# Properties

 Label 6039.2.a.b Level $6039$ Weight $2$ Character orbit 6039.a Self dual yes Analytic conductor $48.222$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6039 = 3^{2} \cdot 11 \cdot 61$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6039.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.2216577807$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.2661761.1 Defining polynomial: $$x^{6} - x^{5} - 6 x^{4} + 3 x^{3} + 9 x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 671) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{5} q^{4} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} ) q^{7} + ( 1 + \beta_{1} - \beta_{4} ) q^{8} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{5} q^{4} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} ) q^{7} + ( 1 + \beta_{1} - \beta_{4} ) q^{8} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{10} + q^{11} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{13} + ( -\beta_{2} - \beta_{4} ) q^{14} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{16} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{17} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{20} -\beta_{3} q^{22} + ( 1 + \beta_{4} + 2 \beta_{5} ) q^{23} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{25} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{26} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{28} + \beta_{4} q^{29} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} + ( -1 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{32} + ( -3 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{34} + ( -1 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{35} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{37} + ( 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{38} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{40} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{41} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{43} -\beta_{5} q^{44} + ( -1 - 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} ) q^{46} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{4} - 4 \beta_{5} ) q^{47} + ( -4 - \beta_{1} + \beta_{2} ) q^{49} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{50} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{52} + ( 3 + \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{53} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{55} + ( \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{56} + ( 1 - \beta_{1} ) q^{58} + ( 2 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{59} - q^{61} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{62} + ( -1 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{64} + ( 2 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{65} + ( -1 - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{67} + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{68} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{70} + ( 3 - \beta_{3} - 4 \beta_{4} ) q^{71} + ( -1 + 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 5 \beta_{5} ) q^{73} + ( -2 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{5} ) q^{74} + ( -2 + \beta_{4} ) q^{76} + ( -1 + \beta_{1} ) q^{77} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 5 \beta_{5} ) q^{79} + ( -4 + 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{80} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{82} + ( -2 - 6 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{83} + ( 1 + \beta_{1} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{85} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{86} + ( 1 + \beta_{1} - \beta_{4} ) q^{88} + ( 4 + \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{89} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{91} + ( -6 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{92} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{94} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{95} + ( -4 - \beta_{1} - 4 \beta_{2} + \beta_{4} ) q^{97} + ( -2 \beta_{1} + \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{4} + q^{5} - 5q^{7} + 6q^{8} + O(q^{10})$$ $$6q + 2q^{4} + q^{5} - 5q^{7} + 6q^{8} - 7q^{10} + 6q^{11} - 4q^{13} - q^{14} - 10q^{16} + 5q^{17} - 3q^{19} + 6q^{20} + 3q^{23} - 11q^{25} - q^{26} + 4q^{28} + q^{29} - 10q^{31} - 3q^{32} - 19q^{34} - 7q^{35} - 19q^{37} + 3q^{38} + 5q^{40} + 7q^{41} - 2q^{43} + 2q^{44} - 7q^{46} - 5q^{47} - 25q^{49} - 17q^{50} + 2q^{52} + 9q^{53} + q^{55} + 4q^{56} + 5q^{58} + 5q^{59} - 6q^{61} - 3q^{62} - 6q^{64} + 11q^{65} - 14q^{67} - 18q^{68} + 7q^{70} + 14q^{71} - 14q^{73} - 6q^{74} - 11q^{76} - 5q^{77} + 5q^{79} - 26q^{80} + q^{82} - 17q^{83} + 2q^{85} + 7q^{86} + 6q^{88} + 25q^{89} - 4q^{91} - 35q^{92} + 30q^{94} - 3q^{95} - 24q^{97} + 2q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 6 x^{4} + 3 x^{3} + 9 x^{2} - x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 5 \nu + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} - 5 \nu^{3} + 2 \nu^{2} + 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + \beta_{4} + 7 \beta_{3} + 8 \beta_{2} + 19 \beta_{1} + 8$$

## 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
 −0.303283 2.36588 −1.34697 0.387870 −1.68584 1.58234
−1.78997 0 1.20400 3.29725 0 −1.30328 1.42482 0 −5.90199
1.2 −1.54773 0 0.395474 −0.422675 0 1.36588 2.48338 0 0.654188
1.3 −0.782747 0 −1.38731 0.742408 0 −2.34697 2.65140 0 −0.581118
1.4 0.255699 0 −1.93462 −2.57819 0 −0.612130 −1.00608 0 −0.659240
1.5 1.57580 0 0.483130 0.593176 0 −2.68584 −2.39028 0 0.934724
1.6 2.28896 0 3.23932 −0.631975 0 0.582341 2.83675 0 −1.44656
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$-1$$
$$61$$ $$1$$

## 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 6039.2.a.b 6
3.b odd 2 1 671.2.a.b 6
33.d even 2 1 7381.2.a.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.b 6 3.b odd 2 1
6039.2.a.b 6 1.a even 1 1 trivial
7381.2.a.h 6 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 7 T_{2}^{4} - 2 T_{2}^{3} + 12 T_{2}^{2} + 5 T_{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6039))$$.