Properties

Label 216.4.a.a
Level $216$
Weight $4$
Character orbit 216.a
Self dual yes
Analytic conductor $12.744$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.7444125612\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 4q^{5} + 3q^{7} + O(q^{10}) \) \( q - 4q^{5} + 3q^{7} - 28q^{11} - 11q^{13} - 44q^{17} + 29q^{19} - 172q^{23} - 109q^{25} - 192q^{29} + 116q^{31} - 12q^{35} - 69q^{37} - 384q^{41} + 328q^{43} - 156q^{47} - 334q^{49} + 392q^{53} + 112q^{55} - 412q^{59} - 425q^{61} + 44q^{65} + 257q^{67} + 1000q^{71} - 359q^{73} - 84q^{77} + 877q^{79} + 328q^{83} + 176q^{85} + 1572q^{89} - 33q^{91} - 116q^{95} - 1483q^{97} + O(q^{100}) \)

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 216.4.a.a 1
3.b odd 2 1 216.4.a.d yes 1
4.b odd 2 1 432.4.a.d 1
8.b even 2 1 1728.4.a.x 1
8.d odd 2 1 1728.4.a.w 1
9.c even 3 2 648.4.i.i 2
9.d odd 6 2 648.4.i.d 2
12.b even 2 1 432.4.a.k 1
24.f even 2 1 1728.4.a.i 1
24.h odd 2 1 1728.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.a 1 1.a even 1 1 trivial
216.4.a.d yes 1 3.b odd 2 1
432.4.a.d 1 4.b odd 2 1
432.4.a.k 1 12.b even 2 1
648.4.i.d 2 9.d odd 6 2
648.4.i.i 2 9.c even 3 2
1728.4.a.i 1 24.f even 2 1
1728.4.a.j 1 24.h odd 2 1
1728.4.a.w 1 8.d odd 2 1
1728.4.a.x 1 8.b even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 4 + T \)
$7$ \( -3 + T \)
$11$ \( 28 + T \)
$13$ \( 11 + T \)
$17$ \( 44 + T \)
$19$ \( -29 + T \)
$23$ \( 172 + T \)
$29$ \( 192 + T \)
$31$ \( -116 + T \)
$37$ \( 69 + T \)
$41$ \( 384 + T \)
$43$ \( -328 + T \)
$47$ \( 156 + T \)
$53$ \( -392 + T \)
$59$ \( 412 + T \)
$61$ \( 425 + T \)
$67$ \( -257 + T \)
$71$ \( -1000 + T \)
$73$ \( 359 + T \)
$79$ \( -877 + T \)
$83$ \( -328 + T \)
$89$ \( -1572 + T \)
$97$ \( 1483 + T \)
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