# Properties

 Label 324.2.h.e Level $324$ Weight $2$ Character orbit 324.h Analytic conductor $2.587$ Analytic rank $0$ Dimension $8$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.58715302549$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{6}$$ Twist minimal: yes 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{2} + ( 2 - 2 \zeta_{24}^{4} ) q^{4} + ( -3 \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{2} + ( 2 - 2 \zeta_{24}^{4} ) q^{4} + ( -3 \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( -1 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{10} + ( -2 - 3 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{13} -4 \zeta_{24}^{4} q^{16} + ( -\zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{17} + ( -2 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{20} + ( -3 \zeta_{24}^{2} + 9 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{25} + ( -\zeta_{24} + \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{26} + ( -2 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{29} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{32} + ( -3 \zeta_{24}^{2} - 5 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{34} + ( -1 - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{37} + ( -2 + 6 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 12 \zeta_{24}^{6} ) q^{40} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{41} + ( -7 + 7 \zeta_{24}^{4} ) q^{49} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{50} + ( 6 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{52} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{53} + ( 7 - 3 \zeta_{24}^{2} - 7 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{58} + ( 6 \zeta_{24}^{2} - 5 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{61} -8 q^{64} + ( 14 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 14 \zeta_{24}^{7} ) q^{65} + ( 6 \zeta_{24} + 8 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{68} + ( 8 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{73} + ( 7 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{74} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{80} + 2 q^{82} + ( 11 + 6 \zeta_{24}^{2} - 11 \zeta_{24}^{4} - 12 \zeta_{24}^{6} ) q^{85} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{89} -8 \zeta_{24}^{4} q^{97} + ( 7 \zeta_{24} - 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{4} + O(q^{10})$$ $$8 q + 8 q^{4} - 8 q^{10} - 8 q^{13} - 16 q^{16} + 36 q^{25} - 20 q^{34} - 8 q^{37} - 8 q^{40} - 28 q^{49} + 16 q^{52} + 28 q^{58} - 20 q^{61} - 64 q^{64} + 64 q^{73} + 16 q^{82} + 44 q^{85} - 32 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{24}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i
−1.22474 0.707107i 0 1.00000 + 1.73205i −2.56961 + 1.48356i 0 0 2.82843i 0 4.19615
107.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 3.79435 2.19067i 0 0 2.82843i 0 −6.19615
107.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.79435 + 2.19067i 0 0 2.82843i 0 −6.19615
107.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.56961 1.48356i 0 0 2.82843i 0 4.19615
215.1 −1.22474 + 0.707107i 0 1.00000 1.73205i −2.56961 1.48356i 0 0 2.82843i 0 4.19615
215.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 3.79435 + 2.19067i 0 0 2.82843i 0 −6.19615
215.3 1.22474 0.707107i 0 1.00000 1.73205i −3.79435 2.19067i 0 0 2.82843i 0 −6.19615
215.4 1.22474 0.707107i 0 1.00000 1.73205i 2.56961 + 1.48356i 0 0 2.82843i 0 4.19615
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 215.4 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 $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
12.b even 2 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.h.e 8
3.b odd 2 1 inner 324.2.h.e 8
4.b odd 2 1 CM 324.2.h.e 8
9.c even 3 1 324.2.b.a 4
9.c even 3 1 inner 324.2.h.e 8
9.d odd 6 1 324.2.b.a 4
9.d odd 6 1 inner 324.2.h.e 8
12.b even 2 1 inner 324.2.h.e 8
36.f odd 6 1 324.2.b.a 4
36.f odd 6 1 inner 324.2.h.e 8
36.h even 6 1 324.2.b.a 4
36.h even 6 1 inner 324.2.h.e 8
72.j odd 6 1 5184.2.c.e 4
72.l even 6 1 5184.2.c.e 4
72.n even 6 1 5184.2.c.e 4
72.p odd 6 1 5184.2.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.b.a 4 9.c even 3 1
324.2.b.a 4 9.d odd 6 1
324.2.b.a 4 36.f odd 6 1
324.2.b.a 4 36.h even 6 1
324.2.h.e 8 1.a even 1 1 trivial
324.2.h.e 8 3.b odd 2 1 inner
324.2.h.e 8 4.b odd 2 1 CM
324.2.h.e 8 9.c even 3 1 inner
324.2.h.e 8 9.d odd 6 1 inner
324.2.h.e 8 12.b even 2 1 inner
324.2.h.e 8 36.f odd 6 1 inner
324.2.h.e 8 36.h even 6 1 inner
5184.2.c.e 4 72.j odd 6 1
5184.2.c.e 4 72.l even 6 1
5184.2.c.e 4 72.n even 6 1
5184.2.c.e 4 72.p odd 6 1

## Hecke kernels

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

 $$T_{5}^{8} - 28 T_{5}^{6} + 615 T_{5}^{4} - 4732 T_{5}^{2} + 28561$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$28561 - 4732 T^{2} + 615 T^{4} - 28 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$( 529 - 92 T + 39 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$17$ $$( 1 + 52 T^{2} + T^{4} )^{2}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$14641 - 9196 T^{2} + 5655 T^{4} - 76 T^{6} + T^{8}$$
$31$ $$T^{8}$$
$37$ $$( -107 + 2 T + T^{2} )^{4}$$
$41$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$( 50 + T^{2} )^{4}$$
$59$ $$T^{8}$$
$61$ $$( 6889 - 830 T + 183 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$( 37 - 16 T + T^{2} )^{4}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 5041 + 196 T^{2} + T^{4} )^{2}$$
$97$ $$( 64 + 8 T + T^{2} )^{4}$$