Properties

Label 6864.2.a.ca
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: 4.4.70164.1
Defining polynomial: \(x^{4} - x^{3} - 10 x^{2} + 10 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + q^{9} - q^{11} - q^{13} + \beta_{1} q^{15} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{19} + ( 1 + \beta_{2} - \beta_{3} ) q^{21} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} + ( \beta_{2} + \beta_{3} ) q^{25} + q^{27} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} + ( 2 + \beta_{1} + \beta_{2} ) q^{31} - q^{33} + ( 2 + \beta_{2} ) q^{35} -2 \beta_{1} q^{37} - q^{39} + ( 3 + \beta_{2} - \beta_{3} ) q^{41} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{43} + \beta_{1} q^{45} + ( 4 - 2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 3 - \beta_{2} - 2 \beta_{3} ) q^{49} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{53} -\beta_{1} q^{55} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{57} + ( 2 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{59} + ( -4 + 2 \beta_{1} - 3 \beta_{2} ) q^{61} + ( 1 + \beta_{2} - \beta_{3} ) q^{63} -\beta_{1} q^{65} + ( 2 + 3 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{69} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{73} + ( \beta_{2} + \beta_{3} ) q^{75} + ( -1 - \beta_{2} + \beta_{3} ) q^{77} + ( 3 + \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{79} + q^{81} + ( 6 - 2 \beta_{1} - 2 \beta_{3} ) q^{83} + ( 7 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{85} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{87} + ( 2 - \beta_{1} - \beta_{2} ) q^{89} + ( -1 - \beta_{2} + \beta_{3} ) q^{91} + ( 2 + \beta_{1} + \beta_{2} ) q^{93} + ( 8 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{95} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + q^{5} + q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + q^{5} + q^{7} + 4q^{9} - 4q^{11} - 4q^{13} + q^{15} - 6q^{17} + q^{21} + 9q^{23} + q^{25} + 4q^{27} - q^{29} + 8q^{31} - 4q^{33} + 7q^{35} - 2q^{37} - 4q^{39} + 9q^{41} + 5q^{43} + q^{45} + 22q^{47} + 9q^{49} - 6q^{51} + 8q^{53} - q^{55} - q^{59} - 11q^{61} + q^{63} - q^{65} + 13q^{67} + 9q^{69} + 8q^{71} - 3q^{73} + q^{75} - q^{77} + 6q^{79} + 4q^{81} + 18q^{83} + 26q^{85} - q^{87} + 8q^{89} - q^{91} + 8q^{93} + 36q^{95} + 10q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 10 x^{2} + 10 x - 2\):

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

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
−3.18590
0.279954
0.699291
3.20666
0 1.00000 0 −3.18590 0 −1.10556 0 1.00000 0
1.2 0 1.00000 0 0.279954 0 4.36642 0 1.00000 0
1.3 0 1.00000 0 0.699291 0 −3.79091 0 1.00000 0
1.4 0 1.00000 0 3.20666 0 1.53005 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.ca 4
4.b odd 2 1 3432.2.a.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.r 4 4.b odd 2 1
6864.2.a.ca 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{4} - T_{5}^{3} - 10 T_{5}^{2} + 10 T_{5} - 2 \)
\( T_{7}^{4} - T_{7}^{3} - 18 T_{7}^{2} + 8 T_{7} + 28 \)
\( T_{17}^{4} + 6 T_{17}^{3} - 22 T_{17}^{2} - 90 T_{17} + 232 \)
\( T_{19}^{4} - 48 T_{19}^{2} - 108 T_{19} + 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( -2 + 10 T - 10 T^{2} - T^{3} + T^{4} \)
$7$ \( 28 + 8 T - 18 T^{2} - T^{3} + T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( 232 - 90 T - 22 T^{2} + 6 T^{3} + T^{4} \)
$19$ \( 72 - 108 T - 48 T^{2} + T^{4} \)
$23$ \( 292 + 168 T - 16 T^{2} - 9 T^{3} + T^{4} \)
$29$ \( -134 + 172 T - 54 T^{2} + T^{3} + T^{4} \)
$31$ \( -8 + 26 T - 4 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( -32 - 80 T - 40 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( -36 + 36 T + 12 T^{2} - 9 T^{3} + T^{4} \)
$43$ \( 2674 + 344 T - 118 T^{2} - 5 T^{3} + T^{4} \)
$47$ \( -896 + 152 T + 108 T^{2} - 22 T^{3} + T^{4} \)
$53$ \( -1616 + 848 T - 88 T^{2} - 8 T^{3} + T^{4} \)
$59$ \( 5488 + 128 T - 172 T^{2} + T^{3} + T^{4} \)
$61$ \( -1184 - 1232 T - 136 T^{2} + 11 T^{3} + T^{4} \)
$67$ \( -3626 + 1378 T - 88 T^{2} - 13 T^{3} + T^{4} \)
$71$ \( -1616 + 848 T - 88 T^{2} - 8 T^{3} + T^{4} \)
$73$ \( 4068 - 288 T - 228 T^{2} + 3 T^{3} + T^{4} \)
$79$ \( 388 + 102 T - 142 T^{2} - 6 T^{3} + T^{4} \)
$83$ \( -1568 + 480 T + 32 T^{2} - 18 T^{3} + T^{4} \)
$89$ \( -224 + 134 T - 4 T^{2} - 8 T^{3} + T^{4} \)
$97$ \( -4064 + 2576 T - 240 T^{2} - 10 T^{3} + T^{4} \)
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