# Properties

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

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{13})$$ Defining polynomial: $$x^{4} - x^{3} + 4 x^{2} + 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( -\beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{5} + ( -3 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{7} + ( 5 + 9 \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( -\beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{5} + ( -3 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{7} + ( 5 + 9 \beta_{2} - \beta_{3} ) q^{8} + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{10} + ( 7 - 7 \beta_{2} ) q^{11} -16 \beta_{2} q^{13} + ( 10 + 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{14} + ( 11 - 5 \beta_{1} + 6 \beta_{2} - 10 \beta_{3} ) q^{16} + ( 4 - 8 \beta_{3} ) q^{17} + ( 18 - 24 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{19} + ( 9 - 5 \beta_{1} - 4 \beta_{2} ) q^{20} + ( -7 \beta_{1} - 7 \beta_{2} + 7 \beta_{3} ) q^{22} + ( 20 + 10 \beta_{2} ) q^{23} + ( 12 + 12 \beta_{2} ) q^{25} + ( -16 - 16 \beta_{2} + 16 \beta_{3} ) q^{26} + ( 27 - 10 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{28} + ( -14 + 28 \beta_{1} - 14 \beta_{2} ) q^{29} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{31} + ( -18 - 11 \beta_{1} - 9 \beta_{2} - 11 \beta_{3} ) q^{32} + ( -20 - 4 \beta_{1} - 24 \beta_{2} - 8 \beta_{3} ) q^{34} + ( -13 - 26 \beta_{2} ) q^{35} + 26 q^{37} + ( -30 - 18 \beta_{1} + 48 \beta_{2} ) q^{38} + ( -9 \beta_{1} + 11 \beta_{2} + 9 \beta_{3} ) q^{40} + ( 4 - 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 6 - 4 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{43} + ( 7 + 35 \beta_{2} + 21 \beta_{3} ) q^{44} + ( 30 - 20 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{46} + ( -2 + 2 \beta_{2} ) q^{47} + 10 \beta_{2} q^{49} + ( 24 - 12 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{50} + ( 16 + 16 \beta_{1} + 32 \beta_{2} + 32 \beta_{3} ) q^{52} + ( 19 - 38 \beta_{3} ) q^{53} + ( -21 + 28 \beta_{1} - 14 \beta_{2} + 14 \beta_{3} ) q^{55} + ( 7 - 27 \beta_{1} + 20 \beta_{2} ) q^{56} + ( 14 \beta_{1} - 98 \beta_{2} - 14 \beta_{3} ) q^{58} + ( -88 - 44 \beta_{2} ) q^{59} + ( -8 - 8 \beta_{2} ) q^{61} + ( 5 + 13 \beta_{2} + 3 \beta_{3} ) q^{62} + ( -71 + 18 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{64} + ( -16 + 32 \beta_{1} - 16 \beta_{2} ) q^{65} + ( -20 \beta_{1} + 10 \beta_{2} + 20 \beta_{3} ) q^{67} + ( -72 + 20 \beta_{1} - 36 \beta_{2} + 20 \beta_{3} ) q^{68} + ( -39 + 13 \beta_{1} - 26 \beta_{2} + 26 \beta_{3} ) q^{70} + ( -36 - 72 \beta_{2} ) q^{71} -19 q^{73} + ( 26 - 26 \beta_{1} ) q^{74} + ( 30 \beta_{1} + 102 \beta_{2} - 30 \beta_{3} ) q^{76} + ( -42 + 42 \beta_{1} - 21 \beta_{2} + 42 \beta_{3} ) q^{77} + ( 24 - 16 \beta_{1} + 8 \beta_{2} - 32 \beta_{3} ) q^{79} + ( 29 + 65 \beta_{2} + 7 \beta_{3} ) q^{80} + ( -10 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 67 - 67 \beta_{2} ) q^{83} + 52 \beta_{2} q^{85} + ( -20 - 6 \beta_{1} - 10 \beta_{2} - 6 \beta_{3} ) q^{86} + ( 105 - 7 \beta_{1} + 98 \beta_{2} - 14 \beta_{3} ) q^{88} + ( 22 - 44 \beta_{3} ) q^{89} + ( -48 + 64 \beta_{1} - 32 \beta_{2} + 32 \beta_{3} ) q^{91} + ( -10 - 30 \beta_{1} + 40 \beta_{2} ) q^{92} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{94} + ( 156 + 78 \beta_{2} ) q^{95} + ( -119 - 119 \beta_{2} ) q^{97} + ( 10 + 10 \beta_{2} - 10 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} - 5 q^{4} + O(q^{10})$$ $$4 q + 3 q^{2} - 5 q^{4} + 26 q^{10} + 42 q^{11} + 32 q^{13} + 39 q^{14} + 7 q^{16} + 39 q^{20} + 21 q^{22} + 60 q^{23} + 24 q^{25} + 78 q^{28} - 87 q^{32} - 52 q^{34} + 104 q^{37} - 234 q^{38} - 13 q^{40} + 60 q^{46} - 12 q^{47} - 20 q^{49} + 36 q^{50} + 80 q^{52} - 39 q^{56} + 182 q^{58} - 264 q^{59} - 16 q^{61} - 230 q^{64} - 156 q^{68} - 39 q^{70} - 76 q^{73} + 78 q^{74} - 234 q^{76} - 52 q^{82} + 402 q^{83} - 104 q^{85} - 78 q^{86} + 189 q^{88} - 150 q^{92} - 6 q^{94} + 468 q^{95} - 238 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 4 x^{2} + 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu^{2} - 4 \nu - 3$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 7$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{3} - 7$$

## Character values

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

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$

## 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
 1.15139 − 1.99426i −0.651388 + 1.12824i 1.15139 + 1.99426i −0.651388 − 1.12824i
−0.151388 + 1.99426i 0 −3.95416 0.603814i −1.80278 3.12250i 0 −5.40833 3.12250i 1.80278 7.79423i 0 6.50000 3.12250i
55.2 1.65139 1.12824i 0 1.45416 3.72631i 1.80278 + 3.12250i 0 5.40833 + 3.12250i −1.80278 7.79423i 0 6.50000 + 3.12250i
271.1 −0.151388 1.99426i 0 −3.95416 + 0.603814i −1.80278 + 3.12250i 0 −5.40833 + 3.12250i 1.80278 + 7.79423i 0 6.50000 + 3.12250i
271.2 1.65139 + 1.12824i 0 1.45416 + 3.72631i 1.80278 3.12250i 0 5.40833 3.12250i −1.80278 + 7.79423i 0 6.50000 3.12250i
 $$n$$: 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
9.d odd 6 1 inner
12.b even 2 1 inner
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.m 4
3.b odd 2 1 324.3.f.l 4
4.b odd 2 1 324.3.f.l 4
9.c even 3 1 108.3.d.c 4
9.c even 3 1 324.3.f.l 4
9.d odd 6 1 108.3.d.c 4
9.d odd 6 1 inner 324.3.f.m 4
12.b even 2 1 inner 324.3.f.m 4
36.f odd 6 1 108.3.d.c 4
36.f odd 6 1 inner 324.3.f.m 4
36.h even 6 1 108.3.d.c 4
36.h even 6 1 324.3.f.l 4
72.j odd 6 1 1728.3.g.i 4
72.l even 6 1 1728.3.g.i 4
72.n even 6 1 1728.3.g.i 4
72.p odd 6 1 1728.3.g.i 4

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

## Hecke kernels

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

 $$T_{5}^{4} + 13 T_{5}^{2} + 169$$ $$T_{7}^{4} - 39 T_{7}^{2} + 1521$$ $$T_{11}^{2} - 21 T_{11} + 147$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 12 T + 7 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$169 + 13 T^{2} + T^{4}$$
$7$ $$1521 - 39 T^{2} + T^{4}$$
$11$ $$( 147 - 21 T + T^{2} )^{2}$$
$13$ $$( 256 - 16 T + T^{2} )^{2}$$
$17$ $$( -208 + T^{2} )^{2}$$
$19$ $$( 1404 + T^{2} )^{2}$$
$23$ $$( 300 - 30 T + T^{2} )^{2}$$
$29$ $$6492304 + 2548 T^{2} + T^{4}$$
$31$ $$1521 - 39 T^{2} + T^{4}$$
$37$ $$( -26 + T )^{4}$$
$41$ $$2704 + 52 T^{2} + T^{4}$$
$43$ $$24336 - 156 T^{2} + T^{4}$$
$47$ $$( 12 + 6 T + T^{2} )^{2}$$
$53$ $$( -4693 + T^{2} )^{2}$$
$59$ $$( 5808 + 132 T + T^{2} )^{2}$$
$61$ $$( 64 + 8 T + T^{2} )^{2}$$
$67$ $$15210000 - 3900 T^{2} + T^{4}$$
$71$ $$( 3888 + T^{2} )^{2}$$
$73$ $$( 19 + T )^{4}$$
$79$ $$6230016 - 2496 T^{2} + T^{4}$$
$83$ $$( 13467 - 201 T + T^{2} )^{2}$$
$89$ $$( -6292 + T^{2} )^{2}$$
$97$ $$( 14161 + 119 T + T^{2} )^{2}$$