# Properties

 Label 729.1.f.a Level $729$ Weight $1$ Character orbit 729.f Analytic conductor $0.364$ Analytic rank $0$ Dimension $6$ Projective image $D_{3}$ CM discriminant -3 Inner twists $12$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 729.f (of order $$18$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.363818394209$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 243) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.243.1 Artin image: $C_9\times S_3$ Artin field: Galois closure of 18.0.2954312706550833698643.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{18} q^{4} - \zeta_{18}^{8} q^{7} +O(q^{10})$$ q - z * q^4 - z^8 * q^7 $$q - \zeta_{18} q^{4} - \zeta_{18}^{8} q^{7} - \zeta_{18}^{4} q^{13} + \zeta_{18}^{2} q^{16} - \zeta_{18}^{6} q^{19} - \zeta_{18}^{5} q^{25} - q^{28} + \zeta_{18} q^{31} + \zeta_{18}^{3} q^{37} - \zeta_{18}^{2} q^{43} + \zeta_{18}^{5} q^{52} + \zeta_{18}^{8} q^{61} - \zeta_{18}^{3} q^{64} + \zeta_{18}^{4} q^{67} + \zeta_{18}^{6} q^{73} + \zeta_{18}^{7} q^{76} + \zeta_{18}^{5} q^{79} - \zeta_{18}^{3} q^{91} - \zeta_{18}^{2} q^{97} +O(q^{100})$$ q - z * q^4 - z^8 * q^7 - z^4 * q^13 + z^2 * q^16 - z^6 * q^19 - z^5 * q^25 - q^28 + z * q^31 + z^3 * q^37 - z^2 * q^43 + z^5 * q^52 + z^8 * q^61 - z^3 * q^64 + z^4 * q^67 + z^6 * q^73 + z^7 * q^76 + z^5 * q^79 - z^3 * q^91 - z^2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q + 3 q^{19} - 6 q^{28} + 3 q^{37} - 3 q^{64} - 6 q^{73} - 3 q^{91}+O(q^{100})$$ 6 * q + 3 * q^19 - 6 * q^28 + 3 * q^37 - 3 * q^64 - 6 * q^73 - 3 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{18}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 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 0 −0.939693 0.342020i 0 0 0.939693 0.342020i 0 0 0
161.1 0 0 0.766044 + 0.642788i 0 0 −0.766044 + 0.642788i 0 0 0
323.1 0 0 0.173648 + 0.984808i 0 0 −0.173648 + 0.984808i 0 0 0
404.1 0 0 0.173648 0.984808i 0 0 −0.173648 0.984808i 0 0 0
566.1 0 0 0.766044 0.642788i 0 0 −0.766044 0.642788i 0 0 0
647.1 0 0 −0.939693 + 0.342020i 0 0 0.939693 + 0.342020i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 647.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
9.c even 3 2 inner
9.d odd 6 2 inner
27.e even 9 3 inner
27.f odd 18 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.1.f.a 6
3.b odd 2 1 CM 729.1.f.a 6
9.c even 3 2 inner 729.1.f.a 6
9.d odd 6 2 inner 729.1.f.a 6
27.e even 9 1 243.1.b.a 1
27.e even 9 2 243.1.d.a 2
27.e even 9 3 inner 729.1.f.a 6
27.f odd 18 1 243.1.b.a 1
27.f odd 18 2 243.1.d.a 2
27.f odd 18 3 inner 729.1.f.a 6
108.j odd 18 1 3888.1.e.b 1
108.j odd 18 2 3888.1.q.b 2
108.l even 18 1 3888.1.e.b 1
108.l even 18 2 3888.1.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.1.b.a 1 27.e even 9 1
243.1.b.a 1 27.f odd 18 1
243.1.d.a 2 27.e even 9 2
243.1.d.a 2 27.f odd 18 2
729.1.f.a 6 1.a even 1 1 trivial
729.1.f.a 6 3.b odd 2 1 CM
729.1.f.a 6 9.c even 3 2 inner
729.1.f.a 6 9.d odd 6 2 inner
729.1.f.a 6 27.e even 9 3 inner
729.1.f.a 6 27.f odd 18 3 inner
3888.1.e.b 1 108.j odd 18 1
3888.1.e.b 1 108.l even 18 1
3888.1.q.b 2 108.j odd 18 2
3888.1.q.b 2 108.l even 18 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(729, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6} - T^{3} + 1$$
$11$ $$T^{6}$$
$13$ $$T^{6} - T^{3} + 1$$
$17$ $$T^{6}$$
$19$ $$(T^{2} - T + 1)^{3}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6} - T^{3} + 1$$
$37$ $$(T^{2} - T + 1)^{3}$$
$41$ $$T^{6}$$
$43$ $$T^{6} - T^{3} + 1$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$T^{6} + 8T^{3} + 64$$
$67$ $$T^{6} + 8T^{3} + 64$$
$71$ $$T^{6}$$
$73$ $$(T^{2} + 2 T + 4)^{3}$$
$79$ $$T^{6} - T^{3} + 1$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6} - T^{3} + 1$$