Properties

Label 6384.2.a.br
Level $6384$
Weight $2$
Character orbit 6384.a
Self dual yes
Analytic conductor $50.976$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta q^{5} + q^{7} + q^{9} + ( 2 - \beta ) q^{11} + \beta q^{13} + \beta q^{15} + ( 4 - \beta ) q^{17} - q^{19} + q^{21} + ( 2 - \beta ) q^{23} + 7 q^{25} + q^{27} -6 q^{29} + 2 \beta q^{31} + ( 2 - \beta ) q^{33} + \beta q^{35} + 10 q^{37} + \beta q^{39} -2 q^{41} -2 \beta q^{43} + \beta q^{45} + ( -4 + 2 \beta ) q^{47} + q^{49} + ( 4 - \beta ) q^{51} + 2 q^{53} + ( -12 + 2 \beta ) q^{55} - q^{57} -4 q^{59} + ( 2 + 2 \beta ) q^{61} + q^{63} + 12 q^{65} + ( -6 + \beta ) q^{67} + ( 2 - \beta ) q^{69} + ( 4 + 2 \beta ) q^{71} + 10 q^{73} + 7 q^{75} + ( 2 - \beta ) q^{77} + ( 2 - 3 \beta ) q^{79} + q^{81} -8 q^{83} + ( -12 + 4 \beta ) q^{85} -6 q^{87} -2 q^{89} + \beta q^{91} + 2 \beta q^{93} -\beta q^{95} + ( 8 + \beta ) q^{97} + ( 2 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + 4 q^{11} + 8 q^{17} - 2 q^{19} + 2 q^{21} + 4 q^{23} + 14 q^{25} + 2 q^{27} - 12 q^{29} + 4 q^{33} + 20 q^{37} - 4 q^{41} - 8 q^{47} + 2 q^{49} + 8 q^{51} + 4 q^{53} - 24 q^{55} - 2 q^{57} - 8 q^{59} + 4 q^{61} + 2 q^{63} + 24 q^{65} - 12 q^{67} + 4 q^{69} + 8 q^{71} + 20 q^{73} + 14 q^{75} + 4 q^{77} + 4 q^{79} + 2 q^{81} - 16 q^{83} - 24 q^{85} - 12 q^{87} - 4 q^{89} + 16 q^{97} + 4 q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
0 1.00000 0 −3.46410 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 3.46410 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.br 2
4.b odd 2 1 798.2.a.k 2
12.b even 2 1 2394.2.a.x 2
28.d even 2 1 5586.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.a.k 2 4.b odd 2 1
2394.2.a.x 2 12.b even 2 1
5586.2.a.bd 2 28.d even 2 1
6384.2.a.br 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6384))\):

\( T_{5}^{2} - 12 \)
\( T_{11}^{2} - 4 T_{11} - 8 \)
\( T_{13}^{2} - 12 \)
\( T_{17}^{2} - 8 T_{17} + 4 \)
\( T_{23}^{2} - 4 T_{23} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -12 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -8 - 4 T + T^{2} \)
$13$ \( -12 + T^{2} \)
$17$ \( 4 - 8 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -8 - 4 T + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( -48 + T^{2} \)
$37$ \( ( -10 + T )^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( -48 + T^{2} \)
$47$ \( -32 + 8 T + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( -44 - 4 T + T^{2} \)
$67$ \( 24 + 12 T + T^{2} \)
$71$ \( -32 - 8 T + T^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( -104 - 4 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( ( 2 + T )^{2} \)
$97$ \( 52 - 16 T + T^{2} \)
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