# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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