Properties

Label 6864.2.a.by
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.22676.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} + 6 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3432)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 + \beta_{1} ) q^{5} + ( -1 - \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 + \beta_{1} ) q^{5} + ( -1 - \beta_{3} ) q^{7} + q^{9} + q^{11} - q^{13} + ( -1 + \beta_{1} ) q^{15} + ( -\beta_{1} - \beta_{3} ) q^{17} + ( -\beta_{2} + \beta_{3} ) q^{19} + ( -1 - \beta_{3} ) q^{21} + ( -1 + 2 \beta_{2} + \beta_{3} ) q^{23} + ( 2 - 2 \beta_{1} + \beta_{3} ) q^{25} + q^{27} + ( -5 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{31} + q^{33} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{35} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{37} - q^{39} + ( -3 - 2 \beta_{1} + \beta_{3} ) q^{41} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + ( -1 + \beta_{1} ) q^{45} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{49} + ( -\beta_{1} - \beta_{3} ) q^{51} -2 q^{53} + ( -1 + \beta_{1} ) q^{55} + ( -\beta_{2} + \beta_{3} ) q^{57} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{59} + ( -3 + 3 \beta_{2} ) q^{61} + ( -1 - \beta_{3} ) q^{63} + ( 1 - \beta_{1} ) q^{65} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{67} + ( -1 + 2 \beta_{2} + \beta_{3} ) q^{69} + ( -6 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{71} + ( -7 + 2 \beta_{1} + \beta_{3} ) q^{73} + ( 2 - 2 \beta_{1} + \beta_{3} ) q^{75} + ( -1 - \beta_{3} ) q^{77} + ( -2 - \beta_{1} - \beta_{3} ) q^{79} + q^{81} + ( 2 + 2 \beta_{2} - 4 \beta_{3} ) q^{83} + ( -6 + \beta_{2} + \beta_{3} ) q^{85} + ( -5 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( -6 + 3 \beta_{1} + \beta_{2} ) q^{89} + ( 1 + \beta_{3} ) q^{91} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{93} + ( 2 - 2 \beta_{2} ) q^{95} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 3q^{5} - 3q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 3q^{5} - 3q^{7} + 4q^{9} + 4q^{11} - 4q^{13} - 3q^{15} - 2q^{19} - 3q^{21} - 3q^{23} + 5q^{25} + 4q^{27} - 21q^{29} - 10q^{31} + 4q^{33} + q^{35} + 4q^{37} - 4q^{39} - 15q^{41} + 3q^{43} - 3q^{45} - 4q^{47} + 5q^{49} - 8q^{53} - 3q^{55} - 2q^{57} + q^{59} - 9q^{61} - 3q^{63} + 3q^{65} + 5q^{67} - 3q^{69} - 18q^{71} - 27q^{73} + 5q^{75} - 3q^{77} - 8q^{79} + 4q^{81} + 14q^{83} - 24q^{85} - 21q^{87} - 20q^{89} + 3q^{91} - 10q^{93} + 6q^{95} + 4q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} + 6 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 3 \beta_{2} - 1\)

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
−0.266370
1.35449
2.31078
−2.39890
0 1.00000 0 −3.92905 0 −3.57932 0 1.00000 0
1.2 0 1.00000 0 −2.16536 0 3.64193 0 1.00000 0
1.3 0 1.00000 0 1.33970 0 −0.474190 0 1.00000 0
1.4 0 1.00000 0 1.75471 0 −2.58843 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(11\) \(-1\)
\(13\) \(1\)

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 6864.2.a.by 4
4.b odd 2 1 3432.2.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.q 4 4.b odd 2 1
6864.2.a.by 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{4} + 3 T_{5}^{3} - 8 T_{5}^{2} - 12 T_{5} + 20 \)
\( T_{7}^{4} + 3 T_{7}^{3} - 12 T_{7}^{2} - 40 T_{7} - 16 \)
\( T_{17}^{4} - 28 T_{17}^{2} - 36 T_{17} + 16 \)
\( T_{19}^{4} + 2 T_{19}^{3} - 24 T_{19}^{2} - 48 T_{19} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 20 - 12 T - 8 T^{2} + 3 T^{3} + T^{4} \)
$7$ \( -16 - 40 T - 12 T^{2} + 3 T^{3} + T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( 16 - 36 T - 28 T^{2} + T^{4} \)
$19$ \( 32 - 48 T - 24 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( 2096 - 104 T - 96 T^{2} + 3 T^{3} + T^{4} \)
$29$ \( -940 - 72 T + 112 T^{2} + 21 T^{3} + T^{4} \)
$31$ \( -656 - 468 T - 44 T^{2} + 10 T^{3} + T^{4} \)
$37$ \( -976 + 624 T - 96 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( -1640 - 444 T + 26 T^{2} + 15 T^{3} + T^{4} \)
$43$ \( -4 - 36 T - 46 T^{2} - 3 T^{3} + T^{4} \)
$47$ \( 512 - 384 T - 96 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( ( 2 + T )^{4} \)
$59$ \( 800 + 16 T - 64 T^{2} - T^{3} + T^{4} \)
$61$ \( 3240 - 756 T - 126 T^{2} + 9 T^{3} + T^{4} \)
$67$ \( 764 + 336 T - 78 T^{2} - 5 T^{3} + T^{4} \)
$71$ \( -3008 - 1184 T - 8 T^{2} + 18 T^{3} + T^{4} \)
$73$ \( -40 + 388 T + 210 T^{2} + 27 T^{3} + T^{4} \)
$79$ \( -152 - 116 T - 4 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( 9344 + 1760 T - 184 T^{2} - 14 T^{3} + T^{4} \)
$89$ \( -704 - 724 T + 40 T^{2} + 20 T^{3} + T^{4} \)
$97$ \( -976 + 624 T - 96 T^{2} - 4 T^{3} + T^{4} \)
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