Properties

Label 6384.2.a.bw
Level $6384$
Weight $2$
Character orbit 6384.a
Self dual yes
Analytic conductor $50.976$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + q^{3} -\beta_{1} q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -\beta_{1} q^{5} + q^{7} + q^{9} + ( 1 + \beta_{1} + \beta_{2} ) q^{11} + 4 q^{13} -\beta_{1} q^{15} -\beta_{1} q^{17} + q^{19} + q^{21} + ( 1 + \beta_{1} + \beta_{2} ) q^{23} + ( 2 - \beta_{1} + \beta_{2} ) q^{25} + q^{27} + ( 1 + \beta_{2} ) q^{29} + ( 3 + \beta_{1} + \beta_{2} ) q^{31} + ( 1 + \beta_{1} + \beta_{2} ) q^{33} -\beta_{1} q^{35} + 2 q^{37} + 4 q^{39} -2 \beta_{2} q^{41} + ( 1 - \beta_{1} - \beta_{2} ) q^{43} -\beta_{1} q^{45} + ( 1 - \beta_{2} ) q^{47} + q^{49} -\beta_{1} q^{51} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{53} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{55} + q^{57} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{61} + q^{63} -4 \beta_{1} q^{65} + ( -1 - \beta_{1} - \beta_{2} ) q^{67} + ( 1 + \beta_{1} + \beta_{2} ) q^{69} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{71} + ( 4 - 2 \beta_{2} ) q^{73} + ( 2 - \beta_{1} + \beta_{2} ) q^{75} + ( 1 + \beta_{1} + \beta_{2} ) q^{77} + ( -1 + \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{83} + ( 7 - \beta_{1} + \beta_{2} ) q^{85} + ( 1 + \beta_{2} ) q^{87} + ( 4 + 2 \beta_{2} ) q^{89} + 4 q^{91} + ( 3 + \beta_{1} + \beta_{2} ) q^{93} -\beta_{1} q^{95} + ( 9 + \beta_{1} - \beta_{2} ) q^{97} + ( 1 + \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 12 q^{13} + 3 q^{19} + 3 q^{21} + 2 q^{23} + 5 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 2 q^{33} + 6 q^{37} + 12 q^{39} + 2 q^{41} + 4 q^{43} + 4 q^{47} + 3 q^{49} - 14 q^{53} - 16 q^{55} + 3 q^{57} - 4 q^{59} + 2 q^{61} + 3 q^{63} - 2 q^{67} + 2 q^{69} - 8 q^{71} + 14 q^{73} + 5 q^{75} + 2 q^{77} - 2 q^{79} + 3 q^{81} - 4 q^{83} + 20 q^{85} + 2 q^{87} + 10 q^{89} + 12 q^{91} + 8 q^{93} + 28 q^{97} + 2 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 3 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1} + 9\)\()/2\)

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
3.12489
−1.76156
−0.363328
0 1.00000 0 −2.64002 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.864641 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 3.50466 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.bw 3
4.b odd 2 1 3192.2.a.u 3
12.b even 2 1 9576.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.u 3 4.b odd 2 1
6384.2.a.bw 3 1.a even 1 1 trivial
9576.2.a.cb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6384))\):

\( T_{5}^{3} - 10 T_{5} - 8 \)
\( T_{11}^{3} - 2 T_{11}^{2} - 24 T_{11} - 16 \)
\( T_{13} - 4 \)
\( T_{17}^{3} - 10 T_{17} - 8 \)
\( T_{23}^{3} - 2 T_{23}^{2} - 24 T_{23} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( -8 - 10 T + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( -16 - 24 T - 2 T^{2} + T^{3} \)
$13$ \( ( -4 + T )^{3} \)
$17$ \( -8 - 10 T + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -16 - 24 T - 2 T^{2} + T^{3} \)
$29$ \( 44 - 18 T - 2 T^{2} + T^{3} \)
$31$ \( 16 - 4 T - 8 T^{2} + T^{3} \)
$37$ \( ( -2 + T )^{3} \)
$41$ \( -200 - 76 T - 2 T^{2} + T^{3} \)
$43$ \( 64 - 20 T - 4 T^{2} + T^{3} \)
$47$ \( -8 - 14 T - 4 T^{2} + T^{3} \)
$53$ \( -4 + 14 T + 14 T^{2} + T^{3} \)
$59$ \( -256 - 128 T + 4 T^{2} + T^{3} \)
$61$ \( 8 - 132 T - 2 T^{2} + T^{3} \)
$67$ \( 16 - 24 T + 2 T^{2} + T^{3} \)
$71$ \( -16 - 46 T + 8 T^{2} + T^{3} \)
$73$ \( 8 - 12 T - 14 T^{2} + T^{3} \)
$79$ \( -32 - 32 T + 2 T^{2} + T^{3} \)
$83$ \( -232 - 62 T + 4 T^{2} + T^{3} \)
$89$ \( 472 - 44 T - 10 T^{2} + T^{3} \)
$97$ \( -512 + 228 T - 28 T^{2} + T^{3} \)
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