# Properties

 Label 324.3.d.d Level $324$ Weight $3$ Character orbit 324.d Self dual yes Analytic conductor $8.828$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $2$

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## 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: yes Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + ( 4 + \beta ) q^{5} + 8 q^{8} +O(q^{10})$$ $$q + 2 q^{2} + 4 q^{4} + ( 4 + \beta ) q^{5} + 8 q^{8} + ( 8 + 2 \beta ) q^{10} + ( 5 - 4 \beta ) q^{13} + 16 q^{16} + ( -8 - 5 \beta ) q^{17} + ( 16 + 4 \beta ) q^{20} + ( 18 + 8 \beta ) q^{25} + ( 10 - 8 \beta ) q^{26} + ( -20 + 7 \beta ) q^{29} + 32 q^{32} + ( -16 - 10 \beta ) q^{34} + ( 35 - 4 \beta ) q^{37} + ( 32 + 8 \beta ) q^{40} -80 q^{41} + 49 q^{49} + ( 36 + 16 \beta ) q^{50} + ( 20 - 16 \beta ) q^{52} -56 q^{53} + ( -40 + 14 \beta ) q^{58} + ( 11 + 20 \beta ) q^{61} + 64 q^{64} + ( -88 - 11 \beta ) q^{65} + ( -32 - 20 \beta ) q^{68} + ( -55 - 16 \beta ) q^{73} + ( 70 - 8 \beta ) q^{74} + ( 64 + 16 \beta ) q^{80} -160 q^{82} + ( -167 - 28 \beta ) q^{85} + ( -80 + 13 \beta ) q^{89} -130 q^{97} + 98 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + O(q^{10})$$ $$2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + 16 q^{10} + 10 q^{13} + 32 q^{16} - 16 q^{17} + 32 q^{20} + 36 q^{25} + 20 q^{26} - 40 q^{29} + 64 q^{32} - 32 q^{34} + 70 q^{37} + 64 q^{40} - 160 q^{41} + 98 q^{49} + 72 q^{50} + 40 q^{52} - 112 q^{53} - 80 q^{58} + 22 q^{61} + 128 q^{64} - 176 q^{65} - 64 q^{68} - 110 q^{73} + 140 q^{74} + 128 q^{80} - 320 q^{82} - 334 q^{85} - 160 q^{89} - 260 q^{97} + 196 q^{98} + O(q^{100})$$

## Character values

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

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

## 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
 −1.73205 1.73205
2.00000 0 4.00000 −1.19615 0 0 8.00000 0 −2.39230
163.2 2.00000 0 4.00000 9.19615 0 0 8.00000 0 18.3923
 $$n$$: 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 $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.d.d yes 2
3.b odd 2 1 324.3.d.a 2
4.b odd 2 1 CM 324.3.d.d yes 2
9.c even 3 2 324.3.f.k 4
9.d odd 6 2 324.3.f.n 4
12.b even 2 1 324.3.d.a 2
36.f odd 6 2 324.3.f.k 4
36.h even 6 2 324.3.f.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.d.a 2 3.b odd 2 1
324.3.d.a 2 12.b even 2 1
324.3.d.d yes 2 1.a even 1 1 trivial
324.3.d.d yes 2 4.b odd 2 1 CM
324.3.f.k 4 9.c even 3 2
324.3.f.k 4 36.f odd 6 2
324.3.f.n 4 9.d odd 6 2
324.3.f.n 4 36.h even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 8 T_{5} - 11$$ acting on $$S_{3}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-11 - 8 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$-407 - 10 T + T^{2}$$
$17$ $$-611 + 16 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$-923 + 40 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$793 - 70 T + T^{2}$$
$41$ $$( 80 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$( 56 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$-10679 - 22 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-3887 + 110 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$1837 + 160 T + T^{2}$$
$97$ $$( 130 + T )^{2}$$
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