# Properties

 Label 324.2.h.d Level $324$ Weight $2$ Character orbit 324.h Analytic conductor $2.587$ Analytic rank $0$ Dimension $8$ CM no 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: 8.0.12960000.1 Defining polynomial: $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} + \beta_{7} ) q^{2} + ( -\beta_{1} - \beta_{2} ) q^{4} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{5} + ( -1 - \beta_{2} + 2 \beta_{6} ) q^{7} + ( -2 \beta_{4} + \beta_{7} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{3} + \beta_{7} ) q^{2} + ( -\beta_{1} - \beta_{2} ) q^{4} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{5} + ( -1 - \beta_{2} + 2 \beta_{6} ) q^{7} + ( -2 \beta_{4} + \beta_{7} ) q^{8} + ( -3 + \beta_{1} + \beta_{6} ) q^{10} + ( -\beta_{3} + \beta_{5} + \beta_{7} ) q^{11} + 2 \beta_{2} q^{13} + ( \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{14} + ( 3 + 3 \beta_{2} + \beta_{6} ) q^{16} + ( 2 \beta_{4} + 2 \beta_{7} ) q^{17} + ( 3 \beta_{3} - 2 \beta_{5} - 3 \beta_{7} ) q^{20} + ( -\beta_{1} + \beta_{2} ) q^{22} + ( -4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{23} -2 \beta_{7} q^{26} + ( 7 + \beta_{1} + \beta_{6} ) q^{28} + ( 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{29} + ( -2 \beta_{1} - \beta_{2} ) q^{31} + ( -3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{32} + ( -6 - 6 \beta_{2} + 2 \beta_{6} ) q^{34} + ( 5 \beta_{4} - 5 \beta_{7} ) q^{35} -4 q^{37} + ( 3 \beta_{1} - \beta_{2} ) q^{40} + ( 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{41} + ( 2 + 2 \beta_{2} - 4 \beta_{6} ) q^{43} + ( -2 \beta_{4} - \beta_{7} ) q^{44} + ( -4 - 4 \beta_{1} - 4 \beta_{6} ) q^{46} + ( 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{47} -8 \beta_{2} q^{49} + ( 2 + 2 \beta_{2} - 2 \beta_{6} ) q^{52} + ( -\beta_{4} - \beta_{7} ) q^{53} + ( -1 + 2 \beta_{1} + 2 \beta_{6} ) q^{55} + ( -7 \beta_{3} - 2 \beta_{5} + 7 \beta_{7} ) q^{56} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{58} + ( 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{59} + ( 4 + 4 \beta_{2} ) q^{61} + ( -4 \beta_{4} + \beta_{7} ) q^{62} + ( 7 - 3 \beta_{1} - 3 \beta_{6} ) q^{64} + ( -2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{65} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 6 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{68} + ( -5 - 5 \beta_{2} - 5 \beta_{6} ) q^{70} + ( -6 \beta_{4} + 6 \beta_{7} ) q^{71} + 5 q^{73} + ( 4 \beta_{3} - 4 \beta_{7} ) q^{74} + ( -3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{77} + ( 2 + 2 \beta_{2} - 4 \beta_{6} ) q^{79} + ( 6 \beta_{4} + \beta_{7} ) q^{80} + ( -12 + 4 \beta_{1} + 4 \beta_{6} ) q^{82} + ( -7 \beta_{3} + 7 \beta_{5} + 7 \beta_{7} ) q^{83} + 10 \beta_{2} q^{85} + ( -2 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} ) q^{86} + ( 5 + 5 \beta_{2} - \beta_{6} ) q^{88} + ( 2 \beta_{4} + 2 \beta_{7} ) q^{89} + ( 2 - 4 \beta_{1} - 4 \beta_{6} ) q^{91} + ( 4 \beta_{3} + 8 \beta_{5} - 4 \beta_{7} ) q^{92} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( -11 - 11 \beta_{2} ) q^{97} + 8 \beta_{7} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{4} + O(q^{10})$$ $$8 q + 2 q^{4} - 20 q^{10} - 8 q^{13} + 14 q^{16} - 6 q^{22} + 60 q^{28} - 20 q^{34} - 32 q^{37} + 10 q^{40} - 48 q^{46} + 32 q^{49} + 4 q^{52} - 20 q^{58} + 16 q^{61} + 44 q^{64} - 30 q^{70} + 40 q^{73} - 80 q^{82} - 40 q^{85} + 18 q^{88} + 12 q^{94} - 44 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} + 8 \nu^{4} - 16 \nu^{2} + 19$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{6} + 8 \nu^{4} - 24 \nu^{2} + 1$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} - 12 \nu^{3} + 11 \nu$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{5} - 6 \nu^{3} - 3 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} + 16 \nu^{5} - 40 \nu^{3} + 23 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{6} + 16 \nu^{4} - 40 \nu^{2} - 3$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$-\nu^{7} + 3 \nu^{5} - 8 \nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_{3}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - 5 \beta_{2} + \beta_{1} - 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + 4 \beta_{5} - \beta_{4} - 4 \beta_{3}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} - 3 \beta_{2} + 2 \beta_{1} - 4$$ $$\nu^{5}$$ $$=$$ $$($$$$11 \beta_{7} - 2 \beta_{5} - 13 \beta_{4} - 13 \beta_{3}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-8 \beta_{6} + 8 \beta_{2} + 8 \beta_{1} - 23$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$34 \beta_{7} - 34 \beta_{5} - 29 \beta_{4} - 5 \beta_{3}$$$$)/3$$

## Character values

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

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$-1$$ $$-\beta_{2}$$

## 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
 1.40126 + 0.809017i 0.535233 + 0.309017i −0.535233 − 0.309017i −1.40126 − 0.809017i 1.40126 − 0.809017i 0.535233 − 0.309017i −0.535233 + 0.309017i −1.40126 + 0.809017i
−1.40126 + 0.190983i 0 1.92705 0.535233i 1.93649 1.11803i 0 3.35410 + 1.93649i −2.59808 + 1.11803i 0 −2.50000 + 1.93649i
107.2 −0.535233 1.30902i 0 −1.42705 + 1.40126i 1.93649 1.11803i 0 −3.35410 1.93649i 2.59808 + 1.11803i 0 −2.50000 1.93649i
107.3 0.535233 + 1.30902i 0 −1.42705 + 1.40126i −1.93649 + 1.11803i 0 −3.35410 1.93649i −2.59808 1.11803i 0 −2.50000 1.93649i
107.4 1.40126 0.190983i 0 1.92705 0.535233i −1.93649 + 1.11803i 0 3.35410 + 1.93649i 2.59808 1.11803i 0 −2.50000 + 1.93649i
215.1 −1.40126 0.190983i 0 1.92705 + 0.535233i 1.93649 + 1.11803i 0 3.35410 1.93649i −2.59808 1.11803i 0 −2.50000 1.93649i
215.2 −0.535233 + 1.30902i 0 −1.42705 1.40126i 1.93649 + 1.11803i 0 −3.35410 + 1.93649i 2.59808 1.11803i 0 −2.50000 + 1.93649i
215.3 0.535233 1.30902i 0 −1.42705 1.40126i −1.93649 1.11803i 0 −3.35410 + 1.93649i −2.59808 + 1.11803i 0 −2.50000 + 1.93649i
215.4 1.40126 + 0.190983i 0 1.92705 + 0.535233i −1.93649 1.11803i 0 3.35410 1.93649i 2.59808 + 1.11803i 0 −2.50000 1.93649i
 $$n$$: 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
3.b odd 2 1 inner
4.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.d 8
3.b odd 2 1 inner 324.2.h.d 8
4.b odd 2 1 inner 324.2.h.d 8
9.c even 3 1 108.2.b.a 4
9.c even 3 1 inner 324.2.h.d 8
9.d odd 6 1 108.2.b.a 4
9.d odd 6 1 inner 324.2.h.d 8
12.b even 2 1 inner 324.2.h.d 8
36.f odd 6 1 108.2.b.a 4
36.f odd 6 1 inner 324.2.h.d 8
36.h even 6 1 108.2.b.a 4
36.h even 6 1 inner 324.2.h.d 8
72.j odd 6 1 1728.2.c.c 4
72.l even 6 1 1728.2.c.c 4
72.n even 6 1 1728.2.c.c 4
72.p odd 6 1 1728.2.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.b.a 4 9.c even 3 1
108.2.b.a 4 9.d odd 6 1
108.2.b.a 4 36.f odd 6 1
108.2.b.a 4 36.h even 6 1
324.2.h.d 8 1.a even 1 1 trivial
324.2.h.d 8 3.b odd 2 1 inner
324.2.h.d 8 4.b odd 2 1 inner
324.2.h.d 8 9.c even 3 1 inner
324.2.h.d 8 9.d odd 6 1 inner
324.2.h.d 8 12.b even 2 1 inner
324.2.h.d 8 36.f odd 6 1 inner
324.2.h.d 8 36.h even 6 1 inner
1728.2.c.c 4 72.j odd 6 1
1728.2.c.c 4 72.l even 6 1
1728.2.c.c 4 72.n even 6 1
1728.2.c.c 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 $$S_{2}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{4} - 5 T_{5}^{2} + 25$$ $$T_{7}^{4} - 15 T_{7}^{2} + 225$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 4 T^{2} - 3 T^{4} - T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 25 - 5 T^{2} + T^{4} )^{2}$$
$7$ $$( 225 - 15 T^{2} + T^{4} )^{2}$$
$11$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$13$ $$( 4 + 2 T + T^{2} )^{4}$$
$17$ $$( 20 + T^{2} )^{4}$$
$19$ $$T^{8}$$
$23$ $$( 2304 + 48 T^{2} + T^{4} )^{2}$$
$29$ $$( 400 - 20 T^{2} + T^{4} )^{2}$$
$31$ $$( 225 - 15 T^{2} + T^{4} )^{2}$$
$37$ $$( 4 + T )^{8}$$
$41$ $$( 6400 - 80 T^{2} + T^{4} )^{2}$$
$43$ $$( 3600 - 60 T^{2} + T^{4} )^{2}$$
$47$ $$( 144 + 12 T^{2} + T^{4} )^{2}$$
$53$ $$( 5 + T^{2} )^{4}$$
$59$ $$( 144 + 12 T^{2} + T^{4} )^{2}$$
$61$ $$( 16 - 4 T + T^{2} )^{4}$$
$67$ $$( 3600 - 60 T^{2} + T^{4} )^{2}$$
$71$ $$( -108 + T^{2} )^{4}$$
$73$ $$( -5 + T )^{8}$$
$79$ $$( 3600 - 60 T^{2} + T^{4} )^{2}$$
$83$ $$( 21609 + 147 T^{2} + T^{4} )^{2}$$
$89$ $$( 20 + T^{2} )^{4}$$
$97$ $$( 121 + 11 T + T^{2} )^{4}$$