Properties

Label 6384.2.a.bv
Level $6384$
Weight $2$
Character orbit 6384.a
Self dual yes
Analytic conductor $50.976$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3192)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{7} + q^{9} + ( 1 - \beta_{1} ) q^{11} + ( -2 - \beta_{2} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} ) q^{17} + q^{19} - q^{21} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{23} -5 q^{25} - q^{27} + ( -4 + \beta_{2} ) q^{29} + ( -1 + \beta_{1} + \beta_{2} ) q^{31} + ( -1 + \beta_{1} ) q^{33} + ( -3 + \beta_{1} + \beta_{2} ) q^{37} + ( 2 + \beta_{2} ) q^{39} + 2 \beta_{1} q^{41} + ( 2 + 2 \beta_{1} ) q^{43} + ( 1 + \beta_{1} ) q^{47} + q^{49} + ( -1 + \beta_{1} - \beta_{2} ) q^{51} + ( -8 - \beta_{2} ) q^{53} - q^{57} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + q^{63} + ( -1 - 3 \beta_{1} ) q^{67} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{69} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{71} -2 \beta_{1} q^{73} + 5 q^{75} + ( 1 - \beta_{1} ) q^{77} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{83} + ( 4 - \beta_{2} ) q^{87} + ( -6 - 2 \beta_{2} ) q^{89} + ( -2 - \beta_{2} ) q^{91} + ( 1 - \beta_{1} - \beta_{2} ) q^{93} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{97} + ( 1 - \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 4 q^{11} - 6 q^{13} + 4 q^{17} + 3 q^{19} - 3 q^{21} + 4 q^{23} - 15 q^{25} - 3 q^{27} - 12 q^{29} - 4 q^{31} - 4 q^{33} - 10 q^{37} + 6 q^{39} - 2 q^{41} + 4 q^{43} + 2 q^{47} + 3 q^{49} - 4 q^{51} - 24 q^{53} - 3 q^{57} - 4 q^{59} - 14 q^{61} + 3 q^{63} - 4 q^{69} + 14 q^{71} + 2 q^{73} + 15 q^{75} + 4 q^{77} - 4 q^{79} + 3 q^{81} + 14 q^{83} + 12 q^{87} - 18 q^{89} - 6 q^{91} + 4 q^{93} - 2 q^{97} + 4 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 6\)\()/2\)

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
2.34292
0.470683
−1.81361
0 −1.00000 0 0 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 0 0 1.00000 0 1.00000 0
1.3 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
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)
\(19\) \(-1\)

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.bv 3
4.b odd 2 1 3192.2.a.v 3
12.b even 2 1 9576.2.a.cc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.v 3 4.b odd 2 1
6384.2.a.bv 3 1.a even 1 1 trivial
9576.2.a.cc 3 12.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6384))\):

\( T_{5} \)
\( T_{11}^{3} - 4 T_{11}^{2} - 12 T_{11} + 16 \)
\( T_{13}^{3} + 6 T_{13}^{2} - 16 T_{13} - 64 \)
\( T_{17}^{3} - 4 T_{17}^{2} - 24 T_{17} + 64 \)
\( T_{23}^{3} - 4 T_{23}^{2} - 60 T_{23} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( 16 - 12 T - 4 T^{2} + T^{3} \)
$13$ \( -64 - 16 T + 6 T^{2} + T^{3} \)
$17$ \( 64 - 24 T - 4 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( 256 - 60 T - 4 T^{2} + T^{3} \)
$29$ \( -32 + 20 T + 12 T^{2} + T^{3} \)
$31$ \( -256 - 56 T + 4 T^{2} + T^{3} \)
$37$ \( -344 - 28 T + 10 T^{2} + T^{3} \)
$41$ \( -8 - 68 T + 2 T^{2} + T^{3} \)
$43$ \( 128 - 64 T - 4 T^{2} + T^{3} \)
$47$ \( 16 - 16 T - 2 T^{2} + T^{3} \)
$53$ \( 272 + 164 T + 24 T^{2} + T^{3} \)
$59$ \( 64 - 112 T + 4 T^{2} + T^{3} \)
$61$ \( -664 - 52 T + 14 T^{2} + T^{3} \)
$67$ \( -128 - 156 T + T^{3} \)
$71$ \( 32 - 14 T^{2} + T^{3} \)
$73$ \( 8 - 68 T - 2 T^{2} + T^{3} \)
$79$ \( 352 - 124 T + 4 T^{2} + T^{3} \)
$83$ \( 1024 - 64 T - 14 T^{2} + T^{3} \)
$89$ \( -584 - 4 T + 18 T^{2} + T^{3} \)
$97$ \( 304 - 96 T + 2 T^{2} + T^{3} \)
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