# Properties

 Label 6048.2.a.bt Level 6048 Weight 2 Character orbit 6048.a Self dual yes Analytic conductor 48.294 Analytic rank 0 Dimension 4 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: $$4$$ Coefficient field: 4.4.39528.1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{5} + q^{7} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{5} + q^{7} -\beta_{1} q^{11} + ( 2 + \beta_{1} + \beta_{3} ) q^{13} + ( 1 + \beta_{2} + \beta_{3} ) q^{17} + ( 2 + \beta_{1} ) q^{19} + \beta_{3} q^{23} + ( 4 + \beta_{1} + \beta_{2} ) q^{25} + ( 2 + \beta_{2} ) q^{29} + ( 2 + \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{35} + ( 1 - \beta_{1} - \beta_{2} ) q^{37} + ( -1 + \beta_{2} ) q^{41} + ( -1 - \beta_{2} + \beta_{3} ) q^{43} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{47} + q^{49} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{53} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -1 - 2 \beta_{2} ) q^{59} + ( -\beta_{1} - 2 \beta_{2} ) q^{61} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 2 \beta_{2} - \beta_{3} ) q^{67} + ( -4 - \beta_{3} ) q^{71} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{73} -\beta_{1} q^{77} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{79} + ( 5 + \beta_{1} + \beta_{2} ) q^{83} + ( 5 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{85} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{89} + ( 2 + \beta_{1} + \beta_{3} ) q^{91} + ( 6 + 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{95} + ( 6 - \beta_{1} - 2 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} + 4q^{7} + O(q^{10})$$ $$4q + 2q^{5} + 4q^{7} + 8q^{13} + 2q^{17} + 8q^{19} + 14q^{25} + 6q^{29} + 6q^{31} + 2q^{35} + 6q^{37} - 6q^{41} - 2q^{43} - 2q^{47} + 4q^{49} + 6q^{53} - 12q^{55} + 4q^{61} + 4q^{65} - 4q^{67} - 16q^{71} + 12q^{73} - 2q^{79} + 18q^{83} + 22q^{85} - 8q^{89} + 8q^{91} + 16q^{95} + 24q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} - 2 x + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 7 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 4$$$$)/2$$

## 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
 1.12265 −1.48380 −2.50948 2.87063
0 0 0 −2.73965 0 1.00000 0 0 0
1.2 0 0 0 −1.79835 0 1.00000 0 0 0
1.3 0 0 0 2.29751 0 1.00000 0 0 0
1.4 0 0 0 4.24049 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.bt yes 4
3.b odd 2 1 6048.2.a.bp yes 4
4.b odd 2 1 6048.2.a.bs yes 4
12.b even 2 1 6048.2.a.bk 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.a.bk 4 12.b even 2 1
6048.2.a.bp yes 4 3.b odd 2 1
6048.2.a.bs yes 4 4.b odd 2 1
6048.2.a.bt yes 4 1.a even 1 1 trivial

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6048))$$:

 $$T_{5}^{4} - 2 T_{5}^{3} - 15 T_{5}^{2} + 12 T_{5} + 48$$ $$T_{11}^{4} - 36 T_{11}^{2} + 16 T_{11} + 192$$ $$T_{13}^{4} - 8 T_{13}^{3} - 21 T_{13}^{2} + 184 T_{13} + 88$$ $$T_{17}^{4} - 2 T_{17}^{3} - 48 T_{17}^{2} + 154 T_{17} - 89$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 2 T + 5 T^{2} - 18 T^{3} + 48 T^{4} - 90 T^{5} + 125 T^{6} - 250 T^{7} + 625 T^{8}$$
$7$ $$( 1 - T )^{4}$$
$11$ $$1 + 8 T^{2} + 16 T^{3} + 126 T^{4} + 176 T^{5} + 968 T^{6} + 14641 T^{8}$$
$13$ $$1 - 8 T + 31 T^{2} - 128 T^{3} + 556 T^{4} - 1664 T^{5} + 5239 T^{6} - 17576 T^{7} + 28561 T^{8}$$
$17$ $$1 - 2 T + 20 T^{2} + 52 T^{3} + 13 T^{4} + 884 T^{5} + 5780 T^{6} - 9826 T^{7} + 83521 T^{8}$$
$19$ $$1 - 8 T + 64 T^{2} - 360 T^{3} + 1806 T^{4} - 6840 T^{5} + 23104 T^{6} - 54872 T^{7} + 130321 T^{8}$$
$23$ $$1 + 47 T^{2} - 128 T^{3} + 1008 T^{4} - 2944 T^{5} + 24863 T^{6} + 279841 T^{8}$$
$29$ $$1 - 6 T + 113 T^{2} - 490 T^{3} + 4896 T^{4} - 14210 T^{5} + 95033 T^{6} - 146334 T^{7} + 707281 T^{8}$$
$31$ $$1 - 6 T + 121 T^{2} - 526 T^{3} + 5604 T^{4} - 16306 T^{5} + 116281 T^{6} - 178746 T^{7} + 923521 T^{8}$$
$37$ $$1 - 6 T + 97 T^{2} - 262 T^{3} + 3804 T^{4} - 9694 T^{5} + 132793 T^{6} - 303918 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 6 T + 161 T^{2} + 698 T^{3} + 9852 T^{4} + 28618 T^{5} + 270641 T^{6} + 413526 T^{7} + 2825761 T^{8}$$
$43$ $$1 + 2 T + 100 T^{2} - 104 T^{3} + 4469 T^{4} - 4472 T^{5} + 184900 T^{6} + 159014 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 2 T + 41 T^{2} + 146 T^{3} + 3640 T^{4} + 6862 T^{5} + 90569 T^{6} + 207646 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 6 T + 53 T^{2} + 370 T^{3} - 2412 T^{4} + 19610 T^{5} + 148877 T^{6} - 893262 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 170 T^{2} + 32 T^{3} + 13899 T^{4} + 1888 T^{5} + 591770 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 4 T + 124 T^{2} + 116 T^{3} + 6358 T^{4} + 7076 T^{5} + 461404 T^{6} - 907924 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 4 T + 139 T^{2} + 892 T^{3} + 10912 T^{4} + 59764 T^{5} + 623971 T^{6} + 1203052 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 16 T + 335 T^{2} + 3432 T^{3} + 37440 T^{4} + 243672 T^{5} + 1688735 T^{6} + 5726576 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 12 T + 184 T^{2} - 2148 T^{3} + 16782 T^{4} - 156804 T^{5} + 980536 T^{6} - 4668204 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 2 T + 97 T^{2} - 1498 T^{3} - 1952 T^{4} - 118342 T^{5} + 605377 T^{6} + 986078 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 18 T + 389 T^{2} - 4490 T^{3} + 50748 T^{4} - 372670 T^{5} + 2679821 T^{6} - 10292166 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 8 T + 263 T^{2} + 1500 T^{3} + 30168 T^{4} + 133500 T^{5} + 2083223 T^{6} + 5639752 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 24 T + 460 T^{2} - 5704 T^{3} + 65814 T^{4} - 553288 T^{5} + 4328140 T^{6} - 21904152 T^{7} + 88529281 T^{8}$$