Properties

Label 6864.2.a.bd
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 858)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

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

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
3.70156
−2.70156
0 −1.00000 0 −3.70156 0 1.70156 0 1.00000 0
1.2 0 −1.00000 0 2.70156 0 −4.70156 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.bd 2
4.b odd 2 1 858.2.a.p 2
12.b even 2 1 2574.2.a.be 2
44.c even 2 1 9438.2.a.ch 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
858.2.a.p 2 4.b odd 2 1
2574.2.a.be 2 12.b even 2 1
6864.2.a.bd 2 1.a even 1 1 trivial
9438.2.a.ch 2 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{2} + T_{5} - 10 \)
\( T_{7}^{2} + 3 T_{7} - 8 \)
\( T_{17} - 4 \)
\( T_{19} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -10 + T + T^{2} \)
$7$ \( -8 + 3 T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( 2 - 7 T + T^{2} \)
$29$ \( 2 + 7 T + T^{2} \)
$31$ \( -40 - 2 T + T^{2} \)
$37$ \( -40 - 2 T + T^{2} \)
$41$ \( 2 + 7 T + T^{2} \)
$43$ \( -92 - T + T^{2} \)
$47$ \( ( 4 + T )^{2} \)
$53$ \( -40 - 2 T + T^{2} \)
$59$ \( 100 - 21 T + T^{2} \)
$61$ \( 46 - 15 T + T^{2} \)
$67$ \( -72 + 9 T + T^{2} \)
$71$ \( -16 - 10 T + T^{2} \)
$73$ \( -8 - 3 T + T^{2} \)
$79$ \( -32 + 6 T + T^{2} \)
$83$ \( 40 + 18 T + T^{2} \)
$89$ \( -16 + 10 T + T^{2} \)
$97$ \( -148 + 8 T + T^{2} \)
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