# Properties

 Label 6048.2.a.bg Level 6048 Weight 2 Character orbit 6048.a Self dual Yes Analytic conductor 48.294 Analytic rank 1 Dimension 2 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6048 = 2^{5} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6048.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$48.2935231425$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{5} - q^{7} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{5} - q^{7} + ( -1 - \beta ) q^{11} + 2 \beta q^{13} + 2 \beta q^{17} + ( -1 - 2 \beta ) q^{19} + ( -1 - 3 \beta ) q^{23} + ( -2 + 2 \beta ) q^{25} -2 \beta q^{29} + ( 1 - 4 \beta ) q^{31} + ( -1 - \beta ) q^{35} + ( 3 - 2 \beta ) q^{37} + ( 5 - 5 \beta ) q^{41} + ( -2 - 2 \beta ) q^{43} + ( -6 + 4 \beta ) q^{47} + q^{49} -4 q^{53} + ( -3 - 2 \beta ) q^{55} + ( -2 + 4 \beta ) q^{59} -8 \beta q^{61} + ( 4 + 2 \beta ) q^{65} + ( -4 + 6 \beta ) q^{67} + ( -7 + \beta ) q^{71} + ( -6 + 2 \beta ) q^{73} + ( 1 + \beta ) q^{77} + ( 2 + 2 \beta ) q^{79} + ( -8 - 2 \beta ) q^{83} + ( 4 + 2 \beta ) q^{85} + ( -3 + 3 \beta ) q^{89} -2 \beta q^{91} + ( -5 - 3 \beta ) q^{95} + ( -4 - 4 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - 2q^{7} + O(q^{10})$$ $$2q + 2q^{5} - 2q^{7} - 2q^{11} - 2q^{19} - 2q^{23} - 4q^{25} + 2q^{31} - 2q^{35} + 6q^{37} + 10q^{41} - 4q^{43} - 12q^{47} + 2q^{49} - 8q^{53} - 6q^{55} - 4q^{59} + 8q^{65} - 8q^{67} - 14q^{71} - 12q^{73} + 2q^{77} + 4q^{79} - 16q^{83} + 8q^{85} - 6q^{89} - 10q^{95} - 8q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 0 0 −0.414214 0 −1.00000 0 0 0
1.2 0 0 0 2.41421 0 −1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6048))$$:

 $$T_{5}^{2} - 2 T_{5} - 1$$ $$T_{11}^{2} + 2 T_{11} - 1$$ $$T_{13}^{2} - 8$$ $$T_{17}^{2} - 8$$