Properties

Label 1323.2.a.k
Level $1323$
Weight $2$
Character orbit 1323.a
Self dual yes
Analytic conductor $10.564$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.5642081874\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

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

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 1323.2.a.k 1
3.b odd 2 1 CM 1323.2.a.k 1
7.b odd 2 1 1323.2.a.j 1
7.c even 3 2 189.2.e.b 2
21.c even 2 1 1323.2.a.j 1
21.h odd 6 2 189.2.e.b 2
63.g even 3 2 567.2.g.c 2
63.h even 3 2 567.2.h.d 2
63.j odd 6 2 567.2.h.d 2
63.n odd 6 2 567.2.g.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.b 2 7.c even 3 2
189.2.e.b 2 21.h odd 6 2
567.2.g.c 2 63.g even 3 2
567.2.g.c 2 63.n odd 6 2
567.2.h.d 2 63.h even 3 2
567.2.h.d 2 63.j odd 6 2
1323.2.a.j 1 7.b odd 2 1
1323.2.a.j 1 21.c even 2 1
1323.2.a.k 1 1.a even 1 1 trivial
1323.2.a.k 1 3.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\):

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

Hecke characteristic polynomials

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