Properties

Label 6864.2.a.bc
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( -2 + \beta ) q^{5} + ( 2 + 2 \beta ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -2 + \beta ) q^{5} + ( 2 + 2 \beta ) q^{7} + q^{9} - q^{11} - q^{13} + ( 2 - \beta ) q^{15} + ( -4 - \beta ) q^{17} + 6 q^{19} + ( -2 - 2 \beta ) q^{21} + ( -2 + 2 \beta ) q^{23} + ( 1 - 4 \beta ) q^{25} - q^{27} + 3 \beta q^{29} -3 \beta q^{31} + q^{33} -2 \beta q^{35} + ( 2 + 4 \beta ) q^{37} + q^{39} -12 q^{41} + ( 2 - 5 \beta ) q^{43} + ( -2 + \beta ) q^{45} + 6 \beta q^{47} + ( 5 + 8 \beta ) q^{49} + ( 4 + \beta ) q^{51} + ( 2 + 8 \beta ) q^{53} + ( 2 - \beta ) q^{55} -6 q^{57} + ( 8 + 2 \beta ) q^{59} -2 q^{61} + ( 2 + 2 \beta ) q^{63} + ( 2 - \beta ) q^{65} + ( -4 + 5 \beta ) q^{67} + ( 2 - 2 \beta ) q^{69} + 4 \beta q^{71} + ( 4 + 6 \beta ) q^{73} + ( -1 + 4 \beta ) q^{75} + ( -2 - 2 \beta ) q^{77} + ( 2 + 5 \beta ) q^{79} + q^{81} -8 \beta q^{83} + ( 6 - 2 \beta ) q^{85} -3 \beta q^{87} + ( -14 - \beta ) q^{89} + ( -2 - 2 \beta ) q^{91} + 3 \beta q^{93} + ( -12 + 6 \beta ) q^{95} + 10 q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 4q^{5} + 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 4q^{5} + 4q^{7} + 2q^{9} - 2q^{11} - 2q^{13} + 4q^{15} - 8q^{17} + 12q^{19} - 4q^{21} - 4q^{23} + 2q^{25} - 2q^{27} + 2q^{33} + 4q^{37} + 2q^{39} - 24q^{41} + 4q^{43} - 4q^{45} + 10q^{49} + 8q^{51} + 4q^{53} + 4q^{55} - 12q^{57} + 16q^{59} - 4q^{61} + 4q^{63} + 4q^{65} - 8q^{67} + 4q^{69} + 8q^{73} - 2q^{75} - 4q^{77} + 4q^{79} + 2q^{81} + 12q^{85} - 28q^{89} - 4q^{91} - 24q^{95} + 20q^{97} - 2q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −1.00000 0 −3.41421 0 −0.828427 0 1.00000 0
1.2 0 −1.00000 0 −0.585786 0 4.82843 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.bc 2
4.b odd 2 1 429.2.a.c 2
12.b even 2 1 1287.2.a.g 2
44.c even 2 1 4719.2.a.o 2
52.b odd 2 1 5577.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.c 2 4.b odd 2 1
1287.2.a.g 2 12.b even 2 1
4719.2.a.o 2 44.c even 2 1
5577.2.a.i 2 52.b odd 2 1
6864.2.a.bc 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{2} + 4 T_{5} + 2 \)
\( T_{7}^{2} - 4 T_{7} - 4 \)
\( T_{17}^{2} + 8 T_{17} + 14 \)
\( T_{19} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 2 + 4 T + T^{2} \)
$7$ \( -4 - 4 T + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 14 + 8 T + T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( -4 + 4 T + T^{2} \)
$29$ \( -18 + T^{2} \)
$31$ \( -18 + T^{2} \)
$37$ \( -28 - 4 T + T^{2} \)
$41$ \( ( 12 + T )^{2} \)
$43$ \( -46 - 4 T + T^{2} \)
$47$ \( -72 + T^{2} \)
$53$ \( -124 - 4 T + T^{2} \)
$59$ \( 56 - 16 T + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( -34 + 8 T + T^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( -56 - 8 T + T^{2} \)
$79$ \( -46 - 4 T + T^{2} \)
$83$ \( -128 + T^{2} \)
$89$ \( 194 + 28 T + T^{2} \)
$97$ \( ( -10 + T )^{2} \)
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