Properties

Label 441.6.a.a
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 11 q^{2} + 89 q^{4} - 627 q^{8} + O(q^{10}) \) \( q - 11 q^{2} + 89 q^{4} - 627 q^{8} + 76 q^{11} + 4049 q^{16} - 836 q^{22} + 4952 q^{23} - 3125 q^{25} - 7282 q^{29} - 24475 q^{32} - 8886 q^{37} + 11748 q^{43} + 6764 q^{44} - 54472 q^{46} + 34375 q^{50} - 24550 q^{53} + 80102 q^{58} + 139657 q^{64} + 69364 q^{67} + 2224 q^{71} + 97746 q^{74} + 80168 q^{79} - 129228 q^{86} - 47652 q^{88} + 440728 q^{92} + O(q^{100}) \)

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
−11.0000 0 89.0000 0 0 0 −627.000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.a 1
3.b odd 2 1 49.6.a.b 1
7.b odd 2 1 CM 441.6.a.a 1
12.b even 2 1 784.6.a.g 1
21.c even 2 1 49.6.a.b 1
21.g even 6 2 49.6.c.a 2
21.h odd 6 2 49.6.c.a 2
84.h odd 2 1 784.6.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.b 1 3.b odd 2 1
49.6.a.b 1 21.c even 2 1
49.6.c.a 2 21.g even 6 2
49.6.c.a 2 21.h odd 6 2
441.6.a.a 1 1.a even 1 1 trivial
441.6.a.a 1 7.b odd 2 1 CM
784.6.a.g 1 12.b even 2 1
784.6.a.g 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} + 11 \)
\( T_{5} \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 11 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -76 + T \)
$13$ \( T \)
$17$ \( T \)
$19$ \( T \)
$23$ \( -4952 + T \)
$29$ \( 7282 + T \)
$31$ \( T \)
$37$ \( 8886 + T \)
$41$ \( T \)
$43$ \( -11748 + T \)
$47$ \( T \)
$53$ \( 24550 + T \)
$59$ \( T \)
$61$ \( T \)
$67$ \( -69364 + T \)
$71$ \( -2224 + T \)
$73$ \( T \)
$79$ \( -80168 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( T \)
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