# Properties

 Label 6048.2.a.bv 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: 4.4.22896.1 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} - 4 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} - 5 \nu - 9$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} - 11 \nu - 9$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 7 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 2 \beta_{1} + 11$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} - 6 \beta_{2} + 9 \beta_{1} + 7$$$$)/2$$

## 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
 −2.65924 −0.896343 3.15690 0.398683
0 0 0 −2.83940 0 1.00000 0 0 0
1.2 0 0 0 −0.314350 0 1.00000 0 0 0
1.3 0 0 0 0.766021 0 1.00000 0 0 0
1.4 0 0 0 4.38773 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.bv yes 4
3.b odd 2 1 6048.2.a.bo yes 4
4.b odd 2 1 6048.2.a.br yes 4
12.b even 2 1 6048.2.a.bm 4

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

 $$T_{5}^{4} - 2 T_{5}^{3} - 12 T_{5}^{2} + 6 T_{5} + 3$$ $$T_{11}^{4} - 2 T_{11}^{3} - 24 T_{11}^{2} + 86 T_{11} - 73$$ $$T_{13}^{4} - 36 T_{13}^{2} - 32 T_{13} + 48$$ $$T_{17}^{4} - 8 T_{17}^{3} - 24 T_{17}^{2} + 224 T_{17} - 64$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 2 T + 8 T^{2} - 24 T^{3} + 33 T^{4} - 120 T^{5} + 200 T^{6} - 250 T^{7} + 625 T^{8}$$
$7$ $$( 1 - T )^{4}$$
$11$ $$1 - 2 T + 20 T^{2} + 20 T^{3} + 125 T^{4} + 220 T^{5} + 2420 T^{6} - 2662 T^{7} + 14641 T^{8}$$
$13$ $$1 + 16 T^{2} - 32 T^{3} + 126 T^{4} - 416 T^{5} + 2704 T^{6} + 28561 T^{8}$$
$17$ $$1 - 8 T + 44 T^{2} - 184 T^{3} + 854 T^{4} - 3128 T^{5} + 12716 T^{6} - 39304 T^{7} + 83521 T^{8}$$
$19$ $$1 + 4 T + 34 T^{2} + 72 T^{3} + 699 T^{4} + 1368 T^{5} + 12274 T^{6} + 27436 T^{7} + 130321 T^{8}$$
$23$ $$1 - 2 T + 8 T^{2} - 4 T^{3} + 761 T^{4} - 92 T^{5} + 4232 T^{6} - 24334 T^{7} + 279841 T^{8}$$
$29$ $$1 - 8 T + 80 T^{2} - 296 T^{3} + 2206 T^{4} - 8584 T^{5} + 67280 T^{6} - 195112 T^{7} + 707281 T^{8}$$
$31$ $$1 - 4 T + 70 T^{2} - 448 T^{3} + 2407 T^{4} - 13888 T^{5} + 67270 T^{6} - 119164 T^{7} + 923521 T^{8}$$
$37$ $$1 + 4 T + 118 T^{2} + 344 T^{3} + 5975 T^{4} + 12728 T^{5} + 161542 T^{6} + 202612 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 2 T + 92 T^{2} - 80 T^{3} + 4093 T^{4} - 3280 T^{5} + 154652 T^{6} - 137842 T^{7} + 2825761 T^{8}$$
$43$ $$1 + 8 T + 160 T^{2} + 952 T^{3} + 10046 T^{4} + 40936 T^{5} + 295840 T^{6} + 636056 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 12 T + 200 T^{2} - 1484 T^{3} + 13950 T^{4} - 69748 T^{5} + 441800 T^{6} - 1245876 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 - 4 T + 53 T^{2} )^{4}$$
$59$ $$1 - 12 T + 248 T^{2} - 1916 T^{3} + 21870 T^{4} - 113044 T^{5} + 863288 T^{6} - 2464548 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 8 T + 100 T^{2} + 440 T^{3} + 3478 T^{4} + 26840 T^{5} + 372100 T^{6} + 1815848 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 8 T + 88 T^{2} - 8 T^{3} + 814 T^{4} - 536 T^{5} + 395032 T^{6} - 2406104 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 18 T + 296 T^{2} + 2724 T^{3} + 27369 T^{4} + 193404 T^{5} + 1492136 T^{6} + 6442398 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 8 T + 136 T^{2} - 1032 T^{3} + 11166 T^{4} - 75336 T^{5} + 724744 T^{6} - 3112136 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 160 T^{2} + 848 T^{3} + 11886 T^{4} + 66992 T^{5} + 998560 T^{6} + 38950081 T^{8}$$
$83$ $$1 - 40 T^{2} - 16 T^{3} + 13134 T^{4} - 1328 T^{5} - 275560 T^{6} + 47458321 T^{8}$$
$89$ $$1 - 18 T + 404 T^{2} - 4416 T^{3} + 54549 T^{4} - 393024 T^{5} + 3200084 T^{6} - 12689442 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 8 T + 244 T^{2} - 2200 T^{3} + 29798 T^{4} - 213400 T^{5} + 2295796 T^{6} - 7301384 T^{7} + 88529281 T^{8}$$