# Properties

 Label 6048.2.a.bu 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

# 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.25808.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{5} + q^{7} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{5} + q^{7} + ( 1 + \beta_{1} ) q^{11} + ( -1 + \beta_{1} - \beta_{3} ) q^{13} + ( 1 - \beta_{2} + \beta_{3} ) q^{17} + ( \beta_{2} + \beta_{3} ) q^{19} + ( 2 - \beta_{3} ) q^{23} + ( 3 + \beta_{2} + \beta_{3} ) q^{25} + ( -1 + \beta_{1} - \beta_{3} ) q^{29} + ( 3 + \beta_{1} + \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{35} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{37} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{41} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{43} + ( 1 - \beta_{1} - \beta_{3} ) q^{47} + q^{49} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{55} + ( 5 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{59} + ( -2 + 2 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -3 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{65} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{67} + ( 3 - \beta_{2} + 2 \beta_{3} ) q^{71} + ( -1 - \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{73} + ( 1 + \beta_{1} ) q^{77} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{79} + ( 5 - \beta_{1} + \beta_{3} ) q^{83} + ( -3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{85} + ( 6 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{89} + ( -1 + \beta_{1} - \beta_{3} ) q^{91} + ( 10 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{95} + ( 4 + 4 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} + 4q^{7} + O(q^{10})$$ $$4q + 2q^{5} + 4q^{7} + 2q^{11} - 4q^{13} + 4q^{17} - 4q^{19} + 10q^{23} + 8q^{25} - 4q^{29} + 8q^{31} + 2q^{35} - 8q^{37} + 2q^{41} + 4q^{43} + 8q^{47} + 4q^{49} + 16q^{53} + 8q^{55} + 24q^{59} - 8q^{61} - 4q^{65} - 4q^{67} + 10q^{71} + 4q^{73} + 2q^{77} - 4q^{79} + 20q^{83} - 16q^{85} + 26q^{89} - 4q^{91} + 34q^{95} + 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 10 x^{2} - 6 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu^{2} - \nu + 9$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu^{2} - 7 \nu + 9$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 7 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3 \beta_{2} + 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{3} - 10 \beta_{2} + 7 \beta_{1} + 9$$$$)/2$$

## 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
 3.31526 −2.46810 0.707500 −1.55466
0 0 0 −2.58060 0 1.00000 0 0 0
1.2 0 0 0 −1.34410 0 1.00000 0 0 0
1.3 0 0 0 1.96666 0 1.00000 0 0 0
1.4 0 0 0 3.95805 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.bu yes 4
3.b odd 2 1 6048.2.a.bn yes 4
4.b odd 2 1 6048.2.a.bq yes 4
12.b even 2 1 6048.2.a.bl 4

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

 $$T_{5}^{4} - 2 T_{5}^{3} - 12 T_{5}^{2} + 10 T_{5} + 27$$ $$T_{11}^{4} - 2 T_{11}^{3} - 32 T_{11}^{2} + 114 T_{11} - 73$$ $$T_{13}^{4} + 4 T_{13}^{3} - 40 T_{13}^{2} - 176 T_{13} - 32$$ $$T_{17}^{4} - 4 T_{17}^{3} - 32 T_{17}^{2} + 80 T_{17} + 48$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 2 T + 8 T^{2} - 20 T^{3} + 57 T^{4} - 100 T^{5} + 200 T^{6} - 250 T^{7} + 625 T^{8}$$
$7$ $$( 1 - T )^{4}$$
$11$ $$1 - 2 T + 12 T^{2} + 48 T^{3} - 51 T^{4} + 528 T^{5} + 1452 T^{6} - 2662 T^{7} + 14641 T^{8}$$
$13$ $$1 + 4 T + 12 T^{2} - 20 T^{3} - 58 T^{4} - 260 T^{5} + 2028 T^{6} + 8788 T^{7} + 28561 T^{8}$$
$17$ $$1 - 4 T + 36 T^{2} - 124 T^{3} + 694 T^{4} - 2108 T^{5} + 10404 T^{6} - 19652 T^{7} + 83521 T^{8}$$
$19$ $$1 + 4 T + 26 T^{2} - 40 T^{3} + 3 T^{4} - 760 T^{5} + 9386 T^{6} + 27436 T^{7} + 130321 T^{8}$$
$23$ $$1 - 10 T + 96 T^{2} - 576 T^{3} + 3385 T^{4} - 13248 T^{5} + 50784 T^{6} - 121670 T^{7} + 279841 T^{8}$$
$29$ $$1 + 4 T + 76 T^{2} + 172 T^{3} + 2694 T^{4} + 4988 T^{5} + 63916 T^{6} + 97556 T^{7} + 707281 T^{8}$$
$31$ $$1 - 8 T + 94 T^{2} - 392 T^{3} + 3303 T^{4} - 12152 T^{5} + 90334 T^{6} - 238328 T^{7} + 923521 T^{8}$$
$37$ $$1 + 8 T + 62 T^{2} - 40 T^{3} - 121 T^{4} - 1480 T^{5} + 84878 T^{6} + 405224 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 2 T + 108 T^{2} - 4 T^{3} + 5237 T^{4} - 164 T^{5} + 181548 T^{6} - 137842 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 4 T + 68 T^{2} + 220 T^{3} + 1014 T^{4} + 9460 T^{5} + 125732 T^{6} - 318028 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 8 T + 124 T^{2} - 1016 T^{3} + 7926 T^{4} - 47752 T^{5} + 273916 T^{6} - 830584 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 16 T + 148 T^{2} - 1136 T^{3} + 8534 T^{4} - 60208 T^{5} + 415732 T^{6} - 2382032 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 24 T + 348 T^{2} - 3464 T^{3} + 29158 T^{4} - 204376 T^{5} + 1211388 T^{6} - 4929096 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 8 T + 116 T^{2} + 824 T^{3} + 7478 T^{4} + 50264 T^{5} + 431636 T^{6} + 1815848 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 4 T + 100 T^{2} + 692 T^{3} + 4454 T^{4} + 46364 T^{5} + 448900 T^{6} + 1203052 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 10 T + 192 T^{2} - 1424 T^{3} + 19193 T^{4} - 101104 T^{5} + 967872 T^{6} - 3579110 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 4 T + 44 T^{2} - 268 T^{3} + 10310 T^{4} - 19564 T^{5} + 234476 T^{6} - 1556068 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 4 T + 20 T^{2} + 404 T^{3} + 11814 T^{4} + 31916 T^{5} + 124820 T^{6} + 1972156 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 20 T + 436 T^{2} - 4932 T^{3} + 57734 T^{4} - 409356 T^{5} + 3003604 T^{6} - 11435740 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 26 T + 548 T^{2} - 7140 T^{3} + 80541 T^{4} - 635460 T^{5} + 4340708 T^{6} - 18329194 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 8 T + 196 T^{2} - 1688 T^{3} + 26118 T^{4} - 163736 T^{5} + 1844164 T^{6} - 7301384 T^{7} + 88529281 T^{8}$$
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