Properties

Label 1323.2.a.r
Level $1323$
Weight $2$
Character orbit 1323.a
Self dual yes
Analytic conductor $10.564$
Analytic rank $0$
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: \(0\)
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: $\mathrm{SU}(2)$

$q$-expansion

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

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
2.00000 0 2.00000 −1.00000 0 0 0 0 −2.00000
\(n\): 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.r 1
3.b odd 2 1 1323.2.a.b 1
7.b odd 2 1 189.2.a.d yes 1
21.c even 2 1 189.2.a.a 1
28.d even 2 1 3024.2.a.u 1
35.c odd 2 1 4725.2.a.c 1
63.l odd 6 2 567.2.f.a 2
63.o even 6 2 567.2.f.h 2
84.h odd 2 1 3024.2.a.l 1
105.g even 2 1 4725.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.a 1 21.c even 2 1
189.2.a.d yes 1 7.b odd 2 1
567.2.f.a 2 63.l odd 6 2
567.2.f.h 2 63.o even 6 2
1323.2.a.b 1 3.b odd 2 1
1323.2.a.r 1 1.a even 1 1 trivial
3024.2.a.l 1 84.h odd 2 1
3024.2.a.u 1 28.d even 2 1
4725.2.a.c 1 35.c odd 2 1
4725.2.a.s 1 105.g even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( -4 + T \)
$13$ \( -2 + T \)
$17$ \( -3 + T \)
$19$ \( -8 + T \)
$23$ \( -6 + T \)
$29$ \( -4 + T \)
$31$ \( 6 + T \)
$37$ \( 3 + T \)
$41$ \( -1 + T \)
$43$ \( -11 + T \)
$47$ \( -9 + T \)
$53$ \( 6 + T \)
$59$ \( 15 + T \)
$61$ \( 4 + T \)
$67$ \( 8 + T \)
$71$ \( -12 + T \)
$73$ \( 6 + T \)
$79$ \( 1 + T \)
$83$ \( 9 + T \)
$89$ \( -2 + T \)
$97$ \( 12 + T \)
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