# Properties

 Label 6039.2.a.a Level $6039$ Weight $2$ Character orbit 6039.a Self dual yes Analytic conductor $48.222$ Analytic rank $1$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.24217.1 Defining polynomial: $$x^{5} - 5 x^{3} - x^{2} + 3 x + 1$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{4} + \beta_{2} q^{5} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{7} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{4} + \beta_{2} q^{5} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{7} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{8} + ( 1 - \beta_{3} ) q^{10} - q^{11} + ( -2 + \beta_{1} - \beta_{4} ) q^{13} + ( 1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{14} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{16} + ( 1 - \beta_{2} ) q^{17} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{19} + ( -\beta_{1} - \beta_{2} ) q^{20} + \beta_{1} q^{22} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{23} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{25} + ( -1 + 4 \beta_{1} + \beta_{2} - \beta_{4} ) q^{26} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{28} + ( 3 - \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{29} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{31} + ( -1 - \beta_{3} - 2 \beta_{4} ) q^{32} + ( -1 - \beta_{1} + \beta_{3} ) q^{34} + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{35} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{37} + ( -1 + 7 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{38} + ( -1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{40} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{41} + ( 2 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{43} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{44} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{46} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{4} ) q^{47} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{49} + ( -1 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{50} + ( -2 + 7 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{52} + ( -1 + 3 \beta_{1} + \beta_{2} - 5 \beta_{3} - 3 \beta_{4} ) q^{53} -\beta_{2} q^{55} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{56} + ( -3 - 5 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{58} + ( 2 - 2 \beta_{1} + 3 \beta_{3} ) q^{59} + q^{61} + ( -5 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{62} + ( 2 + 5 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{64} + ( -1 - 2 \beta_{2} ) q^{65} + ( -1 + 4 \beta_{1} - 4 \beta_{4} ) q^{67} + ( -\beta_{1} + \beta_{4} ) q^{68} + ( -3 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{70} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{71} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{73} + ( -6 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{74} + ( -3 + 11 \beta_{1} + 6 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{76} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{77} + ( -5 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{79} + ( 2 - \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{80} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{82} + ( 1 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{83} + ( -2 - \beta_{1} + \beta_{3} ) q^{85} + ( -1 - 8 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{86} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{88} + ( 4 - \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{89} + ( 2 - 4 \beta_{1} - 5 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{91} + ( -3 + 3 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} ) q^{92} + ( -4 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{94} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{95} + ( -1 + 6 \beta_{1} + 4 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} ) q^{97} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 2q^{2} + 2q^{5} - q^{7} + 6q^{8} + O(q^{10})$$ $$5q + 2q^{2} + 2q^{5} - q^{7} + 6q^{8} + 5q^{10} - 5q^{11} - 10q^{13} + 3q^{14} + 2q^{16} + 3q^{17} - 13q^{19} - 2q^{22} - 15q^{25} - 9q^{26} - 12q^{28} + 7q^{29} - 13q^{31} - q^{32} - 3q^{34} + 13q^{35} - 6q^{37} - 9q^{38} - 5q^{40} + 9q^{41} + 2q^{43} - 7q^{46} + 3q^{47} - 6q^{49} - 9q^{50} - 12q^{52} - 3q^{53} - 2q^{55} - 28q^{56} - 15q^{58} + 14q^{59} + 5q^{61} - 31q^{62} + 6q^{64} - 9q^{65} - 5q^{67} - 5q^{70} - 3q^{71} - 4q^{73} - 2q^{74} - 19q^{76} + q^{77} - 27q^{79} + 2q^{80} + 11q^{82} + 3q^{83} - 8q^{85} - 5q^{86} - 6q^{88} + 12q^{89} + 4q^{91} - 13q^{92} - 18q^{94} - 7q^{95} - 5q^{97} - 14q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 2$$ $$\beta_{2}$$ $$=$$ $$-\nu^{4} + 5 \nu^{2} + \nu - 2$$ $$\beta_{3}$$ $$=$$ $$-\nu^{4} + \nu^{3} + 5 \nu^{2} - 3 \nu - 3$$ $$\beta_{4}$$ $$=$$ $$2 \nu^{4} - \nu^{3} - 9 \nu^{2} + 3 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{2} + \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} - \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3 \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{4} + 5 \beta_{3} - 4 \beta_{1} + 8$$

## 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
 −0.369680 2.17442 −0.722813 0.878095 −1.96003
−1.33536 0 −0.216816 −1.70504 0 −3.82358 2.96025 0 2.27684
1.2 −0.714533 0 −1.48944 1.45989 0 1.23480 2.49332 0 −1.04314
1.3 0.339328 0 −1.88486 −0.383484 0 0.840700 −1.31824 0 −0.130127
1.4 1.26073 0 −0.410549 2.13883 0 2.81011 −3.03906 0 2.69649
1.5 2.44983 0 4.00166 0.489803 0 −2.06203 4.90374 0 1.19993
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.a 5
3.b odd 2 1 671.2.a.a 5
33.d even 2 1 7381.2.a.g 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.a 5 3.b odd 2 1
6039.2.a.a 5 1.a even 1 1 trivial
7381.2.a.g 5 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} - 2 T_{2}^{4} - 3 T_{2}^{3} + 4 T_{2}^{2} + 2 T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6039))$$.