# Properties

 Label 324.3.d.b Level $324$ Weight $3$ Character orbit 324.d 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.d (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 4 q^{5} + ( -2 + 4 \zeta_{6} ) q^{7} + 8 q^{8} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 4 q^{5} + ( -2 + 4 \zeta_{6} ) q^{7} + 8 q^{8} -8 \zeta_{6} q^{10} + ( -7 + 14 \zeta_{6} ) q^{11} -22 q^{13} + ( 8 - 4 \zeta_{6} ) q^{14} -16 \zeta_{6} q^{16} -11 q^{17} + ( -9 + 18 \zeta_{6} ) q^{19} + ( -16 + 16 \zeta_{6} ) q^{20} + ( 28 - 14 \zeta_{6} ) q^{22} + ( -14 + 28 \zeta_{6} ) q^{23} -9 q^{25} + 44 \zeta_{6} q^{26} + ( -8 - 8 \zeta_{6} ) q^{28} + 34 q^{29} + ( -4 + 8 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 22 \zeta_{6} q^{34} + ( -8 + 16 \zeta_{6} ) q^{35} -16 q^{37} + ( 36 - 18 \zeta_{6} ) q^{38} + 32 q^{40} + 13 q^{41} + ( -29 + 58 \zeta_{6} ) q^{43} + ( -28 - 28 \zeta_{6} ) q^{44} + ( 56 - 28 \zeta_{6} ) q^{46} + ( 2 - 4 \zeta_{6} ) q^{47} + 37 q^{49} + 18 \zeta_{6} q^{50} + ( 88 - 88 \zeta_{6} ) q^{52} + 52 q^{53} + ( -28 + 56 \zeta_{6} ) q^{55} + ( -16 + 32 \zeta_{6} ) q^{56} -68 \zeta_{6} q^{58} + ( 31 - 62 \zeta_{6} ) q^{59} -16 q^{61} + ( 16 - 8 \zeta_{6} ) q^{62} + 64 q^{64} -88 q^{65} + ( -67 + 134 \zeta_{6} ) q^{67} + ( 44 - 44 \zeta_{6} ) q^{68} + ( 32 - 16 \zeta_{6} ) q^{70} -25 q^{73} + 32 \zeta_{6} q^{74} + ( -36 - 36 \zeta_{6} ) q^{76} -42 q^{77} + ( 16 - 32 \zeta_{6} ) q^{79} -64 \zeta_{6} q^{80} -26 \zeta_{6} q^{82} + ( 20 - 40 \zeta_{6} ) q^{83} -44 q^{85} + ( 116 - 58 \zeta_{6} ) q^{86} + ( -56 + 112 \zeta_{6} ) q^{88} -2 q^{89} + ( 44 - 88 \zeta_{6} ) q^{91} + ( -56 - 56 \zeta_{6} ) q^{92} + ( -8 + 4 \zeta_{6} ) q^{94} + ( -36 + 72 \zeta_{6} ) q^{95} -43 q^{97} -74 \zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} + 8 q^{5} + 16 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} - 4 q^{4} + 8 q^{5} + 16 q^{8} - 8 q^{10} - 44 q^{13} + 12 q^{14} - 16 q^{16} - 22 q^{17} - 16 q^{20} + 42 q^{22} - 18 q^{25} + 44 q^{26} - 24 q^{28} + 68 q^{29} - 32 q^{32} + 22 q^{34} - 32 q^{37} + 54 q^{38} + 64 q^{40} + 26 q^{41} - 84 q^{44} + 84 q^{46} + 74 q^{49} + 18 q^{50} + 88 q^{52} + 104 q^{53} - 68 q^{58} - 32 q^{61} + 24 q^{62} + 128 q^{64} - 176 q^{65} + 44 q^{68} + 48 q^{70} - 50 q^{73} + 32 q^{74} - 108 q^{76} - 84 q^{77} - 64 q^{80} - 26 q^{82} - 88 q^{85} + 174 q^{86} - 4 q^{89} - 168 q^{92} - 12 q^{94} - 86 q^{97} - 74 q^{98} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i 4.00000 0 3.46410i 8.00000 0 −4.00000 6.92820i
163.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 4.00000 0 3.46410i 8.00000 0 −4.00000 + 6.92820i
 $$n$$: 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.d.b 2
3.b odd 2 1 324.3.d.c 2
4.b odd 2 1 inner 324.3.d.b 2
9.c even 3 1 36.3.f.a 2
9.c even 3 1 36.3.f.b yes 2
9.d odd 6 1 108.3.f.a 2
9.d odd 6 1 108.3.f.b 2
12.b even 2 1 324.3.d.c 2
36.f odd 6 1 36.3.f.a 2
36.f odd 6 1 36.3.f.b yes 2
36.h even 6 1 108.3.f.a 2
36.h even 6 1 108.3.f.b 2
72.j odd 6 1 1728.3.o.a 2
72.j odd 6 1 1728.3.o.b 2
72.l even 6 1 1728.3.o.a 2
72.l even 6 1 1728.3.o.b 2
72.n even 6 1 576.3.o.a 2
72.n even 6 1 576.3.o.b 2
72.p odd 6 1 576.3.o.a 2
72.p odd 6 1 576.3.o.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.a 2 9.c even 3 1
36.3.f.a 2 36.f odd 6 1
36.3.f.b yes 2 9.c even 3 1
36.3.f.b yes 2 36.f odd 6 1
108.3.f.a 2 9.d odd 6 1
108.3.f.a 2 36.h even 6 1
108.3.f.b 2 9.d odd 6 1
108.3.f.b 2 36.h even 6 1
324.3.d.b 2 1.a even 1 1 trivial
324.3.d.b 2 4.b odd 2 1 inner
324.3.d.c 2 3.b odd 2 1
324.3.d.c 2 12.b even 2 1
576.3.o.a 2 72.n even 6 1
576.3.o.a 2 72.p odd 6 1
576.3.o.b 2 72.n even 6 1
576.3.o.b 2 72.p odd 6 1
1728.3.o.a 2 72.j odd 6 1
1728.3.o.a 2 72.l even 6 1
1728.3.o.b 2 72.j odd 6 1
1728.3.o.b 2 72.l even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -4 + T )^{2}$$
$7$ $$12 + T^{2}$$
$11$ $$147 + T^{2}$$
$13$ $$( 22 + T )^{2}$$
$17$ $$( 11 + T )^{2}$$
$19$ $$243 + T^{2}$$
$23$ $$588 + T^{2}$$
$29$ $$( -34 + T )^{2}$$
$31$ $$48 + T^{2}$$
$37$ $$( 16 + T )^{2}$$
$41$ $$( -13 + T )^{2}$$
$43$ $$2523 + T^{2}$$
$47$ $$12 + T^{2}$$
$53$ $$( -52 + T )^{2}$$
$59$ $$2883 + T^{2}$$
$61$ $$( 16 + T )^{2}$$
$67$ $$13467 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 25 + T )^{2}$$
$79$ $$768 + T^{2}$$
$83$ $$1200 + T^{2}$$
$89$ $$( 2 + T )^{2}$$
$97$ $$( 43 + T )^{2}$$