Properties

 Label 324.2.h.b Level $324$ Weight $2$ Character orbit 324.h Analytic conductor $2.587$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$2.58715302549$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + 4 \beta_{2} q^{10} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{11} + ( 1 - \beta_{2} ) q^{13} + ( -2 \beta_{1} + \beta_{3} ) q^{14} + ( -4 + 4 \beta_{2} ) q^{16} -2 \beta_{3} q^{17} + ( -3 + 6 \beta_{2} ) q^{19} + 4 \beta_{3} q^{20} + ( 4 - 8 \beta_{2} ) q^{22} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{23} + 3 \beta_{2} q^{25} + ( \beta_{1} - \beta_{3} ) q^{26} + ( -2 - 2 \beta_{2} ) q^{28} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{29} + ( 2 + 2 \beta_{2} ) q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( 4 - 4 \beta_{2} ) q^{34} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{35} - q^{37} + ( -3 \beta_{1} + 6 \beta_{3} ) q^{38} + ( -8 + 8 \beta_{2} ) q^{40} -4 \beta_{1} q^{41} + ( 4 - 2 \beta_{2} ) q^{43} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{44} + ( 8 - 4 \beta_{2} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( -4 + 4 \beta_{2} ) q^{49} + 3 \beta_{3} q^{50} + 2 q^{52} + 4 \beta_{3} q^{53} + ( 8 - 16 \beta_{2} ) q^{55} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{56} + 8 q^{58} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{59} -11 \beta_{2} q^{61} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{62} -8 q^{64} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -7 - 7 \beta_{2} ) q^{67} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{68} + ( -4 - 4 \beta_{2} ) q^{70} - q^{73} -\beta_{1} q^{74} + ( -12 + 6 \beta_{2} ) q^{76} + 6 \beta_{1} q^{77} + ( -2 + \beta_{2} ) q^{79} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{80} -8 \beta_{2} q^{82} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{83} + ( 8 - 8 \beta_{2} ) q^{85} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{86} + ( 16 - 8 \beta_{2} ) q^{88} -2 \beta_{3} q^{89} + ( -1 + 2 \beta_{2} ) q^{91} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{92} + ( 4 - 8 \beta_{2} ) q^{94} + ( -6 \beta_{1} + 12 \beta_{3} ) q^{95} + 13 \beta_{2} q^{97} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} - 6q^{7} + O(q^{10})$$ $$4q + 4q^{4} - 6q^{7} + 8q^{10} + 2q^{13} - 8q^{16} + 6q^{25} - 12q^{28} + 12q^{31} + 8q^{34} - 4q^{37} - 16q^{40} + 12q^{43} + 24q^{46} - 8q^{49} + 8q^{52} + 32q^{58} - 22q^{61} - 32q^{64} - 42q^{67} - 24q^{70} - 4q^{73} - 36q^{76} - 6q^{79} - 16q^{82} + 16q^{85} + 48q^{88} + 26q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −2.44949 + 1.41421i 0 −1.50000 0.866025i 2.82843i 0 2.00000 3.46410i
107.2 1.22474 0.707107i 0 1.00000 1.73205i 2.44949 1.41421i 0 −1.50000 0.866025i 2.82843i 0 2.00000 3.46410i
215.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −2.44949 1.41421i 0 −1.50000 + 0.866025i 2.82843i 0 2.00000 + 3.46410i
215.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.44949 + 1.41421i 0 −1.50000 + 0.866025i 2.82843i 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
3.b odd 2 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.h.b 4
3.b odd 2 1 inner 324.2.h.b 4
4.b odd 2 1 324.2.h.a 4
9.c even 3 1 108.2.b.b 4
9.c even 3 1 324.2.h.a 4
9.d odd 6 1 108.2.b.b 4
9.d odd 6 1 324.2.h.a 4
12.b even 2 1 324.2.h.a 4
36.f odd 6 1 108.2.b.b 4
36.f odd 6 1 inner 324.2.h.b 4
36.h even 6 1 108.2.b.b 4
36.h even 6 1 inner 324.2.h.b 4
72.j odd 6 1 1728.2.c.d 4
72.l even 6 1 1728.2.c.d 4
72.n even 6 1 1728.2.c.d 4
72.p odd 6 1 1728.2.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.b.b 4 9.c even 3 1
108.2.b.b 4 9.d odd 6 1
108.2.b.b 4 36.f odd 6 1
108.2.b.b 4 36.h even 6 1
324.2.h.a 4 4.b odd 2 1
324.2.h.a 4 9.c even 3 1
324.2.h.a 4 9.d odd 6 1
324.2.h.a 4 12.b even 2 1
324.2.h.b 4 1.a even 1 1 trivial
324.2.h.b 4 3.b odd 2 1 inner
324.2.h.b 4 36.f odd 6 1 inner
324.2.h.b 4 36.h even 6 1 inner
1728.2.c.d 4 72.j odd 6 1
1728.2.c.d 4 72.l even 6 1
1728.2.c.d 4 72.n even 6 1
1728.2.c.d 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 $$S_{2}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{4} - 8 T_{5}^{2} + 64$$ $$T_{7}^{2} + 3 T_{7} + 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$64 - 8 T^{2} + T^{4}$$
$7$ $$( 3 + 3 T + T^{2} )^{2}$$
$11$ $$576 + 24 T^{2} + T^{4}$$
$13$ $$( 1 - T + T^{2} )^{2}$$
$17$ $$( 8 + T^{2} )^{2}$$
$19$ $$( 27 + T^{2} )^{2}$$
$23$ $$576 + 24 T^{2} + T^{4}$$
$29$ $$1024 - 32 T^{2} + T^{4}$$
$31$ $$( 12 - 6 T + T^{2} )^{2}$$
$37$ $$( 1 + T )^{4}$$
$41$ $$1024 - 32 T^{2} + T^{4}$$
$43$ $$( 12 - 6 T + T^{2} )^{2}$$
$47$ $$576 + 24 T^{2} + T^{4}$$
$53$ $$( 32 + T^{2} )^{2}$$
$59$ $$576 + 24 T^{2} + T^{4}$$
$61$ $$( 121 + 11 T + T^{2} )^{2}$$
$67$ $$( 147 + 21 T + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 1 + T )^{4}$$
$79$ $$( 3 + 3 T + T^{2} )^{2}$$
$83$ $$9216 + 96 T^{2} + T^{4}$$
$89$ $$( 8 + T^{2} )^{2}$$
$97$ $$( 169 - 13 T + T^{2} )^{2}$$