Properties

Label 6864.2.a.bz
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.8468.1
Defining polynomial: \(x^{4} - x^{3} - 5 x^{2} + 3 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 429)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{3} q^{5} + ( -1 - \beta_{2} - \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{3} q^{5} + ( -1 - \beta_{2} - \beta_{3} ) q^{7} + q^{9} + q^{11} + q^{13} + \beta_{3} q^{15} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{17} + ( -5 - \beta_{2} - \beta_{3} ) q^{19} + ( -1 - \beta_{2} - \beta_{3} ) q^{21} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{23} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{25} + q^{27} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{29} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{31} + q^{33} + ( -4 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{35} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} + q^{39} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{41} + ( -7 - \beta_{1} + 3 \beta_{3} ) q^{43} + \beta_{3} q^{45} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{47} + ( 1 - 2 \beta_{1} - 2 \beta_{2} ) q^{49} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{51} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{53} + \beta_{3} q^{55} + ( -5 - \beta_{2} - \beta_{3} ) q^{57} + ( -4 + 4 \beta_{1} - 2 \beta_{3} ) q^{59} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{61} + ( -1 - \beta_{2} - \beta_{3} ) q^{63} + \beta_{3} q^{65} + ( -2 \beta_{1} - \beta_{3} ) q^{67} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{69} + ( -2 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{71} + ( -1 + \beta_{2} - 3 \beta_{3} ) q^{73} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{75} + ( -1 - \beta_{2} - \beta_{3} ) q^{77} + ( 1 + \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{79} + q^{81} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{83} + ( -5 - \beta_{2} - 7 \beta_{3} ) q^{85} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{87} + ( 2 + \beta_{1} + \beta_{2} ) q^{89} + ( -1 - \beta_{2} - \beta_{3} ) q^{91} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{93} + ( -4 + \beta_{1} + 3 \beta_{2} - 5 \beta_{3} ) q^{95} + ( 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 2q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 2q^{7} + 4q^{9} + 4q^{11} + 4q^{13} - 8q^{17} - 18q^{19} - 2q^{21} + 4q^{25} + 4q^{27} - 10q^{29} - 12q^{31} + 4q^{33} - 22q^{35} - 2q^{37} + 4q^{39} + 2q^{41} - 28q^{43} - 6q^{47} + 8q^{49} - 8q^{51} - 4q^{53} - 18q^{57} - 16q^{59} - 10q^{61} - 2q^{63} - 10q^{71} - 6q^{73} + 4q^{75} - 2q^{77} + 8q^{79} + 4q^{81} + 8q^{83} - 18q^{85} - 10q^{87} + 6q^{89} - 2q^{91} - 12q^{93} - 22q^{95} + 10q^{97} + 4q^{99} + O(q^{100}) \)

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

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

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
1.31743
−1.89122
2.27841
−0.704624
0 1.00000 0 −2.71878 0 4.30059 0 1.00000 0
1.2 0 1.00000 0 −0.776183 0 −2.69175 0 1.00000 0
1.3 0 1.00000 0 −0.477194 0 −0.435561 0 1.00000 0
1.4 0 1.00000 0 3.97216 0 −3.17328 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.bz 4
4.b odd 2 1 429.2.a.h 4
12.b even 2 1 1287.2.a.m 4
44.c even 2 1 4719.2.a.z 4
52.b odd 2 1 5577.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.h 4 4.b odd 2 1
1287.2.a.m 4 12.b even 2 1
4719.2.a.z 4 44.c even 2 1
5577.2.a.m 4 52.b odd 2 1
6864.2.a.bz 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{4} - 12 T_{5}^{2} - 14 T_{5} - 4 \)
\( T_{7}^{4} + 2 T_{7}^{3} - 16 T_{7}^{2} - 44 T_{7} - 16 \)
\( T_{17}^{4} + 8 T_{17}^{3} - 26 T_{17}^{2} - 162 T_{17} + 412 \)
\( T_{19}^{4} + 18 T_{19}^{3} + 104 T_{19}^{2} + 180 T_{19} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( -4 - 14 T - 12 T^{2} + T^{4} \)
$7$ \( -16 - 44 T - 16 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( 412 - 162 T - 26 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( -64 + 180 T + 104 T^{2} + 18 T^{3} + T^{4} \)
$23$ \( -128 - 148 T - 44 T^{2} + T^{4} \)
$29$ \( -116 - 382 T - 30 T^{2} + 10 T^{3} + T^{4} \)
$31$ \( -968 - 458 T - 20 T^{2} + 12 T^{3} + T^{4} \)
$37$ \( 32 + 64 T - 40 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( -88 + 140 T - 52 T^{2} - 2 T^{3} + T^{4} \)
$43$ \( -4664 - 270 T + 202 T^{2} + 28 T^{3} + T^{4} \)
$47$ \( 128 + 136 T - 84 T^{2} + 6 T^{3} + T^{4} \)
$53$ \( 16 - 224 T - 80 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( -11456 - 2704 T - 96 T^{2} + 16 T^{3} + T^{4} \)
$61$ \( -2816 - 1088 T - 80 T^{2} + 10 T^{3} + T^{4} \)
$67$ \( 1648 - 18 T - 88 T^{2} + T^{4} \)
$71$ \( 128 - 136 T - 76 T^{2} + 10 T^{3} + T^{4} \)
$73$ \( -88 + 292 T - 108 T^{2} + 6 T^{3} + T^{4} \)
$79$ \( 4016 + 1382 T - 250 T^{2} - 8 T^{3} + T^{4} \)
$83$ \( 3392 + 624 T - 120 T^{2} - 8 T^{3} + T^{4} \)
$89$ \( -44 + 58 T - 8 T^{2} - 6 T^{3} + T^{4} \)
$97$ \( -12832 + 3712 T - 248 T^{2} - 10 T^{3} + T^{4} \)
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