# Properties

 Label 216.1.r.a Level $216$ Weight $1$ Character orbit 216.r Analytic conductor $0.108$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 216.r (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.107798042729$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.128536820158464.7

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{2} q^{2} + \zeta_{18}^{8} q^{3} + \zeta_{18}^{4} q^{4} -\zeta_{18} q^{6} + \zeta_{18}^{6} q^{8} -\zeta_{18}^{7} q^{9} +O(q^{10})$$ $$q + \zeta_{18}^{2} q^{2} + \zeta_{18}^{8} q^{3} + \zeta_{18}^{4} q^{4} -\zeta_{18} q^{6} + \zeta_{18}^{6} q^{8} -\zeta_{18}^{7} q^{9} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} ) q^{11} -\zeta_{18}^{3} q^{12} + \zeta_{18}^{8} q^{16} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{17} + q^{18} + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{19} + ( -\zeta_{18}^{5} + \zeta_{18}^{6} ) q^{22} -\zeta_{18}^{5} q^{24} + \zeta_{18}^{2} q^{25} + \zeta_{18}^{6} q^{27} -\zeta_{18} q^{32} + ( \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{33} + ( -\zeta_{18}^{3} - \zeta_{18}^{7} ) q^{34} + \zeta_{18}^{2} q^{36} + ( 1 - \zeta_{18}^{7} ) q^{38} + ( \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{41} + ( -\zeta_{18} + \zeta_{18}^{6} ) q^{43} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{44} -\zeta_{18}^{7} q^{48} -\zeta_{18} q^{49} + \zeta_{18}^{4} q^{50} + ( 1 + \zeta_{18}^{4} ) q^{51} + \zeta_{18}^{8} q^{54} + ( \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{57} + ( 1 + \zeta_{18}^{2} ) q^{59} -\zeta_{18}^{3} q^{64} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{66} + ( \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{67} + ( 1 - \zeta_{18}^{5} ) q^{68} + \zeta_{18}^{4} q^{72} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{73} -\zeta_{18} q^{75} + ( 1 + \zeta_{18}^{2} ) q^{76} -\zeta_{18}^{5} q^{81} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{82} -\zeta_{18}^{2} q^{83} + ( -\zeta_{18}^{3} + \zeta_{18}^{8} ) q^{86} + ( 1 - \zeta_{18} ) q^{88} -\zeta_{18}^{6} q^{89} + q^{96} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} ) q^{97} -\zeta_{18}^{3} q^{98} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 3q^{8} + O(q^{10})$$ $$6q - 3q^{8} - 3q^{11} - 3q^{12} + 6q^{18} - 3q^{22} - 3q^{27} - 3q^{33} - 3q^{34} + 6q^{38} - 3q^{41} - 3q^{43} + 6q^{51} - 3q^{57} + 6q^{59} - 3q^{64} - 3q^{67} + 6q^{68} + 6q^{76} - 3q^{86} + 6q^{88} + 3q^{89} + 6q^{96} - 3q^{97} - 3q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$55$$ $$109$$ $$137$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{18}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 0.939693 + 0.342020i −0.766044 − 0.642788i −0.173648 − 0.984808i −0.173648 + 0.984808i −0.766044 + 0.642788i 0.939693 − 0.342020i
0.766044 + 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i 0 −0.939693 0.342020i 0 −0.500000 + 0.866025i 0.766044 0.642788i 0
67.1 0.173648 + 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 0 0.766044 + 0.642788i 0 −0.500000 0.866025i 0.173648 0.984808i 0
115.1 −0.939693 + 0.342020i 0.173648 0.984808i 0.766044 0.642788i 0 0.173648 + 0.984808i 0 −0.500000 + 0.866025i −0.939693 0.342020i 0
139.1 −0.939693 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i 0 0.173648 0.984808i 0 −0.500000 0.866025i −0.939693 + 0.342020i 0
187.1 0.173648 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 0 0.766044 0.642788i 0 −0.500000 + 0.866025i 0.173648 + 0.984808i 0
211.1 0.766044 0.642788i −0.939693 0.342020i 0.173648 0.984808i 0 −0.939693 + 0.342020i 0 −0.500000 0.866025i 0.766044 + 0.642788i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 211.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
27.e even 9 1 inner
216.r odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.1.r.a 6
3.b odd 2 1 648.1.r.a 6
4.b odd 2 1 864.1.bd.a 6
8.b even 2 1 864.1.bd.a 6
8.d odd 2 1 CM 216.1.r.a 6
9.c even 3 1 1944.1.r.a 6
9.c even 3 1 1944.1.r.b 6
9.d odd 6 1 1944.1.r.c 6
9.d odd 6 1 1944.1.r.d 6
12.b even 2 1 2592.1.bd.a 6
24.f even 2 1 648.1.r.a 6
24.h odd 2 1 2592.1.bd.a 6
27.e even 9 1 inner 216.1.r.a 6
27.e even 9 1 1944.1.r.a 6
27.e even 9 1 1944.1.r.b 6
27.f odd 18 1 648.1.r.a 6
27.f odd 18 1 1944.1.r.c 6
27.f odd 18 1 1944.1.r.d 6
72.l even 6 1 1944.1.r.c 6
72.l even 6 1 1944.1.r.d 6
72.p odd 6 1 1944.1.r.a 6
72.p odd 6 1 1944.1.r.b 6
108.j odd 18 1 864.1.bd.a 6
108.l even 18 1 2592.1.bd.a 6
216.r odd 18 1 inner 216.1.r.a 6
216.r odd 18 1 1944.1.r.a 6
216.r odd 18 1 1944.1.r.b 6
216.t even 18 1 864.1.bd.a 6
216.v even 18 1 648.1.r.a 6
216.v even 18 1 1944.1.r.c 6
216.v even 18 1 1944.1.r.d 6
216.x odd 18 1 2592.1.bd.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.1.r.a 6 1.a even 1 1 trivial
216.1.r.a 6 8.d odd 2 1 CM
216.1.r.a 6 27.e even 9 1 inner
216.1.r.a 6 216.r odd 18 1 inner
648.1.r.a 6 3.b odd 2 1
648.1.r.a 6 24.f even 2 1
648.1.r.a 6 27.f odd 18 1
648.1.r.a 6 216.v even 18 1
864.1.bd.a 6 4.b odd 2 1
864.1.bd.a 6 8.b even 2 1
864.1.bd.a 6 108.j odd 18 1
864.1.bd.a 6 216.t even 18 1
1944.1.r.a 6 9.c even 3 1
1944.1.r.a 6 27.e even 9 1
1944.1.r.a 6 72.p odd 6 1
1944.1.r.a 6 216.r odd 18 1
1944.1.r.b 6 9.c even 3 1
1944.1.r.b 6 27.e even 9 1
1944.1.r.b 6 72.p odd 6 1
1944.1.r.b 6 216.r odd 18 1
1944.1.r.c 6 9.d odd 6 1
1944.1.r.c 6 27.f odd 18 1
1944.1.r.c 6 72.l even 6 1
1944.1.r.c 6 216.v even 18 1
1944.1.r.d 6 9.d odd 6 1
1944.1.r.d 6 27.f odd 18 1
1944.1.r.d 6 72.l even 6 1
1944.1.r.d 6 216.v even 18 1
2592.1.bd.a 6 12.b even 2 1
2592.1.bd.a 6 24.h odd 2 1
2592.1.bd.a 6 108.l even 18 1
2592.1.bd.a 6 216.x odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{3} + T^{6}$$
$3$ $$1 + T^{3} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$19$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6}$$
$37$ $$T^{6}$$
$41$ $$1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$43$ $$1 - 3 T + 3 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6}$$
$61$ $$T^{6}$$
$67$ $$1 - 3 T + 3 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$71$ $$T^{6}$$
$73$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$79$ $$T^{6}$$
$83$ $$1 - T^{3} + T^{6}$$
$89$ $$( 1 - T + T^{2} )^{3}$$
$97$ $$1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$