Properties

Label 4864.2.a.k
Level $4864$
Weight $2$
Character orbit 4864.a
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 2q^{5} + q^{7} - 2q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{5} + q^{7} - 2q^{9} + 4q^{11} - q^{13} - 2q^{15} - 5q^{17} + q^{19} + q^{21} + 9q^{23} - q^{25} - 5q^{27} - 5q^{29} - 2q^{31} + 4q^{33} - 2q^{35} - 10q^{37} - q^{39} + 12q^{41} + 10q^{43} + 4q^{45} - 12q^{47} - 6q^{49} - 5q^{51} - 9q^{53} - 8q^{55} + q^{57} + 9q^{59} + 2q^{61} - 2q^{63} + 2q^{65} + 13q^{67} + 9q^{69} - 10q^{71} - 7q^{73} - q^{75} + 4q^{77} - 10q^{79} + q^{81} - 8q^{83} + 10q^{85} - 5q^{87} - 6q^{89} - q^{91} - 2q^{93} - 2q^{95} - 6q^{97} - 8q^{99} + O(q^{100}) \)

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 −2.00000 0 1.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4864.2.a.k 1
4.b odd 2 1 4864.2.a.c 1
8.b even 2 1 4864.2.a.f 1
8.d odd 2 1 4864.2.a.n 1
16.e even 4 2 2432.2.c.a 2
16.f odd 4 2 2432.2.c.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2432.2.c.a 2 16.e even 4 2
2432.2.c.b yes 2 16.f odd 4 2
4864.2.a.c 1 4.b odd 2 1
4864.2.a.f 1 8.b even 2 1
4864.2.a.k 1 1.a even 1 1 trivial
4864.2.a.n 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4864))\):

\( T_{3} - 1 \)
\( T_{5} + 2 \)
\( T_{7} - 1 \)
\( T_{11} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( 2 + T \)
$7$ \( -1 + T \)
$11$ \( -4 + T \)
$13$ \( 1 + T \)
$17$ \( 5 + T \)
$19$ \( -1 + T \)
$23$ \( -9 + T \)
$29$ \( 5 + T \)
$31$ \( 2 + T \)
$37$ \( 10 + T \)
$41$ \( -12 + T \)
$43$ \( -10 + T \)
$47$ \( 12 + T \)
$53$ \( 9 + T \)
$59$ \( -9 + T \)
$61$ \( -2 + T \)
$67$ \( -13 + T \)
$71$ \( 10 + T \)
$73$ \( 7 + T \)
$79$ \( 10 + T \)
$83$ \( 8 + T \)
$89$ \( 6 + T \)
$97$ \( 6 + T \)
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