# 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: $$1$$ Twist minimal: yes 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.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6048.2.a.bg yes 2
3.b odd 2 1 6048.2.a.z 2
4.b odd 2 1 6048.2.a.bj yes 2
12.b even 2 1 6048.2.a.ba yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.a.z 2 3.b odd 2 1
6048.2.a.ba yes 2 12.b even 2 1
6048.2.a.bg yes 2 1.a even 1 1 trivial
6048.2.a.bj yes 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6048))$$:

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$1 - 2 T + 9 T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$1 + 2 T + 21 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$1 + 18 T^{2} + 169 T^{4}$$
$17$ $$1 + 26 T^{2} + 289 T^{4}$$
$19$ $$1 + 2 T + 31 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T + 29 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 + 50 T^{2} + 841 T^{4}$$
$31$ $$1 - 2 T + 31 T^{2} - 62 T^{3} + 961 T^{4}$$
$37$ $$1 - 6 T + 75 T^{2} - 222 T^{3} + 1369 T^{4}$$
$41$ $$1 - 10 T + 57 T^{2} - 410 T^{3} + 1681 T^{4}$$
$43$ $$1 + 4 T + 82 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 + 12 T + 98 T^{2} + 564 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 4 T + 53 T^{2} )^{2}$$
$59$ $$1 + 4 T + 90 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 6 T^{2} + 3721 T^{4}$$
$67$ $$1 + 8 T + 78 T^{2} + 536 T^{3} + 4489 T^{4}$$
$71$ $$1 + 14 T + 189 T^{2} + 994 T^{3} + 5041 T^{4}$$
$73$ $$1 + 12 T + 174 T^{2} + 876 T^{3} + 5329 T^{4}$$
$79$ $$1 - 4 T + 154 T^{2} - 316 T^{3} + 6241 T^{4}$$
$83$ $$1 + 16 T + 222 T^{2} + 1328 T^{3} + 6889 T^{4}$$
$89$ $$1 + 6 T + 169 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$1 + 8 T + 178 T^{2} + 776 T^{3} + 9409 T^{4}$$