Properties

Label 6762.2.a.bd
Level $6762$
Weight $2$
Character orbit 6762.a
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} + q^{8} + q^{9} + 3q^{10} - q^{12} - 5q^{13} - 3q^{15} + q^{16} + q^{18} - 8q^{19} + 3q^{20} - q^{23} - q^{24} + 4q^{25} - 5q^{26} - q^{27} + 3q^{29} - 3q^{30} - 2q^{31} + q^{32} + q^{36} - 7q^{37} - 8q^{38} + 5q^{39} + 3q^{40} - 9q^{41} - q^{43} + 3q^{45} - q^{46} + 3q^{47} - q^{48} + 4q^{50} - 5q^{52} - 12q^{53} - q^{54} + 8q^{57} + 3q^{58} + 6q^{59} - 3q^{60} - 14q^{61} - 2q^{62} + q^{64} - 15q^{65} - 4q^{67} + q^{69} + 6q^{71} + q^{72} + 4q^{73} - 7q^{74} - 4q^{75} - 8q^{76} + 5q^{78} - 16q^{79} + 3q^{80} + q^{81} - 9q^{82} + 12q^{83} - q^{86} - 3q^{87} - 6q^{89} + 3q^{90} - q^{92} + 2q^{93} + 3q^{94} - 24q^{95} - q^{96} + q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

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 6762.2.a.bd 1
7.b odd 2 1 966.2.a.i 1
21.c even 2 1 2898.2.a.j 1
28.d even 2 1 7728.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.i 1 7.b odd 2 1
2898.2.a.j 1 21.c even 2 1
6762.2.a.bd 1 1.a even 1 1 trivial
7728.2.a.a 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5} - 3 \)
\( T_{11} \)
\( T_{13} + 5 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( -3 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( 5 + T \)
$17$ \( T \)
$19$ \( 8 + T \)
$23$ \( 1 + T \)
$29$ \( -3 + T \)
$31$ \( 2 + T \)
$37$ \( 7 + T \)
$41$ \( 9 + T \)
$43$ \( 1 + T \)
$47$ \( -3 + T \)
$53$ \( 12 + T \)
$59$ \( -6 + T \)
$61$ \( 14 + T \)
$67$ \( 4 + T \)
$71$ \( -6 + T \)
$73$ \( -4 + T \)
$79$ \( 16 + T \)
$83$ \( -12 + T \)
$89$ \( 6 + T \)
$97$ \( -1 + T \)
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