Properties

Label 864.2.a.n
Level 864
Weight 2
Character orbit 864.a
Self dual yes
Analytic conductor 6.899
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{5} -\beta q^{7} +O(q^{10})\) \( q -\beta q^{5} -\beta q^{7} + q^{11} + 4 q^{13} + 2 \beta q^{19} -6 q^{23} + 8 q^{25} + 2 \beta q^{29} + \beta q^{31} + 13 q^{35} + 10 q^{37} + 2 \beta q^{41} -2 \beta q^{43} -10 q^{47} + 6 q^{49} + \beta q^{53} -\beta q^{55} -4 q^{59} -4 \beta q^{65} -2 \beta q^{67} + 8 q^{71} -3 q^{73} -\beta q^{77} + 4 \beta q^{79} + 9 q^{83} -2 \beta q^{89} -4 \beta q^{91} -26 q^{95} + 7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 2q^{11} + 8q^{13} - 12q^{23} + 16q^{25} + 26q^{35} + 20q^{37} - 20q^{47} + 12q^{49} - 8q^{59} + 16q^{71} - 6q^{73} + 18q^{83} - 52q^{95} + 14q^{97} + O(q^{100}) \)

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
2.30278
−1.30278
0 0 0 −3.60555 0 −3.60555 0 0 0
1.2 0 0 0 3.60555 0 3.60555 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.a.n yes 2
3.b odd 2 1 864.2.a.m 2
4.b odd 2 1 864.2.a.m 2
8.b even 2 1 1728.2.a.bc 2
8.d odd 2 1 1728.2.a.bd 2
9.c even 3 2 2592.2.i.bb 4
9.d odd 6 2 2592.2.i.bc 4
12.b even 2 1 inner 864.2.a.n yes 2
24.f even 2 1 1728.2.a.bc 2
24.h odd 2 1 1728.2.a.bd 2
36.f odd 6 2 2592.2.i.bc 4
36.h even 6 2 2592.2.i.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.m 2 3.b odd 2 1
864.2.a.m 2 4.b odd 2 1
864.2.a.n yes 2 1.a even 1 1 trivial
864.2.a.n yes 2 12.b even 2 1 inner
1728.2.a.bc 2 8.b even 2 1
1728.2.a.bc 2 24.f even 2 1
1728.2.a.bd 2 8.d odd 2 1
1728.2.a.bd 2 24.h odd 2 1
2592.2.i.bb 4 9.c even 3 2
2592.2.i.bb 4 36.h even 6 2
2592.2.i.bc 4 9.d odd 6 2
2592.2.i.bc 4 36.f odd 6 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(864))\):

\( T_{5}^{2} - 13 \)
\( T_{7}^{2} - 13 \)
\( T_{11} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 - 3 T^{2} + 25 T^{4} \)
$7$ \( 1 + T^{2} + 49 T^{4} \)
$11$ \( ( 1 - T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( 1 - 14 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 6 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 6 T^{2} + 841 T^{4} \)
$31$ \( 1 + 49 T^{2} + 961 T^{4} \)
$37$ \( ( 1 - 10 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 30 T^{2} + 1681 T^{4} \)
$43$ \( 1 + 34 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 10 T + 47 T^{2} )^{2} \)
$53$ \( 1 + 93 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 61 T^{2} )^{2} \)
$67$ \( 1 + 82 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 3 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 50 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 - 9 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 126 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 7 T + 97 T^{2} )^{2} \)
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