# Properties

 Label 2304.2.a.s Level $2304$ Weight $2$ Character orbit 2304.a Self dual yes Analytic conductor $18.398$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 192) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5} -2 \beta q^{7} +O(q^{10})$$ $$q + 2 \beta q^{5} -2 \beta q^{7} -6 q^{17} -4 q^{19} -4 \beta q^{23} + 7 q^{25} -2 \beta q^{29} -2 \beta q^{31} -12 q^{35} + 4 \beta q^{37} -6 q^{41} -4 q^{43} + 4 \beta q^{47} + 5 q^{49} -2 \beta q^{53} -12 q^{59} -4 \beta q^{61} + 4 q^{67} + 4 \beta q^{71} -2 q^{73} + 6 \beta q^{79} -12 \beta q^{85} + 6 q^{89} -8 \beta q^{95} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 12q^{17} - 8q^{19} + 14q^{25} - 24q^{35} - 12q^{41} - 8q^{43} + 10q^{49} - 24q^{59} + 8q^{67} - 4q^{73} + 12q^{89} - 4q^{97} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 0 0 −3.46410 0 3.46410 0 0 0
1.2 0 0 0 3.46410 0 −3.46410 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
8.d odd 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.s 2
3.b odd 2 1 768.2.a.j 2
4.b odd 2 1 2304.2.a.u 2
8.b even 2 1 2304.2.a.u 2
8.d odd 2 1 inner 2304.2.a.s 2
12.b even 2 1 768.2.a.k 2
16.e even 4 2 576.2.d.b 4
16.f odd 4 2 576.2.d.b 4
24.f even 2 1 768.2.a.j 2
24.h odd 2 1 768.2.a.k 2
48.i odd 4 2 192.2.d.a 4
48.k even 4 2 192.2.d.a 4
240.t even 4 2 4800.2.k.j 4
240.z odd 4 2 4800.2.d.j 4
240.bb even 4 2 4800.2.d.o 4
240.bd odd 4 2 4800.2.d.o 4
240.bf even 4 2 4800.2.d.j 4
240.bm odd 4 2 4800.2.k.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.2.d.a 4 48.i odd 4 2
192.2.d.a 4 48.k even 4 2
576.2.d.b 4 16.e even 4 2
576.2.d.b 4 16.f odd 4 2
768.2.a.j 2 3.b odd 2 1
768.2.a.j 2 24.f even 2 1
768.2.a.k 2 12.b even 2 1
768.2.a.k 2 24.h odd 2 1
2304.2.a.s 2 1.a even 1 1 trivial
2304.2.a.s 2 8.d odd 2 1 inner
2304.2.a.u 2 4.b odd 2 1
2304.2.a.u 2 8.b even 2 1
4800.2.d.j 4 240.z odd 4 2
4800.2.d.j 4 240.bf even 4 2
4800.2.d.o 4 240.bb even 4 2
4800.2.d.o 4 240.bd odd 4 2
4800.2.k.j 4 240.t even 4 2
4800.2.k.j 4 240.bm odd 4 2

## Hecke kernels

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

 $$T_{5}^{2} - 12$$ $$T_{7}^{2} - 12$$ $$T_{11}$$ $$T_{13}$$ $$T_{19} + 4$$