# Properties

 Label 324.3.g.a Level 324 Weight 3 Character orbit 324.g Analytic conductor 8.828 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -11 + 11 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -11 + 11 \zeta_{6} ) q^{7} -23 \zeta_{6} q^{13} -37 q^{19} + ( -25 + 25 \zeta_{6} ) q^{25} + 46 \zeta_{6} q^{31} -73 q^{37} + ( 22 - 22 \zeta_{6} ) q^{43} -72 \zeta_{6} q^{49} + ( -47 + 47 \zeta_{6} ) q^{61} + 13 \zeta_{6} q^{67} + 143 q^{73} + ( -11 + 11 \zeta_{6} ) q^{79} + 253 q^{91} + ( 169 - 169 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 11q^{7} + O(q^{10})$$ $$2q - 11q^{7} - 23q^{13} - 74q^{19} - 25q^{25} + 46q^{31} - 146q^{37} + 22q^{43} - 72q^{49} - 47q^{61} + 13q^{67} + 286q^{73} - 11q^{79} + 506q^{91} + 169q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 −5.50000 9.52628i 0 0 0
269.1 0 0 0 0 0 −5.50000 + 9.52628i 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 CM by $$\Q(\sqrt{-3})$$
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.g.a 2
3.b odd 2 1 CM 324.3.g.a 2
4.b odd 2 1 1296.3.q.c 2
9.c even 3 1 108.3.c.a 1
9.c even 3 1 inner 324.3.g.a 2
9.d odd 6 1 108.3.c.a 1
9.d odd 6 1 inner 324.3.g.a 2
12.b even 2 1 1296.3.q.c 2
36.f odd 6 1 432.3.e.a 1
36.f odd 6 1 1296.3.q.c 2
36.h even 6 1 432.3.e.a 1
36.h even 6 1 1296.3.q.c 2
45.h odd 6 1 2700.3.g.b 1
45.j even 6 1 2700.3.g.b 1
45.k odd 12 2 2700.3.b.d 2
45.l even 12 2 2700.3.b.d 2
72.j odd 6 1 1728.3.e.c 1
72.l even 6 1 1728.3.e.b 1
72.n even 6 1 1728.3.e.c 1
72.p odd 6 1 1728.3.e.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.c.a 1 9.c even 3 1
108.3.c.a 1 9.d odd 6 1
324.3.g.a 2 1.a even 1 1 trivial
324.3.g.a 2 3.b odd 2 1 CM
324.3.g.a 2 9.c even 3 1 inner
324.3.g.a 2 9.d odd 6 1 inner
432.3.e.a 1 36.f odd 6 1
432.3.e.a 1 36.h even 6 1
1296.3.q.c 2 4.b odd 2 1
1296.3.q.c 2 12.b even 2 1
1296.3.q.c 2 36.f odd 6 1
1296.3.q.c 2 36.h even 6 1
1728.3.e.b 1 72.l even 6 1
1728.3.e.b 1 72.p odd 6 1
1728.3.e.c 1 72.j odd 6 1
1728.3.e.c 1 72.n even 6 1
2700.3.b.d 2 45.k odd 12 2
2700.3.b.d 2 45.l even 12 2
2700.3.g.b 1 45.h odd 6 1
2700.3.g.b 1 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}$$ $$T_{7}^{2} + 11 T_{7} + 121$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 5 T + 25 T^{2} )( 1 + 5 T + 25 T^{2} )$$
$7$ $$( 1 - 2 T + 49 T^{2} )( 1 + 13 T + 49 T^{2} )$$
$11$ $$( 1 - 11 T + 121 T^{2} )( 1 + 11 T + 121 T^{2} )$$
$13$ $$( 1 + T + 169 T^{2} )( 1 + 22 T + 169 T^{2} )$$
$17$ $$( 1 - 17 T )^{2}( 1 + 17 T )^{2}$$
$19$ $$( 1 + 37 T + 361 T^{2} )^{2}$$
$23$ $$( 1 - 23 T + 529 T^{2} )( 1 + 23 T + 529 T^{2} )$$
$29$ $$( 1 - 29 T + 841 T^{2} )( 1 + 29 T + 841 T^{2} )$$
$31$ $$( 1 - 59 T + 961 T^{2} )( 1 + 13 T + 961 T^{2} )$$
$37$ $$( 1 + 73 T + 1369 T^{2} )^{2}$$
$41$ $$( 1 - 41 T + 1681 T^{2} )( 1 + 41 T + 1681 T^{2} )$$
$43$ $$( 1 - 83 T + 1849 T^{2} )( 1 + 61 T + 1849 T^{2} )$$
$47$ $$( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} )$$
$53$ $$( 1 - 53 T )^{2}( 1 + 53 T )^{2}$$
$59$ $$( 1 - 59 T + 3481 T^{2} )( 1 + 59 T + 3481 T^{2} )$$
$61$ $$( 1 - 74 T + 3721 T^{2} )( 1 + 121 T + 3721 T^{2} )$$
$67$ $$( 1 - 122 T + 4489 T^{2} )( 1 + 109 T + 4489 T^{2} )$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 - 143 T + 5329 T^{2} )^{2}$$
$79$ $$( 1 - 131 T + 6241 T^{2} )( 1 + 142 T + 6241 T^{2} )$$
$83$ $$( 1 - 83 T + 6889 T^{2} )( 1 + 83 T + 6889 T^{2} )$$
$89$ $$( 1 - 89 T )^{2}( 1 + 89 T )^{2}$$
$97$ $$( 1 - 167 T + 9409 T^{2} )( 1 - 2 T + 9409 T^{2} )$$