
Level $768$
Weight $2$
Character orbit 768.a
Self dual yes
Analytic conductor $6.133$
Analytic rank $1$
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] = [768,2,Mod(1,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, 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("768.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(6.13251087523\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
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)
 
\(f(q)\) \(=\) \( q + q^{3} - 2 q^{5} - 2 q^{7} + q^{9} - 4 q^{13} - 2 q^{15} - 2 q^{17} - 4 q^{19} - 2 q^{21} + 4 q^{23} - q^{25} + q^{27} - 6 q^{29} - 2 q^{31} + 4 q^{35} - 8 q^{37} - 4 q^{39} - 2 q^{41} - 4 q^{43}+ \cdots - 2 q^{97}+O(q^{100}) \) 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 1.00000 0 −2.00000 0 −2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 768.2.a.e 1
3.b odd 2 1 2304.2.a.l 1
4.b odd 2 1 768.2.a.a 1
8.b even 2 1 768.2.a.d 1
8.d odd 2 1 768.2.a.h 1
12.b even 2 1 2304.2.a.o 1
16.e even 4 2 96.2.d.a 2
16.f odd 4 2 24.2.d.a ✓ 2
24.f even 2 1 2304.2.a.e 1
24.h odd 2 1 2304.2.a.b 1
48.i odd 4 2 288.2.d.b 2
48.k even 4 2 72.2.d.b 2
80.i odd 4 2 2400.2.d.b 2
80.j even 4 2 600.2.d.c 2
80.k odd 4 2 600.2.k.b 2
80.q even 4 2 2400.2.k.a 2
80.s even 4 2 600.2.d.b 2
80.t odd 4 2 2400.2.d.c 2
112.j even 4 2 1176.2.c.a 2
112.l odd 4 2 4704.2.c.a 2
144.u even 12 4 648.2.n.c 4
144.v odd 12 4 648.2.n.k 4
144.w odd 12 4 2592.2.r.g 4
144.x even 12 4 2592.2.r.f 4
240.t even 4 2 1800.2.k.a 2
240.z odd 4 2 1800.2.d.i 2
240.bb even 4 2 7200.2.d.g 2
240.bd odd 4 2 1800.2.d.b 2
240.bf even 4 2 7200.2.d.d 2
240.bm odd 4 2 7200.2.k.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a ✓ 2 16.f odd 4 2
72.2.d.b 2 48.k even 4 2
96.2.d.a 2 16.e even 4 2
288.2.d.b 2 48.i odd 4 2
600.2.d.b 2 80.s even 4 2
600.2.d.c 2 80.j even 4 2
600.2.k.b 2 80.k odd 4 2
648.2.n.c 4 144.u even 12 4
648.2.n.k 4 144.v odd 12 4
768.2.a.a 1 4.b odd 2 1
768.2.a.d 1 8.b even 2 1
768.2.a.e 1 1.a even 1 1 trivial
768.2.a.h 1 8.d odd 2 1
1176.2.c.a 2 112.j even 4 2
1800.2.d.b 2 240.bd odd 4 2
1800.2.d.i 2 240.z odd 4 2
1800.2.k.a 2 240.t even 4 2
2304.2.a.b 1 24.h odd 2 1
2304.2.a.e 1 24.f even 2 1
2304.2.a.l 1 3.b odd 2 1
2304.2.a.o 1 12.b even 2 1
2400.2.d.b 2 80.i odd 4 2
2400.2.d.c 2 80.t odd 4 2
2400.2.k.a 2 80.q even 4 2
2592.2.r.f 4 144.x even 12 4
2592.2.r.g 4 144.w odd 12 4
4704.2.c.a 2 112.l odd 4 2
7200.2.d.d 2 240.bf even 4 2
7200.2.d.g 2 240.bb even 4 2
7200.2.k.d 2 240.bm odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(768))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{19} + 4 \) 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
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T + 8 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T + 16 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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