# Properties

 Label 6048.2.a.bb Level 6048 Weight 2 Character orbit 6048.a Self dual yes Analytic conductor 48.294 Analytic rank 0 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \beta ) q^{5} + q^{7} +O(q^{10})$$ $$q + ( -1 + 2 \beta ) q^{5} + q^{7} + ( 2 + 2 \beta ) q^{11} + 2 \beta q^{13} + ( 3 - 2 \beta ) q^{17} + ( 4 + 2 \beta ) q^{19} -4 q^{23} + ( 4 - 4 \beta ) q^{25} + 2 \beta q^{29} + ( -4 - 2 \beta ) q^{31} + ( -1 + 2 \beta ) q^{35} + ( -3 + 4 \beta ) q^{37} + ( 1 + 2 \beta ) q^{41} + ( 5 + 2 \beta ) q^{43} + ( 3 + 4 \beta ) q^{47} + q^{49} -2 q^{53} + ( 6 + 2 \beta ) q^{55} + ( -5 + 4 \beta ) q^{59} + ( 6 - 2 \beta ) q^{61} + ( 8 - 2 \beta ) q^{65} + 4 q^{67} + ( -4 + 4 \beta ) q^{71} -4 \beta q^{73} + ( 2 + 2 \beta ) q^{77} + ( 1 - 2 \beta ) q^{79} + ( -11 + 4 \beta ) q^{83} + ( -11 + 8 \beta ) q^{85} -6 q^{89} + 2 \beta q^{91} + ( 4 + 6 \beta ) q^{95} + ( -4 - 10 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + 2q^{7} + O(q^{10})$$ $$2q - 2q^{5} + 2q^{7} + 4q^{11} + 6q^{17} + 8q^{19} - 8q^{23} + 8q^{25} - 8q^{31} - 2q^{35} - 6q^{37} + 2q^{41} + 10q^{43} + 6q^{47} + 2q^{49} - 4q^{53} + 12q^{55} - 10q^{59} + 12q^{61} + 16q^{65} + 8q^{67} - 8q^{71} + 4q^{77} + 2q^{79} - 22q^{83} - 22q^{85} - 12q^{89} + 8q^{95} - 8q^{97} + O(q^{100})$$

## 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 −3.82843 0 1.00000 0 0 0
1.2 0 0 0 1.82843 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.bb yes 2
3.b odd 2 1 6048.2.a.bi yes 2
4.b odd 2 1 6048.2.a.y 2
12.b even 2 1 6048.2.a.bh yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.a.y 2 4.b odd 2 1
6048.2.a.bb yes 2 1.a even 1 1 trivial
6048.2.a.bh yes 2 12.b even 2 1
6048.2.a.bi yes 2 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6048))$$:

 $$T_{5}^{2} + 2 T_{5} - 7$$ $$T_{11}^{2} - 4 T_{11} - 4$$ $$T_{13}^{2} - 8$$ $$T_{17}^{2} - 6 T_{17} + 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 2 T + 3 T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$1 + 18 T^{2} + 169 T^{4}$$
$17$ $$1 - 6 T + 35 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 - 8 T + 46 T^{2} - 152 T^{3} + 361 T^{4}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$1 + 50 T^{2} + 841 T^{4}$$
$31$ $$1 + 8 T + 70 T^{2} + 248 T^{3} + 961 T^{4}$$
$37$ $$1 + 6 T + 51 T^{2} + 222 T^{3} + 1369 T^{4}$$
$41$ $$1 - 2 T + 75 T^{2} - 82 T^{3} + 1681 T^{4}$$
$43$ $$1 - 10 T + 103 T^{2} - 430 T^{3} + 1849 T^{4}$$
$47$ $$1 - 6 T + 71 T^{2} - 282 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 2 T + 53 T^{2} )^{2}$$
$59$ $$1 + 10 T + 111 T^{2} + 590 T^{3} + 3481 T^{4}$$
$61$ $$1 - 12 T + 150 T^{2} - 732 T^{3} + 3721 T^{4}$$
$67$ $$( 1 - 4 T + 67 T^{2} )^{2}$$
$71$ $$1 + 8 T + 126 T^{2} + 568 T^{3} + 5041 T^{4}$$
$73$ $$1 + 114 T^{2} + 5329 T^{4}$$
$79$ $$1 - 2 T + 151 T^{2} - 158 T^{3} + 6241 T^{4}$$
$83$ $$1 + 22 T + 255 T^{2} + 1826 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{2}$$
$97$ $$1 + 8 T + 10 T^{2} + 776 T^{3} + 9409 T^{4}$$