Properties

Label 216.2.a.d
Level $216$
Weight $2$
Character orbit 216.a
Self dual yes
Analytic conductor $1.725$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{5} - 3q^{7} + O(q^{10}) \) \( q + 4q^{5} - 3q^{7} + 4q^{11} + q^{13} - 4q^{17} - q^{19} + 4q^{23} + 11q^{25} - 4q^{31} - 12q^{35} - 9q^{37} - 8q^{43} - 12q^{47} + 2q^{49} - 8q^{53} + 16q^{55} + 4q^{59} - 5q^{61} + 4q^{65} + 11q^{67} + 8q^{71} + q^{73} - 12q^{77} - 5q^{79} + 8q^{83} - 16q^{85} + 12q^{89} - 3q^{91} - 4q^{95} + 5q^{97} + O(q^{100}) \)

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 4.00000 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 216.2.a.d yes 1
3.b odd 2 1 216.2.a.a 1
4.b odd 2 1 432.2.a.h 1
5.b even 2 1 5400.2.a.bp 1
5.c odd 4 2 5400.2.f.v 2
8.b even 2 1 1728.2.a.a 1
8.d odd 2 1 1728.2.a.b 1
9.c even 3 2 648.2.i.a 2
9.d odd 6 2 648.2.i.h 2
12.b even 2 1 432.2.a.a 1
15.d odd 2 1 5400.2.a.bn 1
15.e even 4 2 5400.2.f.e 2
24.f even 2 1 1728.2.a.bb 1
24.h odd 2 1 1728.2.a.ba 1
36.f odd 6 2 1296.2.i.a 2
36.h even 6 2 1296.2.i.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.a 1 3.b odd 2 1
216.2.a.d yes 1 1.a even 1 1 trivial
432.2.a.a 1 12.b even 2 1
432.2.a.h 1 4.b odd 2 1
648.2.i.a 2 9.c even 3 2
648.2.i.h 2 9.d odd 6 2
1296.2.i.a 2 36.f odd 6 2
1296.2.i.q 2 36.h even 6 2
1728.2.a.a 1 8.b even 2 1
1728.2.a.b 1 8.d odd 2 1
1728.2.a.ba 1 24.h odd 2 1
1728.2.a.bb 1 24.f even 2 1
5400.2.a.bn 1 15.d odd 2 1
5400.2.a.bp 1 5.b even 2 1
5400.2.f.e 2 15.e even 4 2
5400.2.f.v 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(216))\).

Hecke characteristic polynomials

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