Properties

Label 1323.2.a.o
Level $1323$
Weight $2$
Character orbit 1323.a
Self dual yes
Analytic conductor $10.564$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + 3q^{5} - 3q^{8} + O(q^{10}) \) \( q + q^{2} - q^{4} + 3q^{5} - 3q^{8} + 3q^{10} - 5q^{11} - 6q^{13} - q^{16} - 6q^{17} - 3q^{19} - 3q^{20} - 5q^{22} - q^{23} + 4q^{25} - 6q^{26} + 2q^{29} + 3q^{31} + 5q^{32} - 6q^{34} + 3q^{37} - 3q^{38} - 9q^{40} - 9q^{41} + 6q^{43} + 5q^{44} - q^{46} + 6q^{47} + 4q^{50} + 6q^{52} + 8q^{53} - 15q^{55} + 2q^{58} + 6q^{59} - 12q^{61} + 3q^{62} + 7q^{64} - 18q^{65} - 14q^{67} + 6q^{68} + 7q^{71} - 6q^{73} + 3q^{74} + 3q^{76} - 10q^{79} - 3q^{80} - 9q^{82} + 6q^{83} - 18q^{85} + 6q^{86} + 15q^{88} + 3q^{89} + q^{92} + 6q^{94} - 9q^{95} - 12q^{97} + O(q^{100}) \)

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
1.00000 0 −1.00000 3.00000 0 0 −3.00000 0 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1323.2.a.o yes 1
3.b odd 2 1 1323.2.a.e 1
7.b odd 2 1 1323.2.a.n yes 1
21.c even 2 1 1323.2.a.f yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.2.a.e 1 3.b odd 2 1
1323.2.a.f yes 1 21.c even 2 1
1323.2.a.n yes 1 7.b odd 2 1
1323.2.a.o yes 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2} - 1 \)
\( T_{5} - 3 \)
\( T_{13} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( -3 + T \)
$7$ \( T \)
$11$ \( 5 + T \)
$13$ \( 6 + T \)
$17$ \( 6 + T \)
$19$ \( 3 + T \)
$23$ \( 1 + T \)
$29$ \( -2 + T \)
$31$ \( -3 + T \)
$37$ \( -3 + T \)
$41$ \( 9 + T \)
$43$ \( -6 + T \)
$47$ \( -6 + T \)
$53$ \( -8 + T \)
$59$ \( -6 + T \)
$61$ \( 12 + T \)
$67$ \( 14 + T \)
$71$ \( -7 + T \)
$73$ \( 6 + T \)
$79$ \( 10 + T \)
$83$ \( -6 + T \)
$89$ \( -3 + T \)
$97$ \( 12 + T \)
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