Properties

Label 1287.2.a.e
Level $1287$
Weight $2$
Character orbit 1287.a
Self dual yes
Analytic conductor $10.277$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1287.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.2767467401\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 429)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + 2q^{5} - 3q^{8} + O(q^{10}) \) \( q + q^{2} - q^{4} + 2q^{5} - 3q^{8} + 2q^{10} + q^{11} + q^{13} - q^{16} + 6q^{17} - 4q^{19} - 2q^{20} + q^{22} + 8q^{23} - q^{25} + q^{26} + 10q^{29} + 5q^{32} + 6q^{34} + 6q^{37} - 4q^{38} - 6q^{40} - 10q^{41} + 4q^{43} - q^{44} + 8q^{46} - 8q^{47} - 7q^{49} - q^{50} - q^{52} + 10q^{53} + 2q^{55} + 10q^{58} + 12q^{59} + 14q^{61} + 7q^{64} + 2q^{65} - 12q^{67} - 6q^{68} - 6q^{73} + 6q^{74} + 4q^{76} + 8q^{79} - 2q^{80} - 10q^{82} - 12q^{83} + 12q^{85} + 4q^{86} - 3q^{88} - 2q^{89} - 8q^{92} - 8q^{94} - 8q^{95} - 14q^{97} - 7q^{98} + O(q^{100}) \)

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
0
1.00000 0 −1.00000 2.00000 0 0 −3.00000 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)
\(13\) \(-1\)

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 1287.2.a.e 1
3.b odd 2 1 429.2.a.b 1
12.b even 2 1 6864.2.a.e 1
33.d even 2 1 4719.2.a.k 1
39.d odd 2 1 5577.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.b 1 3.b odd 2 1
1287.2.a.e 1 1.a even 1 1 trivial
4719.2.a.k 1 33.d even 2 1
5577.2.a.g 1 39.d odd 2 1
6864.2.a.e 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1287))\):

\( T_{2} - 1 \)
\( T_{5} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( -2 + T \)
$7$ \( T \)
$11$ \( -1 + T \)
$13$ \( -1 + T \)
$17$ \( -6 + T \)
$19$ \( 4 + T \)
$23$ \( -8 + T \)
$29$ \( -10 + T \)
$31$ \( T \)
$37$ \( -6 + T \)
$41$ \( 10 + T \)
$43$ \( -4 + T \)
$47$ \( 8 + T \)
$53$ \( -10 + T \)
$59$ \( -12 + T \)
$61$ \( -14 + T \)
$67$ \( 12 + T \)
$71$ \( T \)
$73$ \( 6 + T \)
$79$ \( -8 + T \)
$83$ \( 12 + T \)
$89$ \( 2 + T \)
$97$ \( 14 + T \)
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