Properties

Label 429.2.a.a
Level $429$
Weight $2$
Character orbit 429.a
Self dual yes
Analytic conductor $3.426$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.42558224671\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{8} + q^{9} + q^{11} + q^{12} + q^{13} - q^{16} - 4q^{17} - q^{18} - 8q^{19} - q^{22} - 3q^{24} - 5q^{25} - q^{26} - q^{27} + 4q^{29} - 6q^{31} - 5q^{32} - q^{33} + 4q^{34} - q^{36} - 6q^{37} + 8q^{38} - q^{39} + 6q^{41} - 2q^{43} - q^{44} - 8q^{47} + q^{48} - 7q^{49} + 5q^{50} + 4q^{51} - q^{52} + 6q^{53} + q^{54} + 8q^{57} - 4q^{58} - 14q^{61} + 6q^{62} + 7q^{64} + q^{66} + 14q^{67} + 4q^{68} - 4q^{71} + 3q^{72} + 6q^{73} + 6q^{74} + 5q^{75} + 8q^{76} + q^{78} - 10q^{79} + q^{81} - 6q^{82} - 12q^{83} + 2q^{86} - 4q^{87} + 3q^{88} + 12q^{89} + 6q^{93} + 8q^{94} + 5q^{96} - 2q^{97} + 7q^{98} + q^{99} + O(q^{100}) \)

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 0 1.00000 0 3.00000 1.00000 0
\(n\): 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 429.2.a.a 1
3.b odd 2 1 1287.2.a.d 1
4.b odd 2 1 6864.2.a.w 1
11.b odd 2 1 4719.2.a.j 1
13.b even 2 1 5577.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.a 1 1.a even 1 1 trivial
1287.2.a.d 1 3.b odd 2 1
4719.2.a.j 1 11.b odd 2 1
5577.2.a.f 1 13.b even 2 1
6864.2.a.w 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(429))\):

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

Hecke characteristic polynomials

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