# Properties

 Label 2304.2.a.w Level $2304$ Weight $2$ Character orbit 2304.a Self dual yes Analytic conductor $18.398$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 576) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \beta q^{7} +O(q^{10})$$ $$q -2 \beta q^{7} -4 \beta q^{13} + 8 q^{19} -5 q^{25} + 6 \beta q^{31} + 4 \beta q^{37} + 8 q^{43} + 5 q^{49} + 4 \beta q^{61} + 16 q^{67} + 10 q^{73} -10 \beta q^{79} + 24 q^{91} -14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 16q^{19} - 10q^{25} + 16q^{43} + 10q^{49} + 32q^{67} + 20q^{73} + 48q^{91} - 28q^{97} + O(q^{100})$$

## 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 0 0 −3.46410 0 0 0
1.2 0 0 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
8.d odd 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.w 2
3.b odd 2 1 CM 2304.2.a.w 2
4.b odd 2 1 2304.2.a.v 2
8.b even 2 1 2304.2.a.v 2
8.d odd 2 1 inner 2304.2.a.w 2
12.b even 2 1 2304.2.a.v 2
16.e even 4 2 576.2.d.c 4
16.f odd 4 2 576.2.d.c 4
24.f even 2 1 inner 2304.2.a.w 2
24.h odd 2 1 2304.2.a.v 2
48.i odd 4 2 576.2.d.c 4
48.k even 4 2 576.2.d.c 4

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

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{2} - 12$$ $$T_{11}$$ $$T_{13}^{2} - 48$$ $$T_{19} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 23 T^{2} )^{2}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$1 - 46 T^{2} + 961 T^{4}$$
$37$ $$1 + 26 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 + 53 T^{2} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$1 + 74 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 16 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )^{2}$$
$79$ $$1 - 142 T^{2} + 6241 T^{4}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$( 1 + 14 T + 97 T^{2} )^{2}$$