# Properties

 Label 2304.2.a.z Level $2304$ Weight $2$ Character orbit 2304.a Self dual yes Analytic conductor $18.398$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 1152) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -4 + 2 \beta^{2} ) q^{5} + ( -10 \beta + 2 \beta^{3} ) q^{7} +O(q^{10})$$ $$q + ( -4 + 2 \beta^{2} ) q^{5} + ( -10 \beta + 2 \beta^{3} ) q^{7} + ( -12 \beta + 4 \beta^{3} ) q^{11} + 7 q^{25} + ( -12 + 6 \beta^{2} ) q^{29} + ( -10 \beta + 2 \beta^{3} ) q^{31} + ( 36 \beta - 12 \beta^{3} ) q^{35} + 17 q^{49} + ( 4 - 2 \beta^{2} ) q^{53} + ( 40 \beta - 8 \beta^{3} ) q^{55} + ( -24 \beta + 8 \beta^{3} ) q^{59} + 14 q^{73} + ( 32 - 16 \beta^{2} ) q^{77} + ( 30 \beta - 6 \beta^{3} ) q^{79} + ( 12 \beta - 4 \beta^{3} ) q^{83} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 28q^{25} + 68q^{49} + 56q^{73} + 8q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.517638 −0.517638 1.93185 −1.93185
0 0 0 −3.46410 0 −4.89898 0 0 0
1.2 0 0 0 −3.46410 0 4.89898 0 0 0
1.3 0 0 0 3.46410 0 −4.89898 0 0 0
1.4 0 0 0 3.46410 0 4.89898 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.z 4
3.b odd 2 1 inner 2304.2.a.z 4
4.b odd 2 1 inner 2304.2.a.z 4
8.b even 2 1 inner 2304.2.a.z 4
8.d odd 2 1 inner 2304.2.a.z 4
12.b even 2 1 inner 2304.2.a.z 4
16.e even 4 2 1152.2.d.g 4
16.f odd 4 2 1152.2.d.g 4
24.f even 2 1 inner 2304.2.a.z 4
24.h odd 2 1 CM 2304.2.a.z 4
48.i odd 4 2 1152.2.d.g 4
48.k even 4 2 1152.2.d.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.d.g 4 16.e even 4 2
1152.2.d.g 4 16.f odd 4 2
1152.2.d.g 4 48.i odd 4 2
1152.2.d.g 4 48.k even 4 2
2304.2.a.z 4 1.a even 1 1 trivial
2304.2.a.z 4 3.b odd 2 1 inner
2304.2.a.z 4 4.b odd 2 1 inner
2304.2.a.z 4 8.b even 2 1 inner
2304.2.a.z 4 8.d odd 2 1 inner
2304.2.a.z 4 12.b even 2 1 inner
2304.2.a.z 4 24.f even 2 1 inner
2304.2.a.z 4 24.h odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2304))$$:

 $$T_{5}^{2} - 12$$ $$T_{7}^{2} - 24$$ $$T_{11}^{2} - 32$$ $$T_{13}$$ $$T_{19}$$