# Properties

 Label 324.3.d.h Level $324$ Weight $3$ Character orbit 324.d Analytic conductor $8.828$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 324.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.389136420864.4 Defining polynomial: $$x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( -1 - \beta_{6} ) q^{4} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( -1 + \beta_{4} + \beta_{5} + \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( -1 - \beta_{6} ) q^{4} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( -1 + \beta_{4} + \beta_{5} + \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} ) q^{8} + ( -7 + \beta_{4} + 2 \beta_{6} ) q^{10} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{7} ) q^{11} + ( -4 - \beta_{4} + \beta_{5} - \beta_{6} ) q^{13} + ( 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{14} + ( -5 - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} ) q^{16} + ( 5 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{17} + ( -1 - 3 \beta_{4} + \beta_{5} + 5 \beta_{6} ) q^{19} + ( -\beta_{1} - \beta_{2} - 7 \beta_{3} - 3 \beta_{7} ) q^{20} + ( -6 + 4 \beta_{5} - 2 \beta_{6} ) q^{22} + ( -3 \beta_{1} + 3 \beta_{2} + 7 \beta_{3} - 2 \beta_{7} ) q^{23} + ( 6 - 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{25} + ( 6 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} + 2 \beta_{7} ) q^{26} + ( 22 - 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{28} + ( -8 \beta_{1} + \beta_{2} - \beta_{3} ) q^{29} + ( 2 - 10 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} ) q^{31} + ( -17 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{7} ) q^{32} + ( -15 - 9 \beta_{4} + 8 \beta_{6} ) q^{34} + ( -3 \beta_{1} + 3 \beta_{2} + 19 \beta_{3} + 10 \beta_{7} ) q^{35} + ( -28 + \beta_{4} - \beta_{5} + \beta_{6} ) q^{37} + ( -2 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{38} + ( 15 + 2 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} ) q^{40} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -7 + 19 \beta_{4} + 7 \beta_{5} - 5 \beta_{6} ) q^{43} + ( 26 \beta_{1} - 6 \beta_{2} - 10 \beta_{3} + 2 \beta_{7} ) q^{44} + ( -26 + 4 \beta_{5} - 6 \beta_{6} ) q^{46} + ( -8 \beta_{1} + 8 \beta_{2} + 24 \beta_{3} ) q^{47} + ( -29 + 10 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} ) q^{49} + ( 24 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} + 8 \beta_{7} ) q^{50} + ( 21 - 4 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} ) q^{52} + ( -18 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{53} + ( 7 - 7 \beta_{4} - 7 \beta_{5} - 7 \beta_{6} ) q^{55} + ( 22 \beta_{1} - 10 \beta_{2} + 18 \beta_{3} + 6 \beta_{7} ) q^{56} + ( -13 + 7 \beta_{4} + 2 \beta_{6} ) q^{58} + ( -4 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} ) q^{59} + ( -4 - 7 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} ) q^{61} + ( -28 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} + 4 \beta_{7} ) q^{62} + ( -7 + 18 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{64} + ( -11 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 1 - 13 \beta_{4} - \beta_{5} + 11 \beta_{6} ) q^{67} + ( -17 \beta_{1} - 17 \beta_{2} - 15 \beta_{3} + \beta_{7} ) q^{68} + ( -50 - 20 \beta_{5} - 6 \beta_{6} ) q^{70} + ( -5 \beta_{1} + 5 \beta_{2} + \beta_{3} - 14 \beta_{7} ) q^{71} + ( 47 - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{73} + ( -6 \beta_{1} + 2 \beta_{2} - 27 \beta_{3} - 2 \beta_{7} ) q^{74} + ( 62 + 12 \beta_{4} + 4 \beta_{5} - 10 \beta_{6} ) q^{76} + ( -34 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} ) q^{77} + ( -3 + 19 \beta_{4} + 3 \beta_{5} - 13 \beta_{6} ) q^{79} + ( 35 \beta_{1} - 13 \beta_{2} + 9 \beta_{3} - 5 \beta_{7} ) q^{80} + ( 12 - 4 \beta_{6} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{7} ) q^{83} + ( 46 - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{85} + ( 66 \beta_{1} + 10 \beta_{2} - 14 \beta_{3} - 14 \beta_{7} ) q^{86} + ( 70 - 20 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} ) q^{88} + ( 11 \beta_{1} - 20 \beta_{2} + 20 \beta_{3} ) q^{89} + ( -3 - \beta_{4} + 3 \beta_{5} + 7 \beta_{6} ) q^{91} + ( 30 \beta_{1} - 2 \beta_{2} - 30 \beta_{3} + 6 \beta_{7} ) q^{92} + ( -80 - 16 \beta_{6} ) q^{94} + ( -7 \beta_{1} + 7 \beta_{2} + 35 \beta_{3} + 14 \beta_{7} ) q^{95} + ( 116 - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{97} + ( -60 \beta_{1} + 20 \beta_{2} - 19 \beta_{3} - 20 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 10 q^{4} + O(q^{10})$$ $$8 q - 10 q^{4} - 50 q^{10} - 32 q^{13} - 46 q^{16} - 36 q^{22} + 48 q^{25} + 180 q^{28} - 122 q^{34} - 224 q^{37} + 154 q^{40} - 204 q^{46} - 232 q^{49} + 154 q^{52} - 86 q^{58} - 32 q^{61} - 10 q^{64} - 492 q^{70} + 376 q^{73} + 516 q^{76} + 88 q^{82} + 368 q^{85} + 516 q^{88} - 672 q^{94} + 928 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + \nu^{3} + 4 \nu$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{5} - 20 \nu^{3} + 64 \nu$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 24 \nu^{3} + 80 \nu$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + \nu^{2} + 8$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} - \nu^{4} + 12 \nu^{2} + 16$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{4} + 24 \nu^{2} + 64$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{7} - 7 \nu^{5} - 16 \nu$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} - \beta_{4} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} + 3 \beta_{3} - 3 \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{6} - \beta_{5} + 9 \beta_{4} - 13$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{7} - 7 \beta_{3} - \beta_{2} + 37 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$13 \beta_{6} - 19 \beta_{5} - 21 \beta_{4} + 9$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-38 \beta_{7} + 11 \beta_{3} - 3 \beta_{2} - 81 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 1.52274 − 1.29664i 1.52274 + 1.29664i 0.656712 − 1.88911i 0.656712 + 1.88911i −0.656712 − 1.88911i −0.656712 + 1.88911i −1.52274 − 1.29664i −1.52274 + 1.29664i
−1.52274 1.29664i 0 0.637459 + 3.94888i 7.82300 0 12.3894i 4.14959 6.83966i 0 −11.9124 10.1436i
163.2 −1.52274 + 1.29664i 0 0.637459 3.94888i 7.82300 0 12.3894i 4.14959 + 6.83966i 0 −11.9124 + 10.1436i
163.3 −0.656712 1.88911i 0 −3.13746 + 2.48120i 0.894797 0 1.58166i 6.74766 + 4.29756i 0 −0.587624 1.69037i
163.4 −0.656712 + 1.88911i 0 −3.13746 2.48120i 0.894797 0 1.58166i 6.74766 4.29756i 0 −0.587624 + 1.69037i
163.5 0.656712 1.88911i 0 −3.13746 2.48120i −0.894797 0 1.58166i −6.74766 + 4.29756i 0 −0.587624 + 1.69037i
163.6 0.656712 + 1.88911i 0 −3.13746 + 2.48120i −0.894797 0 1.58166i −6.74766 4.29756i 0 −0.587624 1.69037i
163.7 1.52274 1.29664i 0 0.637459 3.94888i −7.82300 0 12.3894i −4.14959 6.83966i 0 −11.9124 + 10.1436i
163.8 1.52274 + 1.29664i 0 0.637459 + 3.94888i −7.82300 0 12.3894i −4.14959 + 6.83966i 0 −11.9124 10.1436i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 163.8 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
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.d.h 8
3.b odd 2 1 inner 324.3.d.h 8
4.b odd 2 1 inner 324.3.d.h 8
9.c even 3 2 324.3.f.s 16
9.d odd 6 2 324.3.f.s 16
12.b even 2 1 inner 324.3.d.h 8
36.f odd 6 2 324.3.f.s 16
36.h even 6 2 324.3.f.s 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.d.h 8 1.a even 1 1 trivial
324.3.d.h 8 3.b odd 2 1 inner
324.3.d.h 8 4.b odd 2 1 inner
324.3.d.h 8 12.b even 2 1 inner
324.3.f.s 16 9.c even 3 2
324.3.f.s 16 9.d odd 6 2
324.3.f.s 16 36.f odd 6 2
324.3.f.s 16 36.h even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 62 T_{5}^{2} + 49$$ acting on $$S_{3}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 80 T^{2} + 24 T^{4} + 5 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 49 - 62 T^{2} + T^{4} )^{2}$$
$7$ $$( 384 + 156 T^{2} + T^{4} )^{2}$$
$11$ $$( 18816 + 276 T^{2} + T^{4} )^{2}$$
$13$ $$( -41 + 8 T + T^{2} )^{4}$$
$17$ $$( 52441 - 758 T^{2} + T^{4} )^{2}$$
$19$ $$( 169344 + 828 T^{2} + T^{4} )^{2}$$
$23$ $$( 384 + 756 T^{2} + T^{4} )^{2}$$
$29$ $$( 29929 - 422 T^{2} + T^{4} )^{2}$$
$31$ $$( 301056 + 2544 T^{2} + T^{4} )^{2}$$
$37$ $$( 727 + 56 T + T^{2} )^{4}$$
$41$ $$( 4096 - 176 T^{2} + T^{4} )^{2}$$
$43$ $$( 28201344 + 10812 T^{2} + T^{4} )^{2}$$
$47$ $$( 6291456 + 5376 T^{2} + T^{4} )^{2}$$
$53$ $$( 861184 - 5744 T^{2} + T^{4} )^{2}$$
$59$ $$( 393216 + 1344 T^{2} + T^{4} )^{2}$$
$61$ $$( -2777 + 8 T + T^{2} )^{4}$$
$67$ $$( 322944 + 5052 T^{2} + T^{4} )^{2}$$
$71$ $$( 39567744 + 12852 T^{2} + T^{4} )^{2}$$
$73$ $$( 1981 - 94 T + T^{2} )^{4}$$
$79$ $$( 921984 + 9468 T^{2} + T^{4} )^{2}$$
$83$ $$( 301056 + 1104 T^{2} + T^{4} )^{2}$$
$89$ $$( 52374169 - 15926 T^{2} + T^{4} )^{2}$$
$97$ $$( 13228 - 232 T + T^{2} )^{4}$$