Properties

Label 6864.2.a.bh
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 - \beta ) q^{5} + ( 1 - \beta ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 - \beta ) q^{5} + ( 1 - \beta ) q^{7} + q^{9} - q^{11} + q^{13} + ( -1 - \beta ) q^{15} + 2 \beta q^{17} + ( -2 + 2 \beta ) q^{19} + ( 1 - \beta ) q^{21} + ( 5 - \beta ) q^{23} + 3 \beta q^{25} + q^{27} + ( -3 - 3 \beta ) q^{29} -4 \beta q^{31} - q^{33} + ( 3 + \beta ) q^{35} + ( 2 - 4 \beta ) q^{37} + q^{39} + ( -5 - \beta ) q^{41} + ( -1 + \beta ) q^{43} + ( -1 - \beta ) q^{45} + 8 q^{47} + ( -2 - \beta ) q^{49} + 2 \beta q^{51} + ( 2 + 4 \beta ) q^{53} + ( 1 + \beta ) q^{55} + ( -2 + 2 \beta ) q^{57} + ( 7 + \beta ) q^{59} + ( 3 - \beta ) q^{61} + ( 1 - \beta ) q^{63} + ( -1 - \beta ) q^{65} + ( 13 - \beta ) q^{67} + ( 5 - \beta ) q^{69} + ( 6 - 6 \beta ) q^{71} + ( -9 - \beta ) q^{73} + 3 \beta q^{75} + ( -1 + \beta ) q^{77} + ( -10 + 2 \beta ) q^{79} + q^{81} + ( 14 - 2 \beta ) q^{83} + ( -8 - 4 \beta ) q^{85} + ( -3 - 3 \beta ) q^{87} + ( -8 + 6 \beta ) q^{89} + ( 1 - \beta ) q^{91} -4 \beta q^{93} + ( -6 - 2 \beta ) q^{95} + ( 2 - 8 \beta ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 3q^{5} + q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 3q^{5} + q^{7} + 2q^{9} - 2q^{11} + 2q^{13} - 3q^{15} + 2q^{17} - 2q^{19} + q^{21} + 9q^{23} + 3q^{25} + 2q^{27} - 9q^{29} - 4q^{31} - 2q^{33} + 7q^{35} + 2q^{39} - 11q^{41} - q^{43} - 3q^{45} + 16q^{47} - 5q^{49} + 2q^{51} + 8q^{53} + 3q^{55} - 2q^{57} + 15q^{59} + 5q^{61} + q^{63} - 3q^{65} + 25q^{67} + 9q^{69} + 6q^{71} - 19q^{73} + 3q^{75} - q^{77} - 18q^{79} + 2q^{81} + 26q^{83} - 20q^{85} - 9q^{87} - 10q^{89} + q^{91} - 4q^{93} - 14q^{95} - 4q^{97} - 2q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
0 1.00000 0 −3.56155 0 −1.56155 0 1.00000 0
1.2 0 1.00000 0 0.561553 0 2.56155 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.bh 2
4.b odd 2 1 858.2.a.n 2
12.b even 2 1 2574.2.a.bg 2
44.c even 2 1 9438.2.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
858.2.a.n 2 4.b odd 2 1
2574.2.a.bg 2 12.b even 2 1
6864.2.a.bh 2 1.a even 1 1 trivial
9438.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{2} + 3 T_{5} - 2 \)
\( T_{7}^{2} - T_{7} - 4 \)
\( T_{17}^{2} - 2 T_{17} - 16 \)
\( T_{19}^{2} + 2 T_{19} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -2 + 3 T + T^{2} \)
$7$ \( -4 - T + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -16 - 2 T + T^{2} \)
$19$ \( -16 + 2 T + T^{2} \)
$23$ \( 16 - 9 T + T^{2} \)
$29$ \( -18 + 9 T + T^{2} \)
$31$ \( -64 + 4 T + T^{2} \)
$37$ \( -68 + T^{2} \)
$41$ \( 26 + 11 T + T^{2} \)
$43$ \( -4 + T + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( -52 - 8 T + T^{2} \)
$59$ \( 52 - 15 T + T^{2} \)
$61$ \( 2 - 5 T + T^{2} \)
$67$ \( 152 - 25 T + T^{2} \)
$71$ \( -144 - 6 T + T^{2} \)
$73$ \( 86 + 19 T + T^{2} \)
$79$ \( 64 + 18 T + T^{2} \)
$83$ \( 152 - 26 T + T^{2} \)
$89$ \( -128 + 10 T + T^{2} \)
$97$ \( -268 + 4 T + T^{2} \)
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