Properties

Label 1344.2.a.p
Level $1344$
Weight $2$
Character orbit 1344.a
Self dual yes
Analytic conductor $10.732$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - q^{7} + q^{9} + 2q^{11} + 2q^{13} + 4q^{17} + 4q^{19} - q^{21} - 6q^{23} - 5q^{25} + q^{27} + 2q^{29} + 2q^{33} + 6q^{37} + 2q^{39} + 8q^{41} + 8q^{43} - 4q^{47} + q^{49} + 4q^{51} + 6q^{53} + 4q^{57} + 14q^{61} - q^{63} - 4q^{67} - 6q^{69} - 2q^{71} - 2q^{73} - 5q^{75} - 2q^{77} + 4q^{79} + q^{81} - 12q^{83} + 2q^{87} - 2q^{91} + 6q^{97} + 2q^{99} + O(q^{100}) \)

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 1.00000 0 0 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1344.2.a.p 1
3.b odd 2 1 4032.2.a.q 1
4.b odd 2 1 1344.2.a.e 1
7.b odd 2 1 9408.2.a.w 1
8.b even 2 1 672.2.a.c 1
8.d odd 2 1 672.2.a.g yes 1
12.b even 2 1 4032.2.a.x 1
16.e even 4 2 5376.2.c.y 2
16.f odd 4 2 5376.2.c.j 2
24.f even 2 1 2016.2.a.i 1
24.h odd 2 1 2016.2.a.d 1
28.d even 2 1 9408.2.a.ch 1
56.e even 2 1 4704.2.a.k 1
56.h odd 2 1 4704.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.c 1 8.b even 2 1
672.2.a.g yes 1 8.d odd 2 1
1344.2.a.e 1 4.b odd 2 1
1344.2.a.p 1 1.a even 1 1 trivial
2016.2.a.d 1 24.h odd 2 1
2016.2.a.i 1 24.f even 2 1
4032.2.a.q 1 3.b odd 2 1
4032.2.a.x 1 12.b even 2 1
4704.2.a.k 1 56.e even 2 1
4704.2.a.z 1 56.h odd 2 1
5376.2.c.j 2 16.f odd 4 2
5376.2.c.y 2 16.e even 4 2
9408.2.a.w 1 7.b odd 2 1
9408.2.a.ch 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5} \)
\( T_{11} - 2 \)
\( T_{13} - 2 \)
\( T_{19} - 4 \)

Hecke characteristic polynomials

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