# Properties

 Label 324.2.e.b Level $324$ Weight $2$ Character orbit 324.e Analytic conductor $2.587$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -5 + 5 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -5 + 5 \zeta_{6} ) q^{7} + 7 \zeta_{6} q^{13} - q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{31} - q^{37} + ( -8 + 8 \zeta_{6} ) q^{43} -18 \zeta_{6} q^{49} + ( 13 - 13 \zeta_{6} ) q^{61} -11 \zeta_{6} q^{67} + 17 q^{73} + ( 13 - 13 \zeta_{6} ) q^{79} -35 q^{91} + ( -5 + 5 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 5q^{7} + O(q^{10})$$ $$2q - 5q^{7} + 7q^{13} - 2q^{19} + 5q^{25} + 4q^{31} - 2q^{37} - 8q^{43} - 18q^{49} + 13q^{61} - 11q^{67} + 34q^{73} + 13q^{79} - 70q^{91} - 5q^{97} + O(q^{100})$$

## 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 0 0 −2.50000 4.33013i 0 0 0
217.1 0 0 0 0 0 −2.50000 + 4.33013i 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 324.2.e.b 2
3.b odd 2 1 CM 324.2.e.b 2
4.b odd 2 1 1296.2.i.j 2
9.c even 3 1 108.2.a.a 1
9.c even 3 1 inner 324.2.e.b 2
9.d odd 6 1 108.2.a.a 1
9.d odd 6 1 inner 324.2.e.b 2
12.b even 2 1 1296.2.i.j 2
36.f odd 6 1 432.2.a.d 1
36.f odd 6 1 1296.2.i.j 2
36.h even 6 1 432.2.a.d 1
36.h even 6 1 1296.2.i.j 2
45.h odd 6 1 2700.2.a.b 1
45.j even 6 1 2700.2.a.b 1
45.k odd 12 2 2700.2.d.g 2
45.l even 12 2 2700.2.d.g 2
63.l odd 6 1 5292.2.a.j 1
63.o even 6 1 5292.2.a.j 1
72.j odd 6 1 1728.2.a.p 1
72.l even 6 1 1728.2.a.m 1
72.n even 6 1 1728.2.a.p 1
72.p odd 6 1 1728.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.a.a 1 9.c even 3 1
108.2.a.a 1 9.d odd 6 1
324.2.e.b 2 1.a even 1 1 trivial
324.2.e.b 2 3.b odd 2 1 CM
324.2.e.b 2 9.c even 3 1 inner
324.2.e.b 2 9.d odd 6 1 inner
432.2.a.d 1 36.f odd 6 1
432.2.a.d 1 36.h even 6 1
1296.2.i.j 2 4.b odd 2 1
1296.2.i.j 2 12.b even 2 1
1296.2.i.j 2 36.f odd 6 1
1296.2.i.j 2 36.h even 6 1
1728.2.a.m 1 72.l even 6 1
1728.2.a.m 1 72.p odd 6 1
1728.2.a.p 1 72.j odd 6 1
1728.2.a.p 1 72.n even 6 1
2700.2.a.b 1 45.h odd 6 1
2700.2.a.b 1 45.j even 6 1
2700.2.d.g 2 45.k odd 12 2
2700.2.d.g 2 45.l even 12 2
5292.2.a.j 1 63.l odd 6 1
5292.2.a.j 1 63.o even 6 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{2} + 5 T_{7} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$25 + 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$49 - 7 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$( 1 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -17 + T )^{2}$$
$79$ $$169 - 13 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$25 + 5 T + T^{2}$$