# Properties

 Label 324.1.g.a Level $324$ Weight $1$ Character orbit 324.g Analytic conductor $0.162$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 324.g (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.161697064093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.108.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.314928.1

## $q$-expansion

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$$.

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 0.500000 + 0.866025i 0 0 0
269.1 0 0 0 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 324.1.g.a 2
3.b odd 2 1 CM 324.1.g.a 2
4.b odd 2 1 1296.1.q.a 2
9.c even 3 1 108.1.c.a 1
9.c even 3 1 inner 324.1.g.a 2
9.d odd 6 1 108.1.c.a 1
9.d odd 6 1 inner 324.1.g.a 2
12.b even 2 1 1296.1.q.a 2
27.e even 9 6 2916.1.k.c 6
27.f odd 18 6 2916.1.k.c 6
36.f odd 6 1 432.1.e.a 1
36.f odd 6 1 1296.1.q.a 2
36.h even 6 1 432.1.e.a 1
36.h even 6 1 1296.1.q.a 2
45.h odd 6 1 2700.1.g.b 1
45.j even 6 1 2700.1.g.b 1
45.k odd 12 2 2700.1.b.b 2
45.l even 12 2 2700.1.b.b 2
72.j odd 6 1 1728.1.e.a 1
72.l even 6 1 1728.1.e.b 1
72.n even 6 1 1728.1.e.a 1
72.p odd 6 1 1728.1.e.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.1.c.a 1 9.c even 3 1
108.1.c.a 1 9.d odd 6 1
324.1.g.a 2 1.a even 1 1 trivial
324.1.g.a 2 3.b odd 2 1 CM
324.1.g.a 2 9.c even 3 1 inner
324.1.g.a 2 9.d odd 6 1 inner
432.1.e.a 1 36.f odd 6 1
432.1.e.a 1 36.h even 6 1
1296.1.q.a 2 4.b odd 2 1
1296.1.q.a 2 12.b even 2 1
1296.1.q.a 2 36.f odd 6 1
1296.1.q.a 2 36.h even 6 1
1728.1.e.a 1 72.j odd 6 1
1728.1.e.a 1 72.n even 6 1
1728.1.e.b 1 72.l even 6 1
1728.1.e.b 1 72.p odd 6 1
2700.1.b.b 2 45.k odd 12 2
2700.1.b.b 2 45.l even 12 2
2700.1.g.b 1 45.h odd 6 1
2700.1.g.b 1 45.j even 6 1
2916.1.k.c 6 27.e even 9 6
2916.1.k.c 6 27.f odd 18 6

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

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