Properties

Label 6762.2.a.bc
Level $6762$
Weight $2$
Character orbit 6762.a
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} + q^{8} + q^{9} + 2q^{10} + 4q^{11} - q^{12} - 2q^{13} - 2q^{15} + q^{16} - 6q^{17} + q^{18} + 2q^{20} + 4q^{22} + q^{23} - q^{24} - q^{25} - 2q^{26} - q^{27} - 2q^{29} - 2q^{30} - 4q^{31} + q^{32} - 4q^{33} - 6q^{34} + q^{36} + 6q^{37} + 2q^{39} + 2q^{40} + 6q^{41} + 12q^{43} + 4q^{44} + 2q^{45} + q^{46} + 12q^{47} - q^{48} - q^{50} + 6q^{51} - 2q^{52} + 6q^{53} - q^{54} + 8q^{55} - 2q^{58} + 4q^{59} - 2q^{60} + 10q^{61} - 4q^{62} + q^{64} - 4q^{65} - 4q^{66} + 4q^{67} - 6q^{68} - q^{69} - 16q^{71} + q^{72} - 2q^{73} + 6q^{74} + q^{75} + 2q^{78} + 8q^{79} + 2q^{80} + q^{81} + 6q^{82} + 16q^{83} - 12q^{85} + 12q^{86} + 2q^{87} + 4q^{88} - 6q^{89} + 2q^{90} + q^{92} + 4q^{93} + 12q^{94} - q^{96} + 2q^{97} + 4q^{99} + O(q^{100}) \)

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
0
1.00000 −1.00000 1.00000 2.00000 −1.00000 0 1.00000 1.00000 2.00000
\(n\): 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.bc 1
7.b odd 2 1 966.2.a.j 1
21.c even 2 1 2898.2.a.f 1
28.d even 2 1 7728.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.j 1 7.b odd 2 1
2898.2.a.f 1 21.c even 2 1
6762.2.a.bc 1 1.a even 1 1 trivial
7728.2.a.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5} - 2 \)
\( T_{11} - 4 \)
\( T_{13} + 2 \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( -2 + T \)
$7$ \( T \)
$11$ \( -4 + T \)
$13$ \( 2 + T \)
$17$ \( 6 + T \)
$19$ \( T \)
$23$ \( -1 + T \)
$29$ \( 2 + T \)
$31$ \( 4 + T \)
$37$ \( -6 + T \)
$41$ \( -6 + T \)
$43$ \( -12 + T \)
$47$ \( -12 + T \)
$53$ \( -6 + T \)
$59$ \( -4 + T \)
$61$ \( -10 + T \)
$67$ \( -4 + T \)
$71$ \( 16 + T \)
$73$ \( 2 + T \)
$79$ \( -8 + T \)
$83$ \( -16 + T \)
$89$ \( 6 + T \)
$97$ \( -2 + T \)
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