Properties

Label 6864.2.a.bx
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: 4.4.83476.1
Defining polynomial: \(x^{4} - 12 x^{2} - 2 x + 1\)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{3} q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{3} q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{7} + q^{9} + q^{11} - q^{13} -\beta_{3} q^{15} + ( 3 - \beta_{1} ) q^{17} + ( 1 - \beta_{2} ) q^{19} + ( -1 + \beta_{1} + \beta_{3} ) q^{21} + ( 1 + \beta_{1} - \beta_{3} ) q^{23} + ( 2 - \beta_{2} ) q^{25} - q^{27} + ( 1 - \beta_{1} ) q^{29} + ( -\beta_{1} - \beta_{2} ) q^{31} - q^{33} + ( -6 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{35} + ( 2 + 2 \beta_{1} ) q^{37} + q^{39} + ( 5 + \beta_{1} + \beta_{3} ) q^{41} + ( -3 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + \beta_{3} q^{45} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{47} + ( 5 - 2 \beta_{3} ) q^{49} + ( -3 + \beta_{1} ) q^{51} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{53} + \beta_{3} q^{55} + ( -1 + \beta_{2} ) q^{57} + 2 \beta_{3} q^{59} + ( 4 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{61} + ( 1 - \beta_{1} - \beta_{3} ) q^{63} -\beta_{3} q^{65} + ( -6 + \beta_{3} ) q^{67} + ( -1 - \beta_{1} + \beta_{3} ) q^{69} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{71} + ( 5 + \beta_{2} - 2 \beta_{3} ) q^{73} + ( -2 + \beta_{2} ) q^{75} + ( 1 - \beta_{1} - \beta_{3} ) q^{77} + ( 1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{79} + q^{81} + ( -2 + 2 \beta_{1} ) q^{83} + ( 1 - \beta_{1} + 3 \beta_{3} ) q^{85} + ( -1 + \beta_{1} ) q^{87} + ( -2 - \beta_{1} + \beta_{2} ) q^{89} + ( -1 + \beta_{1} + \beta_{3} ) q^{91} + ( \beta_{1} + \beta_{2} ) q^{93} + ( -6 + \beta_{1} - \beta_{2} + 7 \beta_{3} ) q^{95} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 2q^{5} + 2q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 2q^{5} + 2q^{7} + 4q^{9} + 4q^{11} - 4q^{13} - 2q^{15} + 12q^{17} + 4q^{19} - 2q^{21} + 2q^{23} + 8q^{25} - 4q^{27} + 4q^{29} - 4q^{33} - 22q^{35} + 8q^{37} + 4q^{39} + 22q^{41} - 14q^{43} + 2q^{45} + 6q^{47} + 16q^{49} - 12q^{51} + 10q^{53} + 2q^{55} - 4q^{57} + 4q^{59} + 18q^{61} + 2q^{63} - 2q^{65} - 22q^{67} - 2q^{69} + 2q^{71} + 16q^{73} - 8q^{75} + 2q^{77} - 2q^{79} + 4q^{81} - 8q^{83} + 10q^{85} - 4q^{87} - 8q^{89} - 2q^{91} - 10q^{95} - 10q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 12 x^{2} - 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 12 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 12 \beta_{1} + 1\)

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
0.217439
3.53354
−3.36409
−0.386887
0 −1.00000 0 −3.59899 0 4.38155 0 1.00000 0
1.2 0 −1.00000 0 0.716998 0 −3.25054 0 1.00000 0
1.3 0 −1.00000 0 1.29726 0 3.06684 0 1.00000 0
1.4 0 −1.00000 0 3.58473 0 −2.19785 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.bx 4
4.b odd 2 1 3432.2.a.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.v 4 4.b odd 2 1
6864.2.a.bx 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{4} - 2 T_{5}^{3} - 12 T_{5}^{2} + 26 T_{5} - 12 \)
\( T_{7}^{4} - 2 T_{7}^{3} - 20 T_{7}^{2} + 20 T_{7} + 96 \)
\( T_{17}^{4} - 12 T_{17}^{3} + 42 T_{17}^{2} - 34 T_{17} - 32 \)
\( T_{19}^{4} - 4 T_{19}^{3} - 64 T_{19}^{2} + 140 T_{19} + 1128 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( -12 + 26 T - 12 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( 96 + 20 T - 20 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( -32 - 34 T + 42 T^{2} - 12 T^{3} + T^{4} \)
$19$ \( 1128 + 140 T - 64 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( 200 + 28 T - 28 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( -12 + 22 T - 6 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( 700 + 194 T - 88 T^{2} + T^{4} \)
$37$ \( -128 + 144 T - 24 T^{2} - 8 T^{3} + T^{4} \)
$41$ \( 360 - 428 T + 160 T^{2} - 22 T^{3} + T^{4} \)
$43$ \( -2248 - 814 T - 26 T^{2} + 14 T^{3} + T^{4} \)
$47$ \( 160 + 248 T - 60 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( -64 + 536 T - 68 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( -192 + 208 T - 48 T^{2} - 4 T^{3} + T^{4} \)
$61$ \( -10368 + 2720 T - 88 T^{2} - 18 T^{3} + T^{4} \)
$67$ \( 576 + 530 T + 168 T^{2} + 22 T^{3} + T^{4} \)
$71$ \( 128 + 136 T - 92 T^{2} - 2 T^{3} + T^{4} \)
$73$ \( -3888 + 1028 T - 4 T^{2} - 16 T^{3} + T^{4} \)
$79$ \( 5488 + 294 T - 238 T^{2} + 2 T^{3} + T^{4} \)
$83$ \( -192 - 176 T - 24 T^{2} + 8 T^{3} + T^{4} \)
$89$ \( 388 - 366 T - 52 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( -4256 - 1536 T - 112 T^{2} + 10 T^{3} + T^{4} \)
show more
show less