# Properties

 Label 6048.2.a.p Level 6048 Weight 2 Character orbit 6048.a Self dual yes Analytic conductor 48.294 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Learn more about

## 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} - q^{7} + O(q^{10})$$ $$q + q^{5} - q^{7} + 2q^{11} + q^{17} + 4q^{19} - 4q^{23} - 4q^{25} - 8q^{29} - 4q^{31} - q^{35} - 7q^{37} - 5q^{41} - q^{43} - 9q^{47} + q^{49} + 2q^{53} + 2q^{55} + 3q^{59} - 2q^{61} + 4q^{67} + 12q^{71} - 16q^{73} - 2q^{77} + 15q^{79} - 3q^{83} + q^{85} + 6q^{89} + 4q^{95} - 4q^{97} + O(q^{100})$$

## 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
 0
0 0 0 1.00000 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.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6048.2.a.p yes 1
3.b odd 2 1 6048.2.a.f 1
4.b odd 2 1 6048.2.a.s yes 1
12.b even 2 1 6048.2.a.i yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.a.f 1 3.b odd 2 1
6048.2.a.i yes 1 12.b even 2 1
6048.2.a.p yes 1 1.a even 1 1 trivial
6048.2.a.s yes 1 4.b odd 2 1

## Atkin-Lehner signs

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

## Hecke kernels

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

 $$T_{5} - 1$$ $$T_{11} - 2$$ $$T_{13}$$ $$T_{17} - 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$1 - T + 5 T^{2}$$
$7$ $$1 + T$$
$11$ $$1 - 2 T + 11 T^{2}$$
$13$ $$1 + 13 T^{2}$$
$17$ $$1 - T + 17 T^{2}$$
$19$ $$1 - 4 T + 19 T^{2}$$
$23$ $$1 + 4 T + 23 T^{2}$$
$29$ $$1 + 8 T + 29 T^{2}$$
$31$ $$1 + 4 T + 31 T^{2}$$
$37$ $$1 + 7 T + 37 T^{2}$$
$41$ $$1 + 5 T + 41 T^{2}$$
$43$ $$1 + T + 43 T^{2}$$
$47$ $$1 + 9 T + 47 T^{2}$$
$53$ $$1 - 2 T + 53 T^{2}$$
$59$ $$1 - 3 T + 59 T^{2}$$
$61$ $$1 + 2 T + 61 T^{2}$$
$67$ $$1 - 4 T + 67 T^{2}$$
$71$ $$1 - 12 T + 71 T^{2}$$
$73$ $$1 + 16 T + 73 T^{2}$$
$79$ $$1 - 15 T + 79 T^{2}$$
$83$ $$1 + 3 T + 83 T^{2}$$
$89$ $$1 - 6 T + 89 T^{2}$$
$97$ $$1 + 4 T + 97 T^{2}$$
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