Properties

Label 6864.2.a.cc
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: 4.4.23252.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} + 2\)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( 1 + \beta_{2} ) q^{5} -\beta_{1} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 1 + \beta_{2} ) q^{5} -\beta_{1} q^{7} + q^{9} - q^{11} - q^{13} + ( 1 + \beta_{2} ) q^{15} + ( 1 - \beta_{2} + \beta_{3} ) q^{17} -\beta_{1} q^{19} -\beta_{1} q^{21} + ( 2 - \beta_{3} ) q^{23} + ( 3 + 2 \beta_{2} - \beta_{3} ) q^{25} + q^{27} + ( 1 + \beta_{1} + \beta_{2} ) q^{29} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{31} - q^{33} + ( -2 - 3 \beta_{1} + 3 \beta_{3} ) q^{35} + ( 4 - \beta_{1} + 3 \beta_{3} ) q^{37} - q^{39} + ( 2 + \beta_{1} - 2 \beta_{3} ) q^{41} + ( -3 + \beta_{2} - \beta_{3} ) q^{43} + ( 1 + \beta_{2} ) q^{45} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{47} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 1 - \beta_{2} + \beta_{3} ) q^{51} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -1 - \beta_{2} ) q^{55} -\beta_{1} q^{57} + ( 2 + 2 \beta_{2} - 4 \beta_{3} ) q^{59} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{61} -\beta_{1} q^{63} + ( -1 - \beta_{2} ) q^{65} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{67} + ( 2 - \beta_{3} ) q^{69} + ( 6 + \beta_{1} - 3 \beta_{3} ) q^{71} + ( 4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{73} + ( 3 + 2 \beta_{2} - \beta_{3} ) q^{75} + \beta_{1} q^{77} + ( 3 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{79} + q^{81} + ( -2 + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -4 - \beta_{1} ) q^{85} + ( 1 + \beta_{1} + \beta_{2} ) q^{87} + ( 5 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{89} + \beta_{1} q^{91} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{93} + ( -2 - 3 \beta_{1} + 3 \beta_{3} ) q^{95} + ( 4 - \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{5} - 2q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{5} - 2q^{7} + 4q^{9} - 4q^{11} - 4q^{13} + 4q^{15} + 4q^{17} - 2q^{19} - 2q^{21} + 8q^{23} + 12q^{25} + 4q^{27} + 6q^{29} + 8q^{31} - 4q^{33} - 14q^{35} + 14q^{37} - 4q^{39} + 10q^{41} - 12q^{43} + 4q^{45} - 2q^{47} + 16q^{49} + 4q^{51} + 12q^{53} - 4q^{55} - 2q^{57} + 8q^{59} - 10q^{61} - 2q^{63} - 4q^{65} + 8q^{67} + 8q^{69} + 26q^{71} + 14q^{73} + 12q^{75} + 2q^{77} + 16q^{79} + 4q^{81} - 8q^{83} - 18q^{85} + 6q^{87} + 18q^{89} + 2q^{91} + 8q^{93} - 14q^{95} + 14q^{97} - 4q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 6 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + \beta_{1} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 6\)\()/2\)

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
0.565882
−0.634868
−1.88474
2.95372
0 1.00000 0 −2.24566 0 2.40254 0 1.00000 0
1.2 0 1.00000 0 −0.962075 0 −1.88053 0 1.00000 0
1.3 0 1.00000 0 3.43697 0 2.70832 0 1.00000 0
1.4 0 1.00000 0 3.77076 0 −5.23034 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.cc 4
4.b odd 2 1 3432.2.a.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.t 4 4.b odd 2 1
6864.2.a.cc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{4} - 4 T_{5}^{3} - 8 T_{5}^{2} + 26 T_{5} + 28 \)
\( T_{7}^{4} + 2 T_{7}^{3} - 20 T_{7}^{2} - 4 T_{7} + 64 \)
\( T_{17}^{4} - 4 T_{17}^{3} - 14 T_{17}^{2} + 6 T_{17} + 4 \)
\( T_{19}^{4} + 2 T_{19}^{3} - 20 T_{19}^{2} - 4 T_{19} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 28 + 26 T - 8 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 64 - 4 T - 20 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( 4 + 6 T - 14 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( 64 - 4 T - 20 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( -16 + 12 T + 12 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( -28 + 66 T - 34 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( 224 + 374 T - 52 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( 1568 + 560 T - 32 T^{2} - 14 T^{3} + T^{4} \)
$41$ \( 392 + 196 T - 16 T^{2} - 10 T^{3} + T^{4} \)
$43$ \( -16 + 18 T + 34 T^{2} + 12 T^{3} + T^{4} \)
$47$ \( 352 + 216 T - 92 T^{2} + 2 T^{3} + T^{4} \)
$53$ \( -496 + 1184 T - 96 T^{2} - 12 T^{3} + T^{4} \)
$59$ \( -3776 + 1936 T - 176 T^{2} - 8 T^{3} + T^{4} \)
$61$ \( 448 - 96 T - 56 T^{2} + 10 T^{3} + T^{4} \)
$67$ \( 3448 + 446 T - 120 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( -32 + 280 T + 148 T^{2} - 26 T^{3} + T^{4} \)
$73$ \( -1304 + 644 T - 36 T^{2} - 14 T^{3} + T^{4} \)
$79$ \( 896 + 1198 T - 58 T^{2} - 16 T^{3} + T^{4} \)
$83$ \( -64 + 208 T - 104 T^{2} + 8 T^{3} + T^{4} \)
$89$ \( -1364 + 202 T + 76 T^{2} - 18 T^{3} + T^{4} \)
$97$ \( 4384 + 1232 T - 120 T^{2} - 14 T^{3} + T^{4} \)
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