# Properties

 Label 243.1.d.a Level $243$ Weight $1$ Character orbit 243.d Analytic conductor $0.121$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$243 = 3^{5}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 243.d (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.121272798070$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.243.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.177147.1

## $q$-expansion

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/243\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{6}^{2}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 −0.500000 0.866025i 0 0 0.500000 0.866025i 0 0 0
161.1 0 0 −0.500000 + 0.866025i 0 0 0.500000 + 0.866025i 0 0 0
 $$n$$: 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.1.b.a 1 9.c even 3 1
243.1.b.a 1 9.d odd 6 1
243.1.d.a 2 1.a even 1 1 trivial
243.1.d.a 2 3.b odd 2 1 CM
243.1.d.a 2 9.c even 3 1 inner
243.1.d.a 2 9.d odd 6 1 inner
729.1.f.a 6 27.e even 9 6
729.1.f.a 6 27.f odd 18 6
3888.1.e.b 1 36.f odd 6 1
3888.1.e.b 1 36.h even 6 1
3888.1.q.b 2 4.b odd 2 1
3888.1.q.b 2 12.b even 2 1
3888.1.q.b 2 36.f odd 6 1
3888.1.q.b 2 36.h even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(243, [\chi])$$.

## Hecke characteristic polynomials

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