# Properties

 Label 324.2.e.d Level $324$ Weight $2$ Character orbit 324.e Analytic conductor $2.587$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.58715302549$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 3 \zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} + ( -6 + 6 \zeta_{6} ) q^{11} -5 \zeta_{6} q^{13} + 3 q^{17} + 2 q^{19} + 6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( 3 - 3 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} -6 q^{35} + 5 q^{37} -6 \zeta_{6} q^{41} + ( 10 - 10 \zeta_{6} ) q^{43} + 3 \zeta_{6} q^{49} + 6 q^{53} -18 q^{55} -12 \zeta_{6} q^{59} + ( -5 + 5 \zeta_{6} ) q^{61} + ( 15 - 15 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} -6 q^{71} - q^{73} -12 \zeta_{6} q^{77} + ( 10 - 10 \zeta_{6} ) q^{79} + 9 \zeta_{6} q^{85} + 3 q^{89} + 10 q^{91} + 6 \zeta_{6} q^{95} + ( 10 - 10 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{5} - 2 q^{7} + O(q^{10})$$ $$2 q + 3 q^{5} - 2 q^{7} - 6 q^{11} - 5 q^{13} + 6 q^{17} + 4 q^{19} + 6 q^{23} - 4 q^{25} + 3 q^{29} + 4 q^{31} - 12 q^{35} + 10 q^{37} - 6 q^{41} + 10 q^{43} + 3 q^{49} + 12 q^{53} - 36 q^{55} - 12 q^{59} - 5 q^{61} + 15 q^{65} - 2 q^{67} - 12 q^{71} - 2 q^{73} - 12 q^{77} + 10 q^{79} + 9 q^{85} + 6 q^{89} + 20 q^{91} + 6 q^{95} + 10 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 1.50000 2.59808i 0 −1.00000 1.73205i 0 0 0
217.1 0 0 0 1.50000 + 2.59808i 0 −1.00000 + 1.73205i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.e.d 2
3.b odd 2 1 324.2.e.a 2
4.b odd 2 1 1296.2.i.p 2
9.c even 3 1 324.2.a.b 1
9.c even 3 1 inner 324.2.e.d 2
9.d odd 6 1 324.2.a.d yes 1
9.d odd 6 1 324.2.e.a 2
12.b even 2 1 1296.2.i.d 2
36.f odd 6 1 1296.2.a.a 1
36.f odd 6 1 1296.2.i.p 2
36.h even 6 1 1296.2.a.j 1
36.h even 6 1 1296.2.i.d 2
45.h odd 6 1 8100.2.a.a 1
45.j even 6 1 8100.2.a.f 1
45.k odd 12 2 8100.2.d.j 2
45.l even 12 2 8100.2.d.a 2
72.j odd 6 1 5184.2.a.g 1
72.l even 6 1 5184.2.a.d 1
72.n even 6 1 5184.2.a.bc 1
72.p odd 6 1 5184.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.a.b 1 9.c even 3 1
324.2.a.d yes 1 9.d odd 6 1
324.2.e.a 2 3.b odd 2 1
324.2.e.a 2 9.d odd 6 1
324.2.e.d 2 1.a even 1 1 trivial
324.2.e.d 2 9.c even 3 1 inner
1296.2.a.a 1 36.f odd 6 1
1296.2.a.j 1 36.h even 6 1
1296.2.i.d 2 12.b even 2 1
1296.2.i.d 2 36.h even 6 1
1296.2.i.p 2 4.b odd 2 1
1296.2.i.p 2 36.f odd 6 1
5184.2.a.d 1 72.l even 6 1
5184.2.a.g 1 72.j odd 6 1
5184.2.a.z 1 72.p odd 6 1
5184.2.a.bc 1 72.n even 6 1
8100.2.a.a 1 45.h odd 6 1
8100.2.a.f 1 45.j even 6 1
8100.2.d.a 2 45.l even 12 2
8100.2.d.j 2 45.k odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{7}^{2} + 2 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$9 - 3 T + T^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$( -5 + T )^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$100 - 10 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$25 + 5 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$( 1 + T )^{2}$$
$79$ $$100 - 10 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -3 + T )^{2}$$
$97$ $$100 - 10 T + T^{2}$$