Properties

Label 1368.2.a.j
Level $1368$
Weight $2$
Character orbit 1368.a
Self dual yes
Analytic conductor $10.924$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.9235349965\)
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

\(f(q)\) \(=\) \( q + 4 q^{5} + O(q^{10}) \) \( q + 4 q^{5} - 6 q^{11} + 2 q^{13} + 4 q^{17} - q^{19} + 6 q^{23} + 11 q^{25} + 10 q^{29} + 8 q^{31} - 10 q^{37} - 6 q^{41} - 4 q^{43} + 6 q^{47} - 7 q^{49} + 2 q^{53} - 24 q^{55} + 4 q^{59} + 10 q^{61} + 8 q^{65} - 12 q^{67} + 12 q^{71} - 6 q^{73} - 4 q^{79} + 14 q^{83} + 16 q^{85} - 6 q^{89} - 4 q^{95} - 2 q^{97} + O(q^{100}) \)

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.

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 4.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(1\)

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 1368.2.a.j yes 1
3.b odd 2 1 1368.2.a.a 1
4.b odd 2 1 2736.2.a.x 1
12.b even 2 1 2736.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.a.a 1 3.b odd 2 1
1368.2.a.j yes 1 1.a even 1 1 trivial
2736.2.a.b 1 12.b even 2 1
2736.2.a.x 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1368))\):

\( T_{5} - 4 \)
\( T_{7} \)
\( T_{11} + 6 \)

Hecke characteristic polynomials

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