Properties

Label 3744.2.a.p
Level $3744$
Weight $2$
Character orbit 3744.a
Self dual yes
Analytic conductor $29.896$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1248)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} + 2q^{7} + O(q^{10}) \) \( q + 2q^{5} + 2q^{7} + 6q^{11} - q^{13} + 2q^{17} - 6q^{19} - q^{25} + 6q^{29} + 6q^{31} + 4q^{35} + 2q^{37} + 10q^{41} + 8q^{43} - 6q^{47} - 3q^{49} - 6q^{53} + 12q^{55} + 6q^{59} - 10q^{61} - 2q^{65} + 2q^{67} + 14q^{71} - 14q^{73} + 12q^{77} + 4q^{79} - 6q^{83} + 4q^{85} - 6q^{89} - 2q^{91} - 12q^{95} - 14q^{97} + O(q^{100}) \)

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 2.00000 0 2.00000 0 0 0
\(n\): 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 3744.2.a.p 1
3.b odd 2 1 1248.2.a.g yes 1
4.b odd 2 1 3744.2.a.k 1
8.b even 2 1 7488.2.a.q 1
8.d odd 2 1 7488.2.a.l 1
12.b even 2 1 1248.2.a.a 1
24.f even 2 1 2496.2.a.z 1
24.h odd 2 1 2496.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.a.a 1 12.b even 2 1
1248.2.a.g yes 1 3.b odd 2 1
2496.2.a.m 1 24.h odd 2 1
2496.2.a.z 1 24.f even 2 1
3744.2.a.k 1 4.b odd 2 1
3744.2.a.p 1 1.a even 1 1 trivial
7488.2.a.l 1 8.d odd 2 1
7488.2.a.q 1 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3744))\):

\( T_{5} - 2 \)
\( T_{7} - 2 \)
\( T_{11} - 6 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

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