Properties

Label 6384.2.a.cb
Level $6384$
Weight $2$
Character orbit 6384.a
Self dual yes
Analytic conductor $50.976$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.9248.1
Defining polynomial: \(x^{4} - 5 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} + q^{7} + q^{9} + ( \beta_{1} - \beta_{2} ) q^{11} + \beta_{1} q^{15} + ( -1 - \beta_{2} - \beta_{3} ) q^{17} + q^{19} + q^{21} + ( -1 + 2 \beta_{2} - \beta_{3} ) q^{23} + ( 2 + \beta_{3} ) q^{25} + q^{27} + ( 1 - \beta_{1} + \beta_{3} ) q^{29} + ( 2 + \beta_{1} - \beta_{2} ) q^{31} + ( \beta_{1} - \beta_{2} ) q^{33} + \beta_{1} q^{35} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( -2 + 2 \beta_{2} ) q^{41} + ( 3 + 2 \beta_{2} + \beta_{3} ) q^{43} + \beta_{1} q^{45} + ( 2 + 2 \beta_{1} - 3 \beta_{2} ) q^{47} + q^{49} + ( -1 - \beta_{2} - \beta_{3} ) q^{51} + ( 3 - \beta_{1} - \beta_{3} ) q^{53} + ( 6 + 2 \beta_{3} ) q^{55} + q^{57} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + q^{63} + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{67} + ( -1 + 2 \beta_{2} - \beta_{3} ) q^{69} + ( 3 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{71} + ( -2 + 2 \beta_{2} ) q^{73} + ( 2 + \beta_{3} ) q^{75} + ( \beta_{1} - \beta_{2} ) q^{77} + ( 5 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{79} + q^{81} + ( 5 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{83} + ( -1 - 4 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{85} + ( 1 - \beta_{1} + \beta_{3} ) q^{87} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( 2 + \beta_{1} - \beta_{2} ) q^{93} + \beta_{1} q^{95} + ( -2 + \beta_{1} + \beta_{2} ) q^{97} + ( \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{3} + 4 q^{7} + 4 q^{9} - 4 q^{17} + 4 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} + 4 q^{27} + 4 q^{29} + 8 q^{31} + 4 q^{37} - 8 q^{41} + 12 q^{43} + 8 q^{47} + 4 q^{49} - 4 q^{51} + 12 q^{53} + 24 q^{55} + 4 q^{57} + 8 q^{61} + 4 q^{63} + 8 q^{67} - 4 q^{69} + 12 q^{71} - 8 q^{73} + 8 q^{75} + 20 q^{79} + 4 q^{81} + 20 q^{83} - 4 q^{85} + 4 q^{87} + 8 q^{93} - 8 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{2} + 5 \beta_{1}\)\()/2\)

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
−2.13578
0.662153
−0.662153
2.13578
0 1.00000 0 −3.33513 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 −1.69614 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 1.69614 0 1.00000 0 1.00000 0
1.4 0 1.00000 0 3.33513 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.cb 4
4.b odd 2 1 3192.2.a.x 4
12.b even 2 1 9576.2.a.cj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.x 4 4.b odd 2 1
6384.2.a.cb 4 1.a even 1 1 trivial
9576.2.a.cj 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6384))\):

\( T_{5}^{4} - 14 T_{5}^{2} + 32 \)
\( T_{11}^{4} - 20 T_{11}^{2} + 32 \)
\( T_{13} \)
\( T_{17}^{4} + 4 T_{17}^{3} - 38 T_{17}^{2} - 152 T_{17} + 16 \)
\( T_{23}^{4} + 4 T_{23}^{3} - 68 T_{23}^{2} - 416 T_{23} - 608 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 32 - 14 T^{2} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( 32 - 20 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 16 - 152 T - 38 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( -608 - 416 T - 68 T^{2} + 4 T^{3} + T^{4} \)
$29$ \( 104 + 24 T - 42 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( -32 + 48 T + 4 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( 16 - 16 T - 56 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( -16 - 128 T - 16 T^{2} + 8 T^{3} + T^{4} \)
$43$ \( -1664 + 608 T - 20 T^{2} - 12 T^{3} + T^{4} \)
$47$ \( 2416 + 456 T - 98 T^{2} - 8 T^{3} + T^{4} \)
$53$ \( -472 + 248 T + 6 T^{2} - 12 T^{3} + T^{4} \)
$59$ \( 2048 - 112 T^{2} + T^{4} \)
$61$ \( 208 + 288 T - 56 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 2144 + 864 T - 132 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( -416 + 304 T - 26 T^{2} - 12 T^{3} + T^{4} \)
$73$ \( -16 - 128 T - 16 T^{2} + 8 T^{3} + T^{4} \)
$79$ \( -5312 + 1504 T + 4 T^{2} - 20 T^{3} + T^{4} \)
$83$ \( -1696 + 368 T + 70 T^{2} - 20 T^{3} + T^{4} \)
$89$ \( 1328 - 544 T - 192 T^{2} + T^{4} \)
$97$ \( 32 - 80 T - 4 T^{2} + 8 T^{3} + T^{4} \)
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