Properties

Label 6384.2.a.m
Level $6384$
Weight $2$
Character orbit 6384.a
Self dual yes
Analytic conductor $50.976$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6384.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.9764966504\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(19\) \(1\)

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 6384.2.a.m 1
4.b odd 2 1 798.2.a.i 1
12.b even 2 1 2394.2.a.b 1
28.d even 2 1 5586.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.a.i 1 4.b odd 2 1
2394.2.a.b 1 12.b even 2 1
5586.2.a.t 1 28.d even 2 1
6384.2.a.m 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6384))\):

\( T_{5} - 2 \)
\( T_{11} \)
\( T_{13} - 2 \)
\( T_{17} + 2 \)
\( T_{23} + 4 \)

Hecke characteristic polynomials

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