Properties

Label 441.2.a.e
Level 441
Weight 2
Character orbit 441.a
Self dual yes
Analytic conductor 3.521
Analytic rank 0
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: \(0\)
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.e 1
3.b odd 2 1 CM 441.2.a.e 1
4.b odd 2 1 7056.2.a.bf 1
7.b odd 2 1 441.2.a.d 1
7.c even 3 2 441.2.e.c 2
7.d odd 6 2 63.2.e.a 2
12.b even 2 1 7056.2.a.bf 1
21.c even 2 1 441.2.a.d 1
21.g even 6 2 63.2.e.a 2
21.h odd 6 2 441.2.e.c 2
28.d even 2 1 7056.2.a.y 1
28.f even 6 2 1008.2.s.j 2
63.i even 6 2 567.2.h.c 2
63.k odd 6 2 567.2.g.d 2
63.s even 6 2 567.2.g.d 2
63.t odd 6 2 567.2.h.c 2
84.h odd 2 1 7056.2.a.y 1
84.j odd 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.d odd 6 2
63.2.e.a 2 21.g even 6 2
441.2.a.d 1 7.b odd 2 1
441.2.a.d 1 21.c even 2 1
441.2.a.e 1 1.a even 1 1 trivial
441.2.a.e 1 3.b odd 2 1 CM
441.2.e.c 2 7.c even 3 2
441.2.e.c 2 21.h odd 6 2
567.2.g.d 2 63.k odd 6 2
567.2.g.d 2 63.s even 6 2
567.2.h.c 2 63.i even 6 2
567.2.h.c 2 63.t odd 6 2
1008.2.s.j 2 28.f even 6 2
1008.2.s.j 2 84.j odd 6 2
7056.2.a.y 1 28.d even 2 1
7056.2.a.y 1 84.h odd 2 1
7056.2.a.bf 1 4.b odd 2 1
7056.2.a.bf 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\):

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

Hecke characteristic polynomials

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