Properties

Label 441.2.a.d
Level $441$
Weight $2$
Character orbit 441.a
Self dual yes
Analytic conductor $3.521$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(3.52140272914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{4} + O(q^{10}) \) \( q - 2q^{4} - 7q^{13} + 4q^{16} - 7q^{19} - 5q^{25} - 7q^{31} - q^{37} + 5q^{43} + 14q^{52} + 14q^{61} - 8q^{64} + 11q^{67} - 7q^{73} + 14q^{76} - 13q^{79} + 14q^{97} + O(q^{100}) \)

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
0 0 −2.00000 0 0 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.a.d 1
3.b odd 2 1 CM 441.2.a.d 1
4.b odd 2 1 7056.2.a.y 1
7.b odd 2 1 441.2.a.e 1
7.c even 3 2 63.2.e.a 2
7.d odd 6 2 441.2.e.c 2
12.b even 2 1 7056.2.a.y 1
21.c even 2 1 441.2.a.e 1
21.g even 6 2 441.2.e.c 2
21.h odd 6 2 63.2.e.a 2
28.d even 2 1 7056.2.a.bf 1
28.g odd 6 2 1008.2.s.j 2
63.g even 3 2 567.2.g.d 2
63.h even 3 2 567.2.h.c 2
63.j odd 6 2 567.2.h.c 2
63.n odd 6 2 567.2.g.d 2
84.h odd 2 1 7056.2.a.bf 1
84.n even 6 2 1008.2.s.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.e.a 2 7.c even 3 2
63.2.e.a 2 21.h odd 6 2
441.2.a.d 1 1.a even 1 1 trivial
441.2.a.d 1 3.b odd 2 1 CM
441.2.a.e 1 7.b odd 2 1
441.2.a.e 1 21.c even 2 1
441.2.e.c 2 7.d odd 6 2
441.2.e.c 2 21.g even 6 2
567.2.g.d 2 63.g even 3 2
567.2.g.d 2 63.n odd 6 2
567.2.h.c 2 63.h even 3 2
567.2.h.c 2 63.j odd 6 2
1008.2.s.j 2 28.g odd 6 2
1008.2.s.j 2 84.n even 6 2
7056.2.a.y 1 4.b odd 2 1
7056.2.a.y 1 12.b even 2 1
7056.2.a.bf 1 28.d even 2 1
7056.2.a.bf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} \)
\( T_{5} \)
\( T_{13} + 7 \)

Hecke characteristic polynomials

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