Properties

Label 6864.2.a.bl
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + \beta q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta q^{5} + \beta q^{7} + q^{9} + q^{11} - q^{13} + \beta q^{15} -4 q^{17} + 2 \beta q^{19} + \beta q^{21} + \beta q^{23} + ( 3 + \beta ) q^{25} + q^{27} -\beta q^{29} + ( -2 - 2 \beta ) q^{31} + q^{33} + ( 8 + \beta ) q^{35} -2 q^{37} - q^{39} + ( 2 + \beta ) q^{41} + ( 6 - \beta ) q^{43} + \beta q^{45} + 4 q^{47} + ( 1 + \beta ) q^{49} -4 q^{51} -10 q^{53} + \beta q^{55} + 2 \beta q^{57} + ( 8 + \beta ) q^{59} + ( -2 + 3 \beta ) q^{61} + \beta q^{63} -\beta q^{65} + ( -2 - 3 \beta ) q^{67} + \beta q^{69} + ( 8 + 2 \beta ) q^{71} + ( 2 - 3 \beta ) q^{73} + ( 3 + \beta ) q^{75} + \beta q^{77} + 10 q^{79} + q^{81} + ( 4 - 2 \beta ) q^{83} -4 \beta q^{85} -\beta q^{87} + 8 q^{89} -\beta q^{91} + ( -2 - 2 \beta ) q^{93} + ( 16 + 2 \beta ) q^{95} -10 q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + q^{5} + q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + q^{5} + q^{7} + 2q^{9} + 2q^{11} - 2q^{13} + q^{15} - 8q^{17} + 2q^{19} + q^{21} + q^{23} + 7q^{25} + 2q^{27} - q^{29} - 6q^{31} + 2q^{33} + 17q^{35} - 4q^{37} - 2q^{39} + 5q^{41} + 11q^{43} + q^{45} + 8q^{47} + 3q^{49} - 8q^{51} - 20q^{53} + q^{55} + 2q^{57} + 17q^{59} - q^{61} + q^{63} - q^{65} - 7q^{67} + q^{69} + 18q^{71} + q^{73} + 7q^{75} + q^{77} + 20q^{79} + 2q^{81} + 6q^{83} - 4q^{85} - q^{87} + 16q^{89} - q^{91} - 6q^{93} + 34q^{95} - 20q^{97} + 2q^{99} + O(q^{100}) \)

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
−2.37228
3.37228
0 1.00000 0 −2.37228 0 −2.37228 0 1.00000 0
1.2 0 1.00000 0 3.37228 0 3.37228 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.bl 2
4.b odd 2 1 858.2.a.o 2
12.b even 2 1 2574.2.a.bc 2
44.c even 2 1 9438.2.a.cb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
858.2.a.o 2 4.b odd 2 1
2574.2.a.bc 2 12.b even 2 1
6864.2.a.bl 2 1.a even 1 1 trivial
9438.2.a.cb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{2} - T_{5} - 8 \)
\( T_{7}^{2} - T_{7} - 8 \)
\( T_{17} + 4 \)
\( T_{19}^{2} - 2 T_{19} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -8 - T + T^{2} \)
$7$ \( -8 - T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( ( 4 + T )^{2} \)
$19$ \( -32 - 2 T + T^{2} \)
$23$ \( -8 - T + T^{2} \)
$29$ \( -8 + T + T^{2} \)
$31$ \( -24 + 6 T + T^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( -2 - 5 T + T^{2} \)
$43$ \( 22 - 11 T + T^{2} \)
$47$ \( ( -4 + T )^{2} \)
$53$ \( ( 10 + T )^{2} \)
$59$ \( 64 - 17 T + T^{2} \)
$61$ \( -74 + T + T^{2} \)
$67$ \( -62 + 7 T + T^{2} \)
$71$ \( 48 - 18 T + T^{2} \)
$73$ \( -74 - T + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( -24 - 6 T + T^{2} \)
$89$ \( ( -8 + T )^{2} \)
$97$ \( ( 10 + T )^{2} \)
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