Properties

Label 1728.2.a.x
Level $1728$
Weight $2$
Character orbit 1728.a
Self dual yes
Analytic conductor $13.798$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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 864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} + 3q^{7} + O(q^{10}) \) \( q + 2q^{5} + 3q^{7} + 6q^{11} + 3q^{13} + 2q^{17} + 3q^{19} - 6q^{23} - q^{25} - 8q^{29} + 6q^{35} - 7q^{37} - 8q^{41} + 12q^{43} - 6q^{47} + 2q^{49} + 4q^{53} + 12q^{55} + 6q^{59} + q^{61} + 6q^{65} + 3q^{67} - 12q^{71} - 15q^{73} + 18q^{77} + 9q^{79} - 12q^{83} + 4q^{85} + 10q^{89} + 9q^{91} + 6q^{95} + 9q^{97} + O(q^{100}) \)

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 2.00000 0 3.00000 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

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 1728.2.a.x 1
3.b odd 2 1 1728.2.a.h 1
4.b odd 2 1 1728.2.a.u 1
8.b even 2 1 864.2.a.d yes 1
8.d odd 2 1 864.2.a.a 1
12.b even 2 1 1728.2.a.e 1
24.f even 2 1 864.2.a.i yes 1
24.h odd 2 1 864.2.a.l yes 1
72.j odd 6 2 2592.2.i.c 2
72.l even 6 2 2592.2.i.g 2
72.n even 6 2 2592.2.i.r 2
72.p odd 6 2 2592.2.i.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.a 1 8.d odd 2 1
864.2.a.d yes 1 8.b even 2 1
864.2.a.i yes 1 24.f even 2 1
864.2.a.l yes 1 24.h odd 2 1
1728.2.a.e 1 12.b even 2 1
1728.2.a.h 1 3.b odd 2 1
1728.2.a.u 1 4.b odd 2 1
1728.2.a.x 1 1.a even 1 1 trivial
2592.2.i.c 2 72.j odd 6 2
2592.2.i.g 2 72.l even 6 2
2592.2.i.r 2 72.n even 6 2
2592.2.i.v 2 72.p odd 6 2

Hecke kernels

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

\( T_{5} - 2 \)
\( T_{7} - 3 \)
\( T_{11} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -2 + T \)
$7$ \( -3 + T \)
$11$ \( -6 + T \)
$13$ \( -3 + T \)
$17$ \( -2 + T \)
$19$ \( -3 + T \)
$23$ \( 6 + T \)
$29$ \( 8 + T \)
$31$ \( T \)
$37$ \( 7 + T \)
$41$ \( 8 + T \)
$43$ \( -12 + T \)
$47$ \( 6 + T \)
$53$ \( -4 + T \)
$59$ \( -6 + T \)
$61$ \( -1 + T \)
$67$ \( -3 + T \)
$71$ \( 12 + T \)
$73$ \( 15 + T \)
$79$ \( -9 + T \)
$83$ \( 12 + T \)
$89$ \( -10 + T \)
$97$ \( -9 + T \)
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