Properties

Label 1728.2.a.r
Level $1728$
Weight $2$
Character orbit 1728.a
Self dual yes
Analytic conductor $13.798$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} - 3 q^{7} + O(q^{10}) \) \( q + q^{5} - 3 q^{7} + 5 q^{11} - 4 q^{13} - 8 q^{17} + 2 q^{19} - 2 q^{23} - 4 q^{25} - 6 q^{29} + 7 q^{31} - 3 q^{35} + 6 q^{37} - 6 q^{41} - 2 q^{43} - 6 q^{47} + 2 q^{49} - 5 q^{53} + 5 q^{55} - 4 q^{59} + 8 q^{61} - 4 q^{65} - 10 q^{67} + 8 q^{71} + q^{73} - 15 q^{77} - 16 q^{79} - 11 q^{83} - 8 q^{85} + 6 q^{89} + 12 q^{91} + 2 q^{95} - q^{97} + O(q^{100}) \)

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 1.00000 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.r 1
3.b odd 2 1 1728.2.a.i 1
4.b odd 2 1 1728.2.a.s 1
8.b even 2 1 432.2.a.c 1
8.d odd 2 1 216.2.a.b 1
12.b even 2 1 1728.2.a.l 1
24.f even 2 1 216.2.a.c yes 1
24.h odd 2 1 432.2.a.f 1
40.e odd 2 1 5400.2.a.h 1
40.k even 4 2 5400.2.f.z 2
72.j odd 6 2 1296.2.i.g 2
72.l even 6 2 648.2.i.c 2
72.n even 6 2 1296.2.i.l 2
72.p odd 6 2 648.2.i.e 2
120.m even 2 1 5400.2.a.e 1
120.q odd 4 2 5400.2.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.b 1 8.d odd 2 1
216.2.a.c yes 1 24.f even 2 1
432.2.a.c 1 8.b even 2 1
432.2.a.f 1 24.h odd 2 1
648.2.i.c 2 72.l even 6 2
648.2.i.e 2 72.p odd 6 2
1296.2.i.g 2 72.j odd 6 2
1296.2.i.l 2 72.n even 6 2
1728.2.a.i 1 3.b odd 2 1
1728.2.a.l 1 12.b even 2 1
1728.2.a.r 1 1.a even 1 1 trivial
1728.2.a.s 1 4.b odd 2 1
5400.2.a.e 1 120.m even 2 1
5400.2.a.h 1 40.e odd 2 1
5400.2.f.b 2 120.q odd 4 2
5400.2.f.z 2 40.k even 4 2

Hecke kernels

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

\( T_{5} - 1 \)
\( T_{7} + 3 \)
\( T_{11} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( 3 + T \)
$11$ \( -5 + T \)
$13$ \( 4 + T \)
$17$ \( 8 + T \)
$19$ \( -2 + T \)
$23$ \( 2 + T \)
$29$ \( 6 + T \)
$31$ \( -7 + T \)
$37$ \( -6 + T \)
$41$ \( 6 + T \)
$43$ \( 2 + T \)
$47$ \( 6 + T \)
$53$ \( 5 + T \)
$59$ \( 4 + T \)
$61$ \( -8 + T \)
$67$ \( 10 + T \)
$71$ \( -8 + T \)
$73$ \( -1 + T \)
$79$ \( 16 + T \)
$83$ \( 11 + T \)
$89$ \( -6 + T \)
$97$ \( 1 + T \)
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