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

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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

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