Properties

Label 6864.2.a.cb
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.90996.1
Defining polynomial: \(x^{4} - x^{3} - 11 x^{2} - 3 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1716)
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.

\(f(q)\) \(=\) \( q + q^{3} -\beta_{1} q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -\beta_{1} q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} - q^{11} + q^{13} -\beta_{1} q^{15} + ( -\beta_{1} + \beta_{3} ) q^{17} + ( -1 - \beta_{2} ) q^{19} + ( -1 - \beta_{2} ) q^{21} + ( 2 + 2 \beta_{1} + \beta_{3} ) q^{23} + ( 1 - \beta_{3} ) q^{25} + q^{27} + ( 3 + \beta_{1} - \beta_{2} ) q^{29} + ( -2 - \beta_{1} ) q^{31} - q^{33} + ( 3 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} + q^{39} + ( 1 - \beta_{2} - 2 \beta_{3} ) q^{41} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{43} -\beta_{1} q^{45} + ( -3 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{47} + ( 9 + 2 \beta_{1} + 2 \beta_{3} ) q^{49} + ( -\beta_{1} + \beta_{3} ) q^{51} + ( -2 - 2 \beta_{1} ) q^{53} + \beta_{1} q^{55} + ( -1 - \beta_{2} ) q^{57} -2 \beta_{1} q^{59} + ( 5 - \beta_{2} - \beta_{3} ) q^{61} + ( -1 - \beta_{2} ) q^{63} -\beta_{1} q^{65} + ( -2 - \beta_{1} - 2 \beta_{3} ) q^{67} + ( 2 + 2 \beta_{1} + \beta_{3} ) q^{69} + ( -5 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{71} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{73} + ( 1 - \beta_{3} ) q^{75} + ( 1 + \beta_{2} ) q^{77} + ( -4 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{79} + q^{81} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 11 + 4 \beta_{1} + \beta_{2} ) q^{85} + ( 3 + \beta_{1} - \beta_{2} ) q^{87} + ( 11 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{89} + ( -1 - \beta_{2} ) q^{91} + ( -2 - \beta_{1} ) q^{93} + ( 3 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{95} + ( 5 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 2q^{5} - 2q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 2q^{5} - 2q^{7} + 4q^{9} - 4q^{11} + 4q^{13} + 2q^{15} + 2q^{17} - 2q^{19} - 2q^{21} + 4q^{23} + 4q^{25} + 4q^{27} + 12q^{29} - 6q^{31} - 4q^{33} + 10q^{35} - 2q^{37} + 4q^{39} + 6q^{41} - 2q^{43} + 2q^{45} - 6q^{47} + 32q^{49} + 2q^{51} - 4q^{53} - 2q^{55} - 2q^{57} + 4q^{59} + 22q^{61} - 2q^{63} + 2q^{65} - 6q^{67} + 4q^{69} - 14q^{71} + 6q^{73} + 4q^{75} + 2q^{77} - 14q^{79} + 4q^{81} + 34q^{85} + 12q^{87} + 44q^{89} - 2q^{91} - 6q^{93} + 10q^{95} + 18q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 11 x^{2} - 3 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 2 \nu^{2} - 9 \nu + 2 \)\()/2\)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 10 \nu - 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + \nu^{2} + 12 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} - 4 \beta_{1} + 11\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(11 \beta_{3} + 15 \beta_{2} - 4 \beta_{1} + 27\)\()/2\)

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
−0.622070
0.309233
3.94698
−2.63415
0 1.00000 0 −3.29198 0 −3.59300 0 1.00000 0
1.2 0 1.00000 0 0.472388 0 5.15838 0 1.00000 0
1.3 0 1.00000 0 1.59571 0 −4.44027 0 1.00000 0
1.4 0 1.00000 0 3.22388 0 0.874887 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.cb 4
4.b odd 2 1 1716.2.a.i 4
12.b even 2 1 5148.2.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.a.i 4 4.b odd 2 1
5148.2.a.q 4 12.b even 2 1
6864.2.a.cb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{4} - 2 T_{5}^{3} - 10 T_{5}^{2} + 22 T_{5} - 8 \)
\( T_{7}^{4} + 2 T_{7}^{3} - 28 T_{7}^{2} - 60 T_{7} + 72 \)
\( T_{17}^{4} - 2 T_{17}^{3} - 64 T_{17}^{2} + 186 T_{17} + 300 \)
\( T_{19}^{4} + 2 T_{19}^{3} - 28 T_{19}^{2} - 60 T_{19} + 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( -8 + 22 T - 10 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( 72 - 60 T - 28 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( 300 + 186 T - 64 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 72 - 60 T - 28 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( -512 + 380 T - 64 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( -108 + 54 T + 24 T^{2} - 12 T^{3} + T^{4} \)
$31$ \( -4 - 10 T + 2 T^{2} + 6 T^{3} + T^{4} \)
$37$ \( -32 - 112 T - 88 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( 3872 + 484 T - 136 T^{2} - 6 T^{3} + T^{4} \)
$43$ \( -1048 - 654 T - 104 T^{2} + 2 T^{3} + T^{4} \)
$47$ \( 4768 - 888 T - 188 T^{2} + 6 T^{3} + T^{4} \)
$53$ \( 48 - 40 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( -128 + 176 T - 40 T^{2} - 4 T^{3} + T^{4} \)
$61$ \( -832 - 96 T + 136 T^{2} - 22 T^{3} + T^{4} \)
$67$ \( 4196 - 482 T - 154 T^{2} + 6 T^{3} + T^{4} \)
$71$ \( -2304 - 840 T - 20 T^{2} + 14 T^{3} + T^{4} \)
$73$ \( 536 + 140 T - 40 T^{2} - 6 T^{3} + T^{4} \)
$79$ \( -3392 - 1690 T - 100 T^{2} + 14 T^{3} + T^{4} \)
$83$ \( -256 + 944 T - 232 T^{2} + T^{4} \)
$89$ \( 6360 - 3954 T + 670 T^{2} - 44 T^{3} + T^{4} \)
$97$ \( -31072 + 6224 T - 232 T^{2} - 18 T^{3} + T^{4} \)
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