# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( 2 - 2 \zeta_{6} ) q^{5} + ( -8 + 4 \zeta_{6} ) q^{7} + 8 q^{8} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( 2 - 2 \zeta_{6} ) q^{5} + ( -8 + 4 \zeta_{6} ) q^{7} + 8 q^{8} + 4 \zeta_{6} q^{10} + ( 8 - 4 \zeta_{6} ) q^{11} + ( -2 + 2 \zeta_{6} ) q^{13} + ( 8 - 16 \zeta_{6} ) q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} + 10 q^{17} + ( -12 + 24 \zeta_{6} ) q^{19} -8 q^{20} + ( -8 + 16 \zeta_{6} ) q^{22} + ( 16 + 16 \zeta_{6} ) q^{23} + 21 \zeta_{6} q^{25} -4 \zeta_{6} q^{26} + ( 16 + 16 \zeta_{6} ) q^{28} + 26 \zeta_{6} q^{29} + ( -4 - 4 \zeta_{6} ) q^{31} -32 \zeta_{6} q^{32} + ( -20 + 20 \zeta_{6} ) q^{34} + ( -8 + 16 \zeta_{6} ) q^{35} + 26 q^{37} + ( -24 - 24 \zeta_{6} ) q^{38} + ( 16 - 16 \zeta_{6} ) q^{40} + ( -58 + 58 \zeta_{6} ) q^{41} + ( -56 + 28 \zeta_{6} ) q^{43} + ( -16 - 16 \zeta_{6} ) q^{44} + ( -64 + 32 \zeta_{6} ) q^{46} + ( 80 - 40 \zeta_{6} ) q^{47} + ( -1 + \zeta_{6} ) q^{49} -42 q^{50} + 8 q^{52} -74 q^{53} + ( 8 - 16 \zeta_{6} ) q^{55} + ( -64 + 32 \zeta_{6} ) q^{56} -52 q^{58} + ( 52 + 52 \zeta_{6} ) q^{59} -26 \zeta_{6} q^{61} + ( 16 - 8 \zeta_{6} ) q^{62} + 64 q^{64} + 4 \zeta_{6} q^{65} + ( -4 - 4 \zeta_{6} ) q^{67} -40 \zeta_{6} q^{68} + ( -16 - 16 \zeta_{6} ) q^{70} -46 q^{73} + ( -52 + 52 \zeta_{6} ) q^{74} + ( 96 - 48 \zeta_{6} ) q^{76} + ( -48 + 48 \zeta_{6} ) q^{77} + ( 136 - 68 \zeta_{6} ) q^{79} + 32 \zeta_{6} q^{80} -116 \zeta_{6} q^{82} + ( 56 - 28 \zeta_{6} ) q^{83} + ( 20 - 20 \zeta_{6} ) q^{85} + ( 56 - 112 \zeta_{6} ) q^{86} + ( 64 - 32 \zeta_{6} ) q^{88} + 82 q^{89} + ( 8 - 16 \zeta_{6} ) q^{91} + ( 64 - 128 \zeta_{6} ) q^{92} + ( -80 + 160 \zeta_{6} ) q^{94} + ( 24 + 24 \zeta_{6} ) q^{95} -2 \zeta_{6} q^{97} -2 \zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} + 2q^{5} - 12q^{7} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} + 2q^{5} - 12q^{7} + 16q^{8} + 4q^{10} + 12q^{11} - 2q^{13} - 16q^{16} + 20q^{17} - 16q^{20} + 48q^{23} + 21q^{25} - 4q^{26} + 48q^{28} + 26q^{29} - 12q^{31} - 32q^{32} - 20q^{34} + 52q^{37} - 72q^{38} + 16q^{40} - 58q^{41} - 84q^{43} - 48q^{44} - 96q^{46} + 120q^{47} - q^{49} - 84q^{50} + 16q^{52} - 148q^{53} - 96q^{56} - 104q^{58} + 156q^{59} - 26q^{61} + 24q^{62} + 128q^{64} + 4q^{65} - 12q^{67} - 40q^{68} - 48q^{70} - 92q^{73} - 52q^{74} + 144q^{76} - 48q^{77} + 204q^{79} + 32q^{80} - 116q^{82} + 84q^{83} + 20q^{85} + 96q^{88} + 164q^{89} + 72q^{95} - 2q^{97} - 2q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i 1.00000 + 1.73205i 0 −6.00000 3.46410i 8.00000 0 2.00000 3.46410i
271.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 1.00000 1.73205i 0 −6.00000 + 3.46410i 8.00000 0 2.00000 + 3.46410i
 $$n$$: 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
36.f 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.f.d 2
3.b odd 2 1 324.3.f.g 2
4.b odd 2 1 324.3.f.j 2
9.c even 3 1 12.3.d.a 2
9.c even 3 1 324.3.f.j 2
9.d odd 6 1 36.3.d.c 2
9.d odd 6 1 324.3.f.a 2
12.b even 2 1 324.3.f.a 2
36.f odd 6 1 12.3.d.a 2
36.f odd 6 1 inner 324.3.f.d 2
36.h even 6 1 36.3.d.c 2
36.h even 6 1 324.3.f.g 2
45.h odd 6 1 900.3.c.e 2
45.j even 6 1 300.3.c.b 2
45.k odd 12 2 300.3.f.a 4
45.l even 12 2 900.3.f.c 4
63.l odd 6 1 588.3.g.b 2
72.j odd 6 1 576.3.g.e 2
72.l even 6 1 576.3.g.e 2
72.n even 6 1 192.3.g.b 2
72.p odd 6 1 192.3.g.b 2
144.u even 12 2 2304.3.b.l 4
144.v odd 12 2 768.3.b.c 4
144.w odd 12 2 2304.3.b.l 4
144.x even 12 2 768.3.b.c 4
180.n even 6 1 900.3.c.e 2
180.p odd 6 1 300.3.c.b 2
180.v odd 12 2 900.3.f.c 4
180.x even 12 2 300.3.f.a 4
252.bi even 6 1 588.3.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 9.c even 3 1
12.3.d.a 2 36.f odd 6 1
36.3.d.c 2 9.d odd 6 1
36.3.d.c 2 36.h even 6 1
192.3.g.b 2 72.n even 6 1
192.3.g.b 2 72.p odd 6 1
300.3.c.b 2 45.j even 6 1
300.3.c.b 2 180.p odd 6 1
300.3.f.a 4 45.k odd 12 2
300.3.f.a 4 180.x even 12 2
324.3.f.a 2 9.d odd 6 1
324.3.f.a 2 12.b even 2 1
324.3.f.d 2 1.a even 1 1 trivial
324.3.f.d 2 36.f odd 6 1 inner
324.3.f.g 2 3.b odd 2 1
324.3.f.g 2 36.h even 6 1
324.3.f.j 2 4.b odd 2 1
324.3.f.j 2 9.c even 3 1
576.3.g.e 2 72.j odd 6 1
576.3.g.e 2 72.l even 6 1
588.3.g.b 2 63.l odd 6 1
588.3.g.b 2 252.bi even 6 1
768.3.b.c 4 144.v odd 12 2
768.3.b.c 4 144.x even 12 2
900.3.c.e 2 45.h odd 6 1
900.3.c.e 2 180.n even 6 1
900.3.f.c 4 45.l even 12 2
900.3.f.c 4 180.v odd 12 2
2304.3.b.l 4 144.u even 12 2
2304.3.b.l 4 144.w odd 12 2

## Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{7}^{2} + 12 T_{7} + 48$$ $$T_{11}^{2} - 12 T_{11} + 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$48 + 12 T + T^{2}$$
$11$ $$48 - 12 T + T^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$( -10 + T )^{2}$$
$19$ $$432 + T^{2}$$
$23$ $$768 - 48 T + T^{2}$$
$29$ $$676 - 26 T + T^{2}$$
$31$ $$48 + 12 T + T^{2}$$
$37$ $$( -26 + T )^{2}$$
$41$ $$3364 + 58 T + T^{2}$$
$43$ $$2352 + 84 T + T^{2}$$
$47$ $$4800 - 120 T + T^{2}$$
$53$ $$( 74 + T )^{2}$$
$59$ $$8112 - 156 T + T^{2}$$
$61$ $$676 + 26 T + T^{2}$$
$67$ $$48 + 12 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 46 + T )^{2}$$
$79$ $$13872 - 204 T + T^{2}$$
$83$ $$2352 - 84 T + T^{2}$$
$89$ $$( -82 + T )^{2}$$
$97$ $$4 + 2 T + T^{2}$$