Properties

Label 6048.2.a.f
Level 6048
Weight 2
Character orbit 6048.a
Self dual yes
Analytic conductor 48.294
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - q^{5} - q^{7} + O(q^{10}) \) \( q - q^{5} - q^{7} - 2q^{11} - q^{17} + 4q^{19} + 4q^{23} - 4q^{25} + 8q^{29} - 4q^{31} + q^{35} - 7q^{37} + 5q^{41} - q^{43} + 9q^{47} + q^{49} - 2q^{53} + 2q^{55} - 3q^{59} - 2q^{61} + 4q^{67} - 12q^{71} - 16q^{73} + 2q^{77} + 15q^{79} + 3q^{83} + q^{85} - 6q^{89} - 4q^{95} - 4q^{97} + O(q^{100}) \)

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

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 6048.2.a.f 1
3.b odd 2 1 6048.2.a.p yes 1
4.b odd 2 1 6048.2.a.i yes 1
12.b even 2 1 6048.2.a.s yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.a.f 1 1.a even 1 1 trivial
6048.2.a.i yes 1 4.b odd 2 1
6048.2.a.p yes 1 3.b odd 2 1
6048.2.a.s yes 1 12.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6048))\):

\( T_{5} + 1 \)
\( T_{11} + 2 \)
\( T_{13} \)
\( T_{17} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + T + 5 T^{2} \)
$7$ \( 1 + T \)
$11$ \( 1 + 2 T + 11 T^{2} \)
$13$ \( 1 + 13 T^{2} \)
$17$ \( 1 + T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 - 8 T + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 + 7 T + 37 T^{2} \)
$41$ \( 1 - 5 T + 41 T^{2} \)
$43$ \( 1 + T + 43 T^{2} \)
$47$ \( 1 - 9 T + 47 T^{2} \)
$53$ \( 1 + 2 T + 53 T^{2} \)
$59$ \( 1 + 3 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 + 12 T + 71 T^{2} \)
$73$ \( 1 + 16 T + 73 T^{2} \)
$79$ \( 1 - 15 T + 79 T^{2} \)
$83$ \( 1 - 3 T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 + 4 T + 97 T^{2} \)
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