# Properties

 Label 324.3.c.b Level $324$ Weight $3$ Character orbit 324.c Analytic conductor $8.828$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 324.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( 1 - \beta_{2} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( 1 - \beta_{2} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{11} + ( -3 + \beta_{2} ) q^{13} + \beta_{3} q^{17} + \beta_{2} q^{19} + ( \beta_{1} + 2 \beta_{3} ) q^{23} + ( -8 + 3 \beta_{2} ) q^{25} + 7 \beta_{1} q^{29} + ( 5 - 3 \beta_{2} ) q^{31} + ( 7 \beta_{1} + 2 \beta_{3} ) q^{35} + ( -14 - 4 \beta_{2} ) q^{37} + ( -7 \beta_{1} + \beta_{3} ) q^{41} + 23 q^{43} + ( -\beta_{1} + 2 \beta_{3} ) q^{47} + ( 26 - \beta_{2} ) q^{49} -8 \beta_{1} q^{53} + ( 21 + 3 \beta_{2} ) q^{55} + ( 9 \beta_{1} + \beta_{3} ) q^{59} + ( -19 - 3 \beta_{2} ) q^{61} + ( -9 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -61 + 6 \beta_{2} ) q^{67} -2 \beta_{3} q^{71} + ( 20 + 3 \beta_{2} ) q^{73} + ( 9 \beta_{1} - 8 \beta_{3} ) q^{77} + ( -39 - 5 \beta_{2} ) q^{79} + ( -\beta_{1} + 2 \beta_{3} ) q^{83} + ( -12 + 6 \beta_{2} ) q^{85} -8 \beta_{3} q^{89} + ( -77 + 3 \beta_{2} ) q^{91} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{95} + ( 99 - 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{7} + O(q^{10})$$ $$4 q + 2 q^{7} - 10 q^{13} + 2 q^{19} - 26 q^{25} + 14 q^{31} - 64 q^{37} + 92 q^{43} + 102 q^{49} + 90 q^{55} - 82 q^{61} - 232 q^{67} + 86 q^{73} - 166 q^{79} - 36 q^{85} - 302 q^{91} + 392 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{3} + \nu^{2} - \nu + 3$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 5 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{3} + 5 \nu^{2} - 5 \nu - 21$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1} + 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} + 3 \beta_{2} + 5 \beta_{1} + 21$$$$)/18$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} - 5 \beta_{1} + 36$$$$)/9$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
0 0 0 7.57301i 0 9.11684 0 0 0
161.2 0 0 0 2.37686i 0 −8.11684 0 0 0
161.3 0 0 0 2.37686i 0 −8.11684 0 0 0
161.4 0 0 0 7.57301i 0 9.11684 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.c.b 4
3.b odd 2 1 inner 324.3.c.b 4
4.b odd 2 1 1296.3.e.e 4
9.c even 3 1 36.3.g.a 4
9.c even 3 1 108.3.g.a 4
9.d odd 6 1 36.3.g.a 4
9.d odd 6 1 108.3.g.a 4
12.b even 2 1 1296.3.e.e 4
36.f odd 6 1 144.3.q.b 4
36.f odd 6 1 432.3.q.b 4
36.h even 6 1 144.3.q.b 4
36.h even 6 1 432.3.q.b 4
45.h odd 6 1 900.3.p.a 4
45.h odd 6 1 2700.3.p.b 4
45.j even 6 1 900.3.p.a 4
45.j even 6 1 2700.3.p.b 4
45.k odd 12 2 900.3.u.a 8
45.k odd 12 2 2700.3.u.b 8
45.l even 12 2 900.3.u.a 8
45.l even 12 2 2700.3.u.b 8
72.j odd 6 1 576.3.q.d 4
72.j odd 6 1 1728.3.q.g 4
72.l even 6 1 576.3.q.g 4
72.l even 6 1 1728.3.q.h 4
72.n even 6 1 576.3.q.d 4
72.n even 6 1 1728.3.q.g 4
72.p odd 6 1 576.3.q.g 4
72.p odd 6 1 1728.3.q.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 9.c even 3 1
36.3.g.a 4 9.d odd 6 1
108.3.g.a 4 9.c even 3 1
108.3.g.a 4 9.d odd 6 1
144.3.q.b 4 36.f odd 6 1
144.3.q.b 4 36.h even 6 1
324.3.c.b 4 1.a even 1 1 trivial
324.3.c.b 4 3.b odd 2 1 inner
432.3.q.b 4 36.f odd 6 1
432.3.q.b 4 36.h even 6 1
576.3.q.d 4 72.j odd 6 1
576.3.q.d 4 72.n even 6 1
576.3.q.g 4 72.l even 6 1
576.3.q.g 4 72.p odd 6 1
900.3.p.a 4 45.h odd 6 1
900.3.p.a 4 45.j even 6 1
900.3.u.a 8 45.k odd 12 2
900.3.u.a 8 45.l even 12 2
1296.3.e.e 4 4.b odd 2 1
1296.3.e.e 4 12.b even 2 1
1728.3.q.g 4 72.j odd 6 1
1728.3.q.g 4 72.n even 6 1
1728.3.q.h 4 72.l even 6 1
1728.3.q.h 4 72.p odd 6 1
2700.3.p.b 4 45.h odd 6 1
2700.3.p.b 4 45.j even 6 1
2700.3.u.b 8 45.k odd 12 2
2700.3.u.b 8 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 63 T_{5}^{2} + 324$$ acting on $$S_{3}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$324 + 63 T^{2} + T^{4}$$
$7$ $$( -74 - T + T^{2} )^{2}$$
$11$ $$81 + 414 T^{2} + T^{4}$$
$13$ $$( -68 + 5 T + T^{2} )^{2}$$
$17$ $$20736 + 387 T^{2} + T^{4}$$
$19$ $$( -74 - T + T^{2} )^{2}$$
$23$ $$627264 + 1683 T^{2} + T^{4}$$
$29$ $$777924 + 3087 T^{2} + T^{4}$$
$31$ $$( -656 - 7 T + T^{2} )^{2}$$
$37$ $$( -932 + 32 T + T^{2} )^{2}$$
$41$ $$2424249 + 3222 T^{2} + T^{4}$$
$43$ $$( -23 + T )^{4}$$
$47$ $$104976 + 1539 T^{2} + T^{4}$$
$53$ $$1327104 + 4032 T^{2} + T^{4}$$
$59$ $$68121 + 5814 T^{2} + T^{4}$$
$61$ $$( -248 + 41 T + T^{2} )^{2}$$
$67$ $$( 691 + 116 T + T^{2} )^{2}$$
$71$ $$331776 + 1548 T^{2} + T^{4}$$
$73$ $$( -206 - 43 T + T^{2} )^{2}$$
$79$ $$( -134 + 83 T + T^{2} )^{2}$$
$83$ $$104976 + 1539 T^{2} + T^{4}$$
$89$ $$84934656 + 24768 T^{2} + T^{4}$$
$97$ $$( 9307 - 196 T + T^{2} )^{2}$$