Properties

Label 78.2.a.a
Level $78$
Weight $2$
Character orbit 78.a
Self dual yes
Analytic conductor $0.623$
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] = [78,2,Mod(1,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("78.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.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: \(0.622833135766\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} + q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 8 q^{19} + 2 q^{20} - 4 q^{21}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
−1.00000 −1.00000 1.00000 2.00000 1.00000 4.00000 −1.00000 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(13\) \( -1 \)

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 78.2.a.a 1
3.b odd 2 1 234.2.a.c 1
4.b odd 2 1 624.2.a.h 1
5.b even 2 1 1950.2.a.w 1
5.c odd 4 2 1950.2.e.i 2
7.b odd 2 1 3822.2.a.j 1
8.b even 2 1 2496.2.a.t 1
8.d odd 2 1 2496.2.a.b 1
9.c even 3 2 2106.2.e.q 2
9.d odd 6 2 2106.2.e.j 2
11.b odd 2 1 9438.2.a.t 1
12.b even 2 1 1872.2.a.c 1
13.b even 2 1 1014.2.a.d 1
13.c even 3 2 1014.2.e.f 2
13.d odd 4 2 1014.2.b.b 2
13.e even 6 2 1014.2.e.c 2
13.f odd 12 4 1014.2.i.d 4
15.d odd 2 1 5850.2.a.d 1
15.e even 4 2 5850.2.e.bb 2
24.f even 2 1 7488.2.a.bk 1
24.h odd 2 1 7488.2.a.bz 1
39.d odd 2 1 3042.2.a.f 1
39.f even 4 2 3042.2.b.g 2
52.b odd 2 1 8112.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 1.a even 1 1 trivial
234.2.a.c 1 3.b odd 2 1
624.2.a.h 1 4.b odd 2 1
1014.2.a.d 1 13.b even 2 1
1014.2.b.b 2 13.d odd 4 2
1014.2.e.c 2 13.e even 6 2
1014.2.e.f 2 13.c even 3 2
1014.2.i.d 4 13.f odd 12 4
1872.2.a.c 1 12.b even 2 1
1950.2.a.w 1 5.b even 2 1
1950.2.e.i 2 5.c odd 4 2
2106.2.e.j 2 9.d odd 6 2
2106.2.e.q 2 9.c even 3 2
2496.2.a.b 1 8.d odd 2 1
2496.2.a.t 1 8.b even 2 1
3042.2.a.f 1 39.d odd 2 1
3042.2.b.g 2 39.f even 4 2
3822.2.a.j 1 7.b odd 2 1
5850.2.a.d 1 15.d odd 2 1
5850.2.e.bb 2 15.e even 4 2
7488.2.a.bk 1 24.f even 2 1
7488.2.a.bz 1 24.h odd 2 1
8112.2.a.v 1 52.b odd 2 1
9438.2.a.t 1 11.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(78))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 16 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 14 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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