# Properties

 Label 324.3.f.e Level $324$ Weight $3$ Character orbit 324.f Analytic conductor $8.828$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ 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: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + ( 8 - 8 \zeta_{6} ) q^{5} + 8 q^{8} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + ( 8 - 8 \zeta_{6} ) q^{5} + 8 q^{8} -16 q^{10} + ( 10 - 10 \zeta_{6} ) q^{13} -16 \zeta_{6} q^{16} + 16 q^{17} + 32 \zeta_{6} q^{20} -39 \zeta_{6} q^{25} -20 q^{26} -40 \zeta_{6} q^{29} + ( -32 + 32 \zeta_{6} ) q^{32} -32 \zeta_{6} q^{34} -70 q^{37} + ( 64 - 64 \zeta_{6} ) q^{40} + ( 80 - 80 \zeta_{6} ) q^{41} + ( -49 + 49 \zeta_{6} ) q^{49} + ( -78 + 78 \zeta_{6} ) q^{50} + 40 \zeta_{6} q^{52} -56 q^{53} + ( -80 + 80 \zeta_{6} ) q^{58} + 22 \zeta_{6} q^{61} + 64 q^{64} -80 \zeta_{6} q^{65} + ( -64 + 64 \zeta_{6} ) q^{68} + 110 q^{73} + 140 \zeta_{6} q^{74} -128 q^{80} -160 q^{82} + ( 128 - 128 \zeta_{6} ) q^{85} + 160 q^{89} + 130 \zeta_{6} q^{97} + 98 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} + 8 q^{5} + 16 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} - 4 q^{4} + 8 q^{5} + 16 q^{8} - 32 q^{10} + 10 q^{13} - 16 q^{16} + 32 q^{17} + 32 q^{20} - 39 q^{25} - 40 q^{26} - 40 q^{29} - 32 q^{32} - 32 q^{34} - 140 q^{37} + 64 q^{40} + 80 q^{41} - 49 q^{49} - 78 q^{50} + 40 q^{52} - 112 q^{53} - 80 q^{58} + 22 q^{61} + 128 q^{64} - 80 q^{65} - 64 q^{68} + 220 q^{73} + 140 q^{74} - 256 q^{80} - 320 q^{82} + 128 q^{85} + 320 q^{89} + 130 q^{97} + 196 q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 + 1.73205i 0 −2.00000 3.46410i 4.00000 + 6.92820i 0 0 8.00000 0 −16.0000
271.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i 4.00000 6.92820i 0 0 8.00000 0 −16.0000
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.f.e 2
3.b odd 2 1 324.3.f.f 2
4.b odd 2 1 CM 324.3.f.e 2
9.c even 3 1 36.3.d.b yes 1
9.c even 3 1 inner 324.3.f.e 2
9.d odd 6 1 36.3.d.a 1
9.d odd 6 1 324.3.f.f 2
12.b even 2 1 324.3.f.f 2
36.f odd 6 1 36.3.d.b yes 1
36.f odd 6 1 inner 324.3.f.e 2
36.h even 6 1 36.3.d.a 1
36.h even 6 1 324.3.f.f 2
45.h odd 6 1 900.3.c.c 1
45.j even 6 1 900.3.c.b 1
45.k odd 12 2 900.3.f.a 2
45.l even 12 2 900.3.f.b 2
72.j odd 6 1 576.3.g.a 1
72.l even 6 1 576.3.g.a 1
72.n even 6 1 576.3.g.c 1
72.p odd 6 1 576.3.g.c 1
144.u even 12 2 2304.3.b.d 2
144.v odd 12 2 2304.3.b.e 2
144.w odd 12 2 2304.3.b.d 2
144.x even 12 2 2304.3.b.e 2
180.n even 6 1 900.3.c.c 1
180.p odd 6 1 900.3.c.b 1
180.v odd 12 2 900.3.f.b 2
180.x even 12 2 900.3.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.d.a 1 9.d odd 6 1
36.3.d.a 1 36.h even 6 1
36.3.d.b yes 1 9.c even 3 1
36.3.d.b yes 1 36.f odd 6 1
324.3.f.e 2 1.a even 1 1 trivial
324.3.f.e 2 4.b odd 2 1 CM
324.3.f.e 2 9.c even 3 1 inner
324.3.f.e 2 36.f odd 6 1 inner
324.3.f.f 2 3.b odd 2 1
324.3.f.f 2 9.d odd 6 1
324.3.f.f 2 12.b even 2 1
324.3.f.f 2 36.h even 6 1
576.3.g.a 1 72.j odd 6 1
576.3.g.a 1 72.l even 6 1
576.3.g.c 1 72.n even 6 1
576.3.g.c 1 72.p odd 6 1
900.3.c.b 1 45.j even 6 1
900.3.c.b 1 180.p odd 6 1
900.3.c.c 1 45.h odd 6 1
900.3.c.c 1 180.n even 6 1
900.3.f.a 2 45.k odd 12 2
900.3.f.a 2 180.x even 12 2
900.3.f.b 2 45.l even 12 2
900.3.f.b 2 180.v odd 12 2
2304.3.b.d 2 144.u even 12 2
2304.3.b.d 2 144.w odd 12 2
2304.3.b.e 2 144.v odd 12 2
2304.3.b.e 2 144.x even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{2} - 8 T_{5} + 64$$ $$T_{7}$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$64 - 8 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$100 - 10 T + T^{2}$$
$17$ $$( -16 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$1600 + 40 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( 70 + T )^{2}$$
$41$ $$6400 - 80 T + T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$( 56 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$484 - 22 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -110 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -160 + T )^{2}$$
$97$ $$16900 - 130 T + T^{2}$$