Properties

Label 378.2.a.d
Level $378$
Weight $2$
Character orbit 378.a
Self dual yes
Analytic conductor $3.018$
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] = [378,2,Mod(1,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, 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("378.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.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: \(3.01834519640\)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{7} - q^{8} + 5 q^{13} - q^{14} + q^{16} - 3 q^{17} + 2 q^{19} + 9 q^{23} - 5 q^{25} - 5 q^{26} + q^{28} + 3 q^{29} + 5 q^{31} - q^{32} + 3 q^{34} + 2 q^{37} - 2 q^{38} + 6 q^{41} - q^{43} - 9 q^{46} + 6 q^{47} + q^{49} + 5 q^{50} + 5 q^{52} - 3 q^{53} - q^{56} - 3 q^{58} + 3 q^{59} - 10 q^{61} - 5 q^{62} + q^{64} - 13 q^{67} - 3 q^{68} - 9 q^{71} + 2 q^{73} - 2 q^{74} + 2 q^{76} - 10 q^{79} - 6 q^{82} + 12 q^{83} + q^{86} - 15 q^{89} + 5 q^{91} + 9 q^{92} - 6 q^{94} + 8 q^{97} - q^{98}+O(q^{100}) \) 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 0 1.00000 0 0 1.00000 −1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)

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 378.2.a.d 1
3.b odd 2 1 378.2.a.e yes 1
4.b odd 2 1 3024.2.a.o 1
5.b even 2 1 9450.2.a.cl 1
7.b odd 2 1 2646.2.a.f 1
9.c even 3 2 1134.2.f.k 2
9.d odd 6 2 1134.2.f.e 2
12.b even 2 1 3024.2.a.p 1
15.d odd 2 1 9450.2.a.l 1
21.c even 2 1 2646.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.d 1 1.a even 1 1 trivial
378.2.a.e yes 1 3.b odd 2 1
1134.2.f.e 2 9.d odd 6 2
1134.2.f.k 2 9.c even 3 2
2646.2.a.f 1 7.b odd 2 1
2646.2.a.y 1 21.c even 2 1
3024.2.a.o 1 4.b odd 2 1
3024.2.a.p 1 12.b even 2 1
9450.2.a.l 1 15.d odd 2 1
9450.2.a.cl 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(378))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{17} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 5 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 9 \) Copy content Toggle raw display
$29$ \( T - 3 \) Copy content Toggle raw display
$31$ \( T - 5 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T + 3 \) Copy content Toggle raw display
$59$ \( T - 3 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T + 13 \) Copy content Toggle raw display
$71$ \( T + 9 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 15 \) Copy content Toggle raw display
$97$ \( T - 8 \) Copy content Toggle raw display
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