Properties

Label 6864.2.a.cd
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: 4.4.29268.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 5 x + 16\)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 1 + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 1 + \beta_{2} ) q^{7} + q^{9} + q^{11} + q^{13} + ( 1 - \beta_{1} ) q^{15} + ( 2 + \beta_{3} ) q^{17} + ( -1 - \beta_{2} ) q^{19} + ( 1 + \beta_{2} ) q^{21} + ( 1 + \beta_{1} + \beta_{3} ) q^{23} + ( 2 - \beta_{1} + \beta_{3} ) q^{25} + q^{27} + ( 2 + \beta_{1} - \beta_{2} ) q^{29} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{31} + q^{33} + ( 2 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{35} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + q^{39} + ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{47} + ( 1 - 2 \beta_{1} + 2 \beta_{2} ) q^{49} + ( 2 + \beta_{3} ) q^{51} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 1 - \beta_{1} ) q^{55} + ( -1 - \beta_{2} ) q^{57} + ( -2 + 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 2 - 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{61} + ( 1 + \beta_{2} ) q^{63} + ( 1 - \beta_{1} ) q^{65} + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{67} + ( 1 + \beta_{1} + \beta_{3} ) q^{69} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{71} + ( 3 + \beta_{2} + 2 \beta_{3} ) q^{73} + ( 2 - \beta_{1} + \beta_{3} ) q^{75} + ( 1 + \beta_{2} ) q^{77} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{79} + q^{81} + ( -2 + 2 \beta_{2} - 2 \beta_{3} ) q^{83} + ( 3 - 4 \beta_{1} - \beta_{2} ) q^{85} + ( 2 + \beta_{1} - \beta_{2} ) q^{87} + ( 3 + 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{89} + ( 1 + \beta_{2} ) q^{91} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{93} + ( -2 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{95} + ( 8 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{5} + 2q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{5} + 2q^{7} + 4q^{9} + 4q^{11} + 4q^{13} + 4q^{15} + 8q^{17} - 2q^{19} + 2q^{21} + 4q^{23} + 8q^{25} + 4q^{27} + 10q^{29} + 8q^{31} + 4q^{33} + 2q^{35} + 2q^{37} + 4q^{39} + 2q^{41} + 4q^{43} + 4q^{45} - 6q^{47} + 8q^{51} + 12q^{53} + 4q^{55} - 2q^{57} - 12q^{59} + 10q^{61} + 2q^{63} + 4q^{65} - 8q^{67} + 4q^{69} - 6q^{71} + 10q^{73} + 8q^{75} + 2q^{77} + 8q^{79} + 4q^{81} - 12q^{83} + 14q^{85} + 10q^{87} + 10q^{89} + 2q^{91} + 8q^{93} - 2q^{95} + 26q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 9 x^{2} + 5 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 5 \nu - 2 \)\()/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 5 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} + 7 \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + \beta_{1} + 10\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{3} - 5 \beta_{2} + 9 \beta_{1} + 4\)\()/2\)

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
2.85121
−1.25548
−2.43292
1.83719
0 1.00000 0 −2.46130 0 1.27820 0 1.00000 0
1.2 0 1.00000 0 −0.149237 0 −1.16830 0 1.00000 0
1.3 0 1.00000 0 3.11806 0 4.35203 0 1.00000 0
1.4 0 1.00000 0 3.49248 0 −2.46193 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.cd 4
4.b odd 2 1 3432.2.a.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.s 4 4.b odd 2 1
6864.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{4} - 4 T_{5}^{3} - 6 T_{5}^{2} + 26 T_{5} + 4 \)
\( T_{7}^{4} - 2 T_{7}^{3} - 12 T_{7}^{2} + 4 T_{7} + 16 \)
\( T_{17}^{4} - 8 T_{17}^{3} + 118 T_{17} - 212 \)
\( T_{19}^{4} + 2 T_{19}^{3} - 12 T_{19}^{2} - 4 T_{19} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 4 + 26 T - 6 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 16 + 4 T - 12 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( -212 + 118 T - 8 T^{3} + T^{4} \)
$19$ \( 16 - 4 T - 12 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( 16 + 20 T - 24 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( -284 + 122 T + 12 T^{2} - 10 T^{3} + T^{4} \)
$31$ \( 64 + 334 T - 42 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( 256 + 64 T - 48 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( 1864 + 340 T - 120 T^{2} - 2 T^{3} + T^{4} \)
$43$ \( -344 + 506 T - 96 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( -384 - 264 T - 36 T^{2} + 6 T^{3} + T^{4} \)
$53$ \( -2544 + 960 T - 48 T^{2} - 12 T^{3} + T^{4} \)
$59$ \( 1728 - 432 T - 72 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( -3296 + 1280 T - 96 T^{2} - 10 T^{3} + T^{4} \)
$67$ \( 1264 - 1114 T - 150 T^{2} + 8 T^{3} + T^{4} \)
$71$ \( 192 - 120 T - 36 T^{2} + 6 T^{3} + T^{4} \)
$73$ \( -3368 + 1220 T - 84 T^{2} - 10 T^{3} + T^{4} \)
$79$ \( -992 + 454 T - 36 T^{2} - 8 T^{3} + T^{4} \)
$83$ \( 1728 - 432 T - 72 T^{2} + 12 T^{3} + T^{4} \)
$89$ \( 5884 + 1682 T - 210 T^{2} - 10 T^{3} + T^{4} \)
$97$ \( -25088 + 4096 T - 26 T^{3} + T^{4} \)
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