Properties

Label 216.4.a.b
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

\(f(q)\) \(=\) \( q - q^{5} - 9q^{7} + O(q^{10}) \) \( q - q^{5} - 9q^{7} + 17q^{11} - 44q^{13} - 56q^{17} - 94q^{19} + 50q^{23} - 124q^{25} + 30q^{29} - 139q^{31} + 9q^{35} - 174q^{37} - 318q^{41} - 242q^{43} + 630q^{47} - 262q^{49} - 547q^{53} - 17q^{55} + 236q^{59} + 328q^{61} + 44q^{65} + 614q^{67} - 296q^{71} + 433q^{73} - 153q^{77} - 56q^{79} + 1225q^{83} + 56q^{85} - 1506q^{89} + 396q^{91} + 94q^{95} + 1391q^{97} + O(q^{100}) \)

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 −9.00000 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.b 1
3.b odd 2 1 216.4.a.c yes 1
4.b odd 2 1 432.4.a.f 1
8.b even 2 1 1728.4.a.s 1
8.d odd 2 1 1728.4.a.t 1
9.c even 3 2 648.4.i.g 2
9.d odd 6 2 648.4.i.f 2
12.b even 2 1 432.4.a.i 1
24.f even 2 1 1728.4.a.n 1
24.h odd 2 1 1728.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.b 1 1.a even 1 1 trivial
216.4.a.c yes 1 3.b odd 2 1
432.4.a.f 1 4.b odd 2 1
432.4.a.i 1 12.b even 2 1
648.4.i.f 2 9.d odd 6 2
648.4.i.g 2 9.c even 3 2
1728.4.a.m 1 24.h odd 2 1
1728.4.a.n 1 24.f even 2 1
1728.4.a.s 1 8.b even 2 1
1728.4.a.t 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(216))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( 9 + T \)
$11$ \( -17 + T \)
$13$ \( 44 + T \)
$17$ \( 56 + T \)
$19$ \( 94 + T \)
$23$ \( -50 + T \)
$29$ \( -30 + T \)
$31$ \( 139 + T \)
$37$ \( 174 + T \)
$41$ \( 318 + T \)
$43$ \( 242 + T \)
$47$ \( -630 + T \)
$53$ \( 547 + T \)
$59$ \( -236 + T \)
$61$ \( -328 + T \)
$67$ \( -614 + T \)
$71$ \( 296 + T \)
$73$ \( -433 + T \)
$79$ \( 56 + T \)
$83$ \( -1225 + T \)
$89$ \( 1506 + T \)
$97$ \( -1391 + T \)
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