# Properties

 Label 324.3.c.a Level $324$ Weight $3$ Character orbit 324.c Analytic conductor $8.828$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{4}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( -1 + \beta_{2} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( -1 + \beta_{2} ) q^{7} -\beta_{3} q^{11} + ( -1 + 2 \beta_{2} ) q^{13} + ( -5 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -7 + 5 \beta_{2} ) q^{19} + ( 6 \beta_{1} + \beta_{3} ) q^{23} + ( 7 + 3 \beta_{2} ) q^{25} + ( -5 \beta_{1} - 3 \beta_{3} ) q^{29} + ( 8 + 6 \beta_{2} ) q^{31} + ( -6 \beta_{1} + \beta_{3} ) q^{35} + ( 14 + \beta_{2} ) q^{37} + ( 12 \beta_{1} - \beta_{3} ) q^{41} + ( -13 + 9 \beta_{2} ) q^{43} + ( 10 \beta_{1} - 2 \beta_{3} ) q^{47} + ( -21 - 2 \beta_{2} ) q^{49} + ( 4 \beta_{1} + 3 \beta_{3} ) q^{53} + ( 9 + 3 \beta_{2} ) q^{55} + ( -14 \beta_{1} + 2 \beta_{3} ) q^{59} + ( -16 - 9 \beta_{2} ) q^{61} + ( -11 \beta_{1} + 2 \beta_{3} ) q^{65} + ( 5 + 3 \beta_{2} ) q^{67} + ( -2 \beta_{1} - 7 \beta_{3} ) q^{71} + ( -64 - 9 \beta_{2} ) q^{73} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{77} + ( 77 - 7 \beta_{2} ) q^{79} + ( 4 \beta_{1} + 10 \beta_{3} ) q^{83} + ( 72 - 21 \beta_{2} ) q^{85} + ( 19 \beta_{1} - \beta_{3} ) q^{89} + ( 55 - 3 \beta_{2} ) q^{91} + ( -32 \beta_{1} + 5 \beta_{3} ) q^{95} + ( -40 - 22 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{7} + O(q^{10})$$ $$4 q - 4 q^{7} - 4 q^{13} - 28 q^{19} + 28 q^{25} + 32 q^{31} + 56 q^{37} - 52 q^{43} - 84 q^{49} + 36 q^{55} - 64 q^{61} + 20 q^{67} - 256 q^{73} + 308 q^{79} + 288 q^{85} + 220 q^{91} - 160 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu$$ $$\beta_{2}$$ $$=$$ $$3 \nu^{2} + 6$$ $$\beta_{3}$$ $$=$$ $$9 \nu^{3} + 33 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} - 6$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} - 11 \beta_{1}$$$$)/9$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 − 1.93185i − 0.517638i 0.517638i 1.93185i
0 0 0 5.79555i 0 −6.19615 0 0 0
161.2 0 0 0 1.55291i 0 4.19615 0 0 0
161.3 0 0 0 1.55291i 0 4.19615 0 0 0
161.4 0 0 0 5.79555i 0 −6.19615 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.c.a 4
3.b odd 2 1 inner 324.3.c.a 4
4.b odd 2 1 1296.3.e.f 4
9.c even 3 2 324.3.g.d 8
9.d odd 6 2 324.3.g.d 8
12.b even 2 1 1296.3.e.f 4
36.f odd 6 2 1296.3.q.n 8
36.h even 6 2 1296.3.q.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.c.a 4 1.a even 1 1 trivial
324.3.c.a 4 3.b odd 2 1 inner
324.3.g.d 8 9.c even 3 2
324.3.g.d 8 9.d odd 6 2
1296.3.e.f 4 4.b odd 2 1
1296.3.e.f 4 12.b even 2 1
1296.3.q.n 8 36.f odd 6 2
1296.3.q.n 8 36.h even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 36 T_{5}^{2} + 81$$ acting on $$S_{3}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$81 + 36 T^{2} + T^{4}$$
$7$ $$( -26 + 2 T + T^{2} )^{2}$$
$11$ $$324 + 252 T^{2} + T^{4}$$
$13$ $$( -107 + 2 T + T^{2} )^{2}$$
$17$ $$558009 + 1548 T^{2} + T^{4}$$
$19$ $$( -626 + 14 T + T^{2} )^{2}$$
$23$ $$715716 + 1764 T^{2} + T^{4}$$
$29$ $$1996569 + 3708 T^{2} + T^{4}$$
$31$ $$( -908 - 16 T + T^{2} )^{2}$$
$37$ $$( 169 - 28 T + T^{2} )^{2}$$
$41$ $$39204 + 5004 T^{2} + T^{4}$$
$43$ $$( -2018 + 26 T + T^{2} )^{2}$$
$47$ $$944784 + 3888 T^{2} + T^{4}$$
$53$ $$1127844 + 3276 T^{2} + T^{4}$$
$59$ $$685584 + 7056 T^{2} + T^{4}$$
$61$ $$( -1931 + 32 T + T^{2} )^{2}$$
$67$ $$( -218 - 10 T + T^{2} )^{2}$$
$71$ $$171396 + 12996 T^{2} + T^{4}$$
$73$ $$( 1909 + 128 T + T^{2} )^{2}$$
$79$ $$( 4606 - 154 T + T^{2} )^{2}$$
$83$ $$3779136 + 27216 T^{2} + T^{4}$$
$89$ $$2313441 + 12564 T^{2} + T^{4}$$
$97$ $$( -11468 + 80 T + T^{2} )^{2}$$