Properties

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

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{5} + q^{7} + O(q^{10}) \) \( q - 3q^{5} + q^{7} + 3q^{11} - q^{13} - 6q^{17} + 4q^{19} - 3q^{23} + 4q^{25} - 3q^{29} - 5q^{31} - 3q^{35} + 2q^{37} - 3q^{41} + q^{43} - 9q^{47} - 6q^{49} + 6q^{53} - 9q^{55} - 3q^{59} - 13q^{61} + 3q^{65} + 7q^{67} - 12q^{71} - 10q^{73} + 3q^{77} - 11q^{79} - 9q^{83} + 18q^{85} - 6q^{89} - q^{91} - 12q^{95} + 11q^{97} + O(q^{100}) \)

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 −3.00000 0 1.00000 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.b 1
3.b odd 2 1 1296.2.a.k 1
4.b odd 2 1 324.2.a.a 1
8.b even 2 1 5184.2.a.bb 1
8.d odd 2 1 5184.2.a.ba 1
9.c even 3 2 432.2.i.c 2
9.d odd 6 2 144.2.i.a 2
12.b even 2 1 324.2.a.c 1
20.d odd 2 1 8100.2.a.g 1
20.e even 4 2 8100.2.d.c 2
24.f even 2 1 5184.2.a.e 1
24.h odd 2 1 5184.2.a.f 1
36.f odd 6 2 108.2.e.a 2
36.h even 6 2 36.2.e.a 2
60.h even 2 1 8100.2.a.j 1
60.l odd 4 2 8100.2.d.h 2
72.j odd 6 2 576.2.i.e 2
72.l even 6 2 576.2.i.f 2
72.n even 6 2 1728.2.i.c 2
72.p odd 6 2 1728.2.i.d 2
180.n even 6 2 900.2.i.b 2
180.p odd 6 2 2700.2.i.b 2
180.v odd 12 4 900.2.s.b 4
180.x even 12 4 2700.2.s.b 4
252.n even 6 2 5292.2.l.c 2
252.o even 6 2 1764.2.l.c 2
252.r odd 6 2 1764.2.i.c 2
252.s odd 6 2 1764.2.j.b 2
252.u odd 6 2 5292.2.i.c 2
252.bb even 6 2 1764.2.i.a 2
252.bi even 6 2 5292.2.j.a 2
252.bj even 6 2 5292.2.i.a 2
252.bl odd 6 2 5292.2.l.a 2
252.bn odd 6 2 1764.2.l.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 36.h even 6 2
108.2.e.a 2 36.f odd 6 2
144.2.i.a 2 9.d odd 6 2
324.2.a.a 1 4.b odd 2 1
324.2.a.c 1 12.b even 2 1
432.2.i.c 2 9.c even 3 2
576.2.i.e 2 72.j odd 6 2
576.2.i.f 2 72.l even 6 2
900.2.i.b 2 180.n even 6 2
900.2.s.b 4 180.v odd 12 4
1296.2.a.b 1 1.a even 1 1 trivial
1296.2.a.k 1 3.b odd 2 1
1728.2.i.c 2 72.n even 6 2
1728.2.i.d 2 72.p odd 6 2
1764.2.i.a 2 252.bb even 6 2
1764.2.i.c 2 252.r odd 6 2
1764.2.j.b 2 252.s odd 6 2
1764.2.l.a 2 252.bn odd 6 2
1764.2.l.c 2 252.o even 6 2
2700.2.i.b 2 180.p odd 6 2
2700.2.s.b 4 180.x even 12 4
5184.2.a.e 1 24.f even 2 1
5184.2.a.f 1 24.h odd 2 1
5184.2.a.ba 1 8.d odd 2 1
5184.2.a.bb 1 8.b even 2 1
5292.2.i.a 2 252.bj even 6 2
5292.2.i.c 2 252.u odd 6 2
5292.2.j.a 2 252.bi even 6 2
5292.2.l.a 2 252.bl odd 6 2
5292.2.l.c 2 252.n even 6 2
8100.2.a.g 1 20.d odd 2 1
8100.2.a.j 1 60.h even 2 1
8100.2.d.c 2 20.e even 4 2
8100.2.d.h 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1296))\):

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

Hecke characteristic polynomials

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