Properties

Label 1728.2.a.p
Level $1728$
Weight $2$
Character orbit 1728.a
Self dual yes
Analytic conductor $13.798$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 5q^{7} + O(q^{10}) \) \( q + 5q^{7} + 7q^{13} + q^{19} - 5q^{25} - 4q^{31} + q^{37} - 8q^{43} + 18q^{49} + 13q^{61} - 11q^{67} + 17q^{73} - 13q^{79} + 35q^{91} + 5q^{97} + O(q^{100}) \)

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 0 0 0 5.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

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 1728.2.a.p 1
3.b odd 2 1 CM 1728.2.a.p 1
4.b odd 2 1 1728.2.a.m 1
8.b even 2 1 108.2.a.a 1
8.d odd 2 1 432.2.a.d 1
12.b even 2 1 1728.2.a.m 1
24.f even 2 1 432.2.a.d 1
24.h odd 2 1 108.2.a.a 1
40.f even 2 1 2700.2.a.b 1
40.i odd 4 2 2700.2.d.g 2
56.h odd 2 1 5292.2.a.j 1
72.j odd 6 2 324.2.e.b 2
72.l even 6 2 1296.2.i.j 2
72.n even 6 2 324.2.e.b 2
72.p odd 6 2 1296.2.i.j 2
120.i odd 2 1 2700.2.a.b 1
120.w even 4 2 2700.2.d.g 2
168.i even 2 1 5292.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.a.a 1 8.b even 2 1
108.2.a.a 1 24.h odd 2 1
324.2.e.b 2 72.j odd 6 2
324.2.e.b 2 72.n even 6 2
432.2.a.d 1 8.d odd 2 1
432.2.a.d 1 24.f even 2 1
1296.2.i.j 2 72.l even 6 2
1296.2.i.j 2 72.p odd 6 2
1728.2.a.m 1 4.b odd 2 1
1728.2.a.m 1 12.b even 2 1
1728.2.a.p 1 1.a even 1 1 trivial
1728.2.a.p 1 3.b odd 2 1 CM
2700.2.a.b 1 40.f even 2 1
2700.2.a.b 1 120.i odd 2 1
2700.2.d.g 2 40.i odd 4 2
2700.2.d.g 2 120.w even 4 2
5292.2.a.j 1 56.h odd 2 1
5292.2.a.j 1 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1728))\):

\( T_{5} \)
\( T_{7} - 5 \)
\( T_{11} \)

Hecke characteristic polynomials

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