Properties

Label 1296.2.a.i
Level $1296$
Weight $2$
Character orbit 1296.a
Self dual yes
Analytic conductor $10.349$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.3486121020\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} + 3q^{7} + O(q^{10}) \) \( q + q^{5} + 3q^{7} + 5q^{11} - 5q^{13} + 2q^{17} + 4q^{19} - q^{23} - 4q^{25} + 9q^{29} + q^{31} + 3q^{35} - 6q^{37} - 3q^{41} - q^{43} - 3q^{47} + 2q^{49} - 2q^{53} + 5q^{55} + 11q^{59} + 7q^{61} - 5q^{65} + q^{67} + 4q^{71} - 2q^{73} + 15q^{77} - q^{79} + q^{83} + 2q^{85} + 18q^{89} - 15q^{91} + 4q^{95} - 13q^{97} + O(q^{100}) \)

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 1296.2.a.i 1
3.b odd 2 1 1296.2.a.e 1
4.b odd 2 1 648.2.a.c 1
8.b even 2 1 5184.2.a.n 1
8.d odd 2 1 5184.2.a.i 1
9.c even 3 2 432.2.i.a 2
9.d odd 6 2 144.2.i.b 2
12.b even 2 1 648.2.a.a 1
24.f even 2 1 5184.2.a.s 1
24.h odd 2 1 5184.2.a.x 1
36.f odd 6 2 216.2.i.a 2
36.h even 6 2 72.2.i.a 2
72.j odd 6 2 576.2.i.c 2
72.l even 6 2 576.2.i.d 2
72.n even 6 2 1728.2.i.g 2
72.p odd 6 2 1728.2.i.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.a 2 36.h even 6 2
144.2.i.b 2 9.d odd 6 2
216.2.i.a 2 36.f odd 6 2
432.2.i.a 2 9.c even 3 2
576.2.i.c 2 72.j odd 6 2
576.2.i.d 2 72.l even 6 2
648.2.a.a 1 12.b even 2 1
648.2.a.c 1 4.b odd 2 1
1296.2.a.e 1 3.b odd 2 1
1296.2.a.i 1 1.a even 1 1 trivial
1728.2.i.g 2 72.n even 6 2
1728.2.i.h 2 72.p odd 6 2
5184.2.a.i 1 8.d odd 2 1
5184.2.a.n 1 8.b even 2 1
5184.2.a.s 1 24.f even 2 1
5184.2.a.x 1 24.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1296))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( -3 + T \)
$11$ \( -5 + T \)
$13$ \( 5 + T \)
$17$ \( -2 + T \)
$19$ \( -4 + T \)
$23$ \( 1 + T \)
$29$ \( -9 + T \)
$31$ \( -1 + T \)
$37$ \( 6 + T \)
$41$ \( 3 + T \)
$43$ \( 1 + T \)
$47$ \( 3 + T \)
$53$ \( 2 + T \)
$59$ \( -11 + T \)
$61$ \( -7 + T \)
$67$ \( -1 + T \)
$71$ \( -4 + T \)
$73$ \( 2 + T \)
$79$ \( 1 + T \)
$83$ \( -1 + T \)
$89$ \( -18 + T \)
$97$ \( 13 + T \)
show more
show less