Properties

Label 6864.2.a.ce
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.46437524.1
Defining polynomial: \(x^{5} - x^{4} - 20 x^{3} + 8 x^{2} + 70 x + 28\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{1} q^{5} + ( 1 + \beta_{4} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{1} q^{5} + ( 1 + \beta_{4} ) q^{7} + q^{9} - q^{11} + q^{13} -\beta_{1} q^{15} + ( 1 + \beta_{1} - \beta_{4} ) q^{17} + ( -2 - \beta_{2} ) q^{19} + ( -1 - \beta_{4} ) q^{21} + ( 1 - \beta_{4} ) q^{23} + ( 3 + \beta_{1} + \beta_{3} ) q^{25} - q^{27} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} ) q^{31} + q^{33} + ( -1 - \beta_{2} - \beta_{4} ) q^{35} + ( 2 + 2 \beta_{4} ) q^{37} - q^{39} + ( 3 + \beta_{4} ) q^{41} + ( \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{43} + \beta_{1} q^{45} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{47} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{49} + ( -1 - \beta_{1} + \beta_{4} ) q^{51} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{53} -\beta_{1} q^{55} + ( 2 + \beta_{2} ) q^{57} + ( -4 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{59} + ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{61} + ( 1 + \beta_{4} ) q^{63} + \beta_{1} q^{65} + ( -2 + \beta_{1} ) q^{67} + ( -1 + \beta_{4} ) q^{69} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{71} + ( 4 - \beta_{1} - \beta_{3} ) q^{73} + ( -3 - \beta_{1} - \beta_{3} ) q^{75} + ( -1 - \beta_{4} ) q^{77} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{79} + q^{81} + 2 \beta_{1} q^{83} + ( 9 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{87} + ( 3 + 3 \beta_{1} + \beta_{2} + 3 \beta_{4} ) q^{89} + ( 1 + \beta_{4} ) q^{91} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{93} + ( 1 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{95} + ( 7 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{3} + q^{5} + 3q^{7} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{3} + q^{5} + 3q^{7} + 5q^{9} - 5q^{11} + 5q^{13} - q^{15} + 8q^{17} - 10q^{19} - 3q^{21} + 7q^{23} + 16q^{25} - 5q^{27} - 3q^{29} + 4q^{31} + 5q^{33} - 3q^{35} + 6q^{37} - 5q^{39} + 13q^{41} - 3q^{43} + q^{45} - 2q^{47} + 10q^{49} - 8q^{51} + 4q^{53} - q^{55} + 10q^{57} - 21q^{59} - 15q^{61} + 3q^{63} + q^{65} - 9q^{67} - 7q^{69} + 4q^{71} + 19q^{73} - 16q^{75} - 3q^{77} - 8q^{79} + 5q^{81} + 2q^{83} + 46q^{85} + 3q^{87} + 12q^{89} + 3q^{91} - 4q^{93} + 32q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 20 x^{3} + 8 x^{2} + 70 x + 28\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 4 \nu^{3} - 17 \nu^{2} - 60 \nu + 8 \)\()/17\)
\(\beta_{3}\)\(=\)\( \nu^{2} - \nu - 8 \)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{4} + 5 \nu^{3} + 51 \nu^{2} - 58 \nu - 109 \)\()/17\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{1} + 8\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 3 \beta_{2} + 14 \beta_{1} + 5\)
\(\nu^{4}\)\(=\)\(-4 \beta_{4} + 17 \beta_{3} + 5 \beta_{2} + 21 \beta_{1} + 108\)

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
−3.73149
−1.57692
−0.447720
2.49314
4.26299
0 −1.00000 0 −3.73149 0 −0.404219 0 1.00000 0
1.2 0 −1.00000 0 −1.57692 0 5.18377 0 1.00000 0
1.3 0 −1.00000 0 −0.447720 0 −3.31638 0 1.00000 0
1.4 0 −1.00000 0 2.49314 0 2.46927 0 1.00000 0
1.5 0 −1.00000 0 4.26299 0 −0.932447 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.ce 5
4.b odd 2 1 3432.2.a.x 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.x 5 4.b odd 2 1
6864.2.a.ce 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{5} - T_{5}^{4} - 20 T_{5}^{3} + 8 T_{5}^{2} + 70 T_{5} + 28 \)
\( T_{7}^{5} - 3 T_{7}^{4} - 18 T_{7}^{3} + 24 T_{7}^{2} + 52 T_{7} + 16 \)
\( T_{17}^{5} - 8 T_{17}^{4} - 20 T_{17}^{3} + 202 T_{17}^{2} - 56 T_{17} - 448 \)
\( T_{19}^{5} + 10 T_{19}^{4} - 16 T_{19}^{3} - 420 T_{19}^{2} - 1232 T_{19} - 784 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( ( 1 + T )^{5} \)
$5$ \( 28 + 70 T + 8 T^{2} - 20 T^{3} - T^{4} + T^{5} \)
$7$ \( 16 + 52 T + 24 T^{2} - 18 T^{3} - 3 T^{4} + T^{5} \)
$11$ \( ( 1 + T )^{5} \)
$13$ \( ( -1 + T )^{5} \)
$17$ \( -448 - 56 T + 202 T^{2} - 20 T^{3} - 8 T^{4} + T^{5} \)
$19$ \( -784 - 1232 T - 420 T^{2} - 16 T^{3} + 10 T^{4} + T^{5} \)
$23$ \( -56 - 84 T + 76 T^{2} - 2 T^{3} - 7 T^{4} + T^{5} \)
$29$ \( 31468 + 7894 T - 630 T^{2} - 180 T^{3} + 3 T^{4} + T^{5} \)
$31$ \( 2016 + 1428 T + 86 T^{2} - 70 T^{3} - 4 T^{4} + T^{5} \)
$37$ \( 512 + 832 T + 192 T^{2} - 72 T^{3} - 6 T^{4} + T^{5} \)
$41$ \( 72 - 84 T - 20 T^{2} + 46 T^{3} - 13 T^{4} + T^{5} \)
$43$ \( 9184 + 2674 T - 438 T^{2} - 118 T^{3} + 3 T^{4} + T^{5} \)
$47$ \( 14336 + 4352 T - 392 T^{2} - 156 T^{3} + 2 T^{4} + T^{5} \)
$53$ \( -128 - 176 T + 608 T^{2} - 112 T^{3} - 4 T^{4} + T^{5} \)
$59$ \( 28096 - 3056 T - 1368 T^{2} + 32 T^{3} + 21 T^{4} + T^{5} \)
$61$ \( 224 - 304 T - 736 T^{2} - 42 T^{3} + 15 T^{4} + T^{5} \)
$67$ \( 56 - 90 T - 56 T^{2} + 12 T^{3} + 9 T^{4} + T^{5} \)
$71$ \( -90944 + 14800 T + 1664 T^{2} - 296 T^{3} - 4 T^{4} + T^{5} \)
$73$ \( -7712 - 2460 T + 936 T^{2} + 28 T^{3} - 19 T^{4} + T^{5} \)
$79$ \( -21056 + 15212 T - 1994 T^{2} - 312 T^{3} + 8 T^{4} + T^{5} \)
$83$ \( 896 + 1120 T + 64 T^{2} - 80 T^{3} - 2 T^{4} + T^{5} \)
$89$ \( -164704 + 19684 T + 3098 T^{2} - 290 T^{3} - 12 T^{4} + T^{5} \)
$97$ \( 8064 - 5376 T + 16 T^{2} + 260 T^{3} - 32 T^{4} + T^{5} \)
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