Properties

Label 6384.2.a.bs
Level $6384$
Weight $2$
Character orbit 6384.a
Self dual yes
Analytic conductor $50.976$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
0 1.00000 0 −0.732051 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 2.73205 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.bs 2
4.b odd 2 1 3192.2.a.q 2
12.b even 2 1 9576.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.q 2 4.b odd 2 1
6384.2.a.bs 2 1.a even 1 1 trivial
9576.2.a.bj 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6384))\):

\( T_{5}^{2} - 2 T_{5} - 2 \)
\( T_{11} \)
\( T_{13}^{2} + 4 T_{13} - 8 \)
\( T_{17}^{2} + 6 T_{17} + 6 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -2 - 2 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( -8 + 4 T + T^{2} \)
$17$ \( 6 + 6 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( -74 + 2 T + T^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( -44 + 4 T + T^{2} \)
$43$ \( -44 - 4 T + T^{2} \)
$47$ \( -66 - 6 T + T^{2} \)
$53$ \( 22 + 10 T + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 88 - 20 T + T^{2} \)
$71$ \( 6 + 6 T + T^{2} \)
$73$ \( -108 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( -2 + 10 T + T^{2} \)
$89$ \( -44 + 4 T + T^{2} \)
$97$ \( ( -2 + T )^{2} \)
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