Properties

Label 1323.2.a.i
Level $1323$
Weight $2$
Character orbit 1323.a
Self dual yes
Analytic conductor $10.564$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{4} + O(q^{10}) \) \( q - 2q^{4} - 5q^{13} + 4q^{16} + 7q^{19} - 5q^{25} + 4q^{31} + 11q^{37} + 8q^{43} + 10q^{52} + q^{61} - 8q^{64} + 5q^{67} + 7q^{73} - 14q^{76} + 17q^{79} + 19q^{97} + O(q^{100}) \)

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.i 1
3.b odd 2 1 CM 1323.2.a.i 1
7.b odd 2 1 27.2.a.a 1
21.c even 2 1 27.2.a.a 1
28.d even 2 1 432.2.a.e 1
35.c odd 2 1 675.2.a.e 1
35.f even 4 2 675.2.b.f 2
56.e even 2 1 1728.2.a.o 1
56.h odd 2 1 1728.2.a.n 1
63.l odd 6 2 81.2.c.a 2
63.o even 6 2 81.2.c.a 2
77.b even 2 1 3267.2.a.f 1
84.h odd 2 1 432.2.a.e 1
91.b odd 2 1 4563.2.a.e 1
105.g even 2 1 675.2.a.e 1
105.k odd 4 2 675.2.b.f 2
119.d odd 2 1 7803.2.a.k 1
133.c even 2 1 9747.2.a.f 1
168.e odd 2 1 1728.2.a.o 1
168.i even 2 1 1728.2.a.n 1
189.y odd 18 6 729.2.e.f 6
189.be even 18 6 729.2.e.f 6
231.h odd 2 1 3267.2.a.f 1
252.s odd 6 2 1296.2.i.i 2
252.bi even 6 2 1296.2.i.i 2
273.g even 2 1 4563.2.a.e 1
357.c even 2 1 7803.2.a.k 1
399.h odd 2 1 9747.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 7.b odd 2 1
27.2.a.a 1 21.c even 2 1
81.2.c.a 2 63.l odd 6 2
81.2.c.a 2 63.o even 6 2
432.2.a.e 1 28.d even 2 1
432.2.a.e 1 84.h odd 2 1
675.2.a.e 1 35.c odd 2 1
675.2.a.e 1 105.g even 2 1
675.2.b.f 2 35.f even 4 2
675.2.b.f 2 105.k odd 4 2
729.2.e.f 6 189.y odd 18 6
729.2.e.f 6 189.be even 18 6
1296.2.i.i 2 252.s odd 6 2
1296.2.i.i 2 252.bi even 6 2
1323.2.a.i 1 1.a even 1 1 trivial
1323.2.a.i 1 3.b odd 2 1 CM
1728.2.a.n 1 56.h odd 2 1
1728.2.a.n 1 168.i even 2 1
1728.2.a.o 1 56.e even 2 1
1728.2.a.o 1 168.e odd 2 1
3267.2.a.f 1 77.b even 2 1
3267.2.a.f 1 231.h odd 2 1
4563.2.a.e 1 91.b odd 2 1
4563.2.a.e 1 273.g even 2 1
7803.2.a.k 1 119.d odd 2 1
7803.2.a.k 1 357.c even 2 1
9747.2.a.f 1 133.c even 2 1
9747.2.a.f 1 399.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\):

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

Hecke characteristic polynomials

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