Properties

Label 216.1.h.a
Level $216$
Weight $1$
Character orbit 216.h
Self dual yes
Analytic conductor $0.108$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -24
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 216.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.107798042729\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.216.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.139968.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} + q^{11} + q^{14} + q^{16} + q^{20} - q^{22} - q^{28} - 2q^{29} - q^{31} - q^{32} - q^{35} - q^{40} + q^{44} + q^{53} + q^{55} + q^{56} + 2q^{58} - 2q^{59} + q^{62} + q^{64} + q^{70} - q^{73} - q^{77} + 2q^{79} + q^{80} + q^{83} - q^{88} - q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0
−1.00000 0 1.00000 1.00000 0 −1.00000 −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.1.h.a 1
3.b odd 2 1 216.1.h.b yes 1
4.b odd 2 1 864.1.h.b 1
8.b even 2 1 216.1.h.b yes 1
8.d odd 2 1 864.1.h.a 1
9.c even 3 2 648.1.j.b 2
9.d odd 6 2 648.1.j.a 2
12.b even 2 1 864.1.h.a 1
24.f even 2 1 864.1.h.b 1
24.h odd 2 1 CM 216.1.h.a 1
36.f odd 6 2 2592.1.n.a 2
36.h even 6 2 2592.1.n.b 2
72.j odd 6 2 648.1.j.b 2
72.l even 6 2 2592.1.n.a 2
72.n even 6 2 648.1.j.a 2
72.p odd 6 2 2592.1.n.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.1.h.a 1 1.a even 1 1 trivial
216.1.h.a 1 24.h odd 2 1 CM
216.1.h.b yes 1 3.b odd 2 1
216.1.h.b yes 1 8.b even 2 1
648.1.j.a 2 9.d odd 6 2
648.1.j.a 2 72.n even 6 2
648.1.j.b 2 9.c even 3 2
648.1.j.b 2 72.j odd 6 2
864.1.h.a 1 8.d odd 2 1
864.1.h.a 1 12.b even 2 1
864.1.h.b 1 4.b odd 2 1
864.1.h.b 1 24.f even 2 1
2592.1.n.a 2 36.f odd 6 2
2592.1.n.a 2 72.l even 6 2
2592.1.n.b 2 36.h even 6 2
2592.1.n.b 2 72.p odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 1 \) acting on \(S_{1}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

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