# Properties

 Label 324.2.e.c 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) 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.

 $$f(q)$$ $$=$$ $$q + ( 4 - 4 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 4 - 4 \zeta_{6} ) q^{7} -2 \zeta_{6} q^{13} + 8 q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{31} -10 q^{37} + ( -8 + 8 \zeta_{6} ) q^{43} -9 \zeta_{6} q^{49} + ( -14 + 14 \zeta_{6} ) q^{61} + 16 \zeta_{6} q^{67} -10 q^{73} + ( 4 - 4 \zeta_{6} ) q^{79} -8 q^{91} + ( -14 + 14 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} + O(q^{10})$$ $$2q + 4q^{7} - 2q^{13} + 16q^{19} + 5q^{25} + 4q^{31} - 20q^{37} - 8q^{43} - 9q^{49} - 14q^{61} + 16q^{67} - 20q^{73} + 4q^{79} - 16q^{91} - 14q^{97} + O(q^{100})$$

## 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 2.00000 + 3.46410i 0 0 0
217.1 0 0 0 0 0 2.00000 3.46410i 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.c 2
3.b odd 2 1 CM 324.2.e.c 2
4.b odd 2 1 1296.2.i.h 2
9.c even 3 1 36.2.a.a 1
9.c even 3 1 inner 324.2.e.c 2
9.d odd 6 1 36.2.a.a 1
9.d odd 6 1 inner 324.2.e.c 2
12.b even 2 1 1296.2.i.h 2
36.f odd 6 1 144.2.a.a 1
36.f odd 6 1 1296.2.i.h 2
36.h even 6 1 144.2.a.a 1
36.h even 6 1 1296.2.i.h 2
45.h odd 6 1 900.2.a.g 1
45.j even 6 1 900.2.a.g 1
45.k odd 12 2 900.2.d.b 2
45.l even 12 2 900.2.d.b 2
63.g even 3 1 1764.2.k.h 2
63.h even 3 1 1764.2.k.h 2
63.i even 6 1 1764.2.k.g 2
63.j odd 6 1 1764.2.k.h 2
63.k odd 6 1 1764.2.k.g 2
63.l odd 6 1 1764.2.a.e 1
63.n odd 6 1 1764.2.k.h 2
63.o even 6 1 1764.2.a.e 1
63.s even 6 1 1764.2.k.g 2
63.t odd 6 1 1764.2.k.g 2
72.j odd 6 1 576.2.a.e 1
72.l even 6 1 576.2.a.f 1
72.n even 6 1 576.2.a.e 1
72.p odd 6 1 576.2.a.f 1
99.g even 6 1 4356.2.a.g 1
99.h odd 6 1 4356.2.a.g 1
117.n odd 6 1 6084.2.a.i 1
117.t even 6 1 6084.2.a.i 1
117.y odd 12 2 6084.2.b.f 2
117.z even 12 2 6084.2.b.f 2
144.u even 12 2 2304.2.d.a 2
144.v odd 12 2 2304.2.d.a 2
144.w odd 12 2 2304.2.d.q 2
144.x even 12 2 2304.2.d.q 2
180.n even 6 1 3600.2.a.e 1
180.p odd 6 1 3600.2.a.e 1
180.v odd 12 2 3600.2.f.m 2
180.x even 12 2 3600.2.f.m 2
252.s odd 6 1 7056.2.a.bb 1
252.bi even 6 1 7056.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 9.c even 3 1
36.2.a.a 1 9.d odd 6 1
144.2.a.a 1 36.f odd 6 1
144.2.a.a 1 36.h even 6 1
324.2.e.c 2 1.a even 1 1 trivial
324.2.e.c 2 3.b odd 2 1 CM
324.2.e.c 2 9.c even 3 1 inner
324.2.e.c 2 9.d odd 6 1 inner
576.2.a.e 1 72.j odd 6 1
576.2.a.e 1 72.n even 6 1
576.2.a.f 1 72.l even 6 1
576.2.a.f 1 72.p odd 6 1
900.2.a.g 1 45.h odd 6 1
900.2.a.g 1 45.j even 6 1
900.2.d.b 2 45.k odd 12 2
900.2.d.b 2 45.l even 12 2
1296.2.i.h 2 4.b odd 2 1
1296.2.i.h 2 12.b even 2 1
1296.2.i.h 2 36.f odd 6 1
1296.2.i.h 2 36.h even 6 1
1764.2.a.e 1 63.l odd 6 1
1764.2.a.e 1 63.o even 6 1
1764.2.k.g 2 63.i even 6 1
1764.2.k.g 2 63.k odd 6 1
1764.2.k.g 2 63.s even 6 1
1764.2.k.g 2 63.t odd 6 1
1764.2.k.h 2 63.g even 3 1
1764.2.k.h 2 63.h even 3 1
1764.2.k.h 2 63.j odd 6 1
1764.2.k.h 2 63.n odd 6 1
2304.2.d.a 2 144.u even 12 2
2304.2.d.a 2 144.v odd 12 2
2304.2.d.q 2 144.w odd 12 2
2304.2.d.q 2 144.x even 12 2
3600.2.a.e 1 180.n even 6 1
3600.2.a.e 1 180.p odd 6 1
3600.2.f.m 2 180.v odd 12 2
3600.2.f.m 2 180.x even 12 2
4356.2.a.g 1 99.g even 6 1
4356.2.a.g 1 99.h odd 6 1
6084.2.a.i 1 117.n odd 6 1
6084.2.a.i 1 117.t even 6 1
6084.2.b.f 2 117.y odd 12 2
6084.2.b.f 2 117.z even 12 2
7056.2.a.bb 1 252.s odd 6 1
7056.2.a.bb 1 252.bi even 6 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{2} - 4 T_{7} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 - 4 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -8 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$196 + 14 T + T^{2}$$
$67$ $$256 - 16 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$196 + 14 T + T^{2}$$