# Properties

 Label 6027.2.a.j Level 6027 Weight 2 Character orbit 6027.a Self dual Yes Analytic conductor 48.126 Analytic rank 0 Dimension 2 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6027 = 3 \cdot 7^{2} \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6027.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$48.1258372982$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} - q^{3} + ( 2 + \beta ) q^{4} + ( 2 - \beta ) q^{5} + \beta q^{6} + ( -4 - \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} - q^{3} + ( 2 + \beta ) q^{4} + ( 2 - \beta ) q^{5} + \beta q^{6} + ( -4 - \beta ) q^{8} + q^{9} + ( 4 - \beta ) q^{10} + ( -1 - \beta ) q^{11} + ( -2 - \beta ) q^{12} -\beta q^{13} + ( -2 + \beta ) q^{15} + 3 \beta q^{16} + ( 3 - \beta ) q^{17} -\beta q^{18} + 6 q^{19} -\beta q^{20} + ( 4 + 2 \beta ) q^{22} -3 \beta q^{23} + ( 4 + \beta ) q^{24} + ( 3 - 3 \beta ) q^{25} + ( 4 + \beta ) q^{26} - q^{27} -9 q^{29} + ( -4 + \beta ) q^{30} + ( -3 - \beta ) q^{31} + ( -4 - \beta ) q^{32} + ( 1 + \beta ) q^{33} + ( 4 - 2 \beta ) q^{34} + ( 2 + \beta ) q^{36} + 9 q^{37} -6 \beta q^{38} + \beta q^{39} + ( -4 + 3 \beta ) q^{40} - q^{41} + ( -5 - \beta ) q^{43} + ( -6 - 4 \beta ) q^{44} + ( 2 - \beta ) q^{45} + ( 12 + 3 \beta ) q^{46} + ( 3 + 2 \beta ) q^{47} -3 \beta q^{48} + 12 q^{50} + ( -3 + \beta ) q^{51} + ( -4 - 3 \beta ) q^{52} + ( -6 + 5 \beta ) q^{53} + \beta q^{54} + 2 q^{55} -6 q^{57} + 9 \beta q^{58} + ( -4 - 4 \beta ) q^{59} + \beta q^{60} + ( -5 + 3 \beta ) q^{61} + ( 4 + 4 \beta ) q^{62} + ( 4 - \beta ) q^{64} + ( 4 - \beta ) q^{65} + ( -4 - 2 \beta ) q^{66} + ( -2 - \beta ) q^{67} + 2 q^{68} + 3 \beta q^{69} + ( -1 - 5 \beta ) q^{71} + ( -4 - \beta ) q^{72} + ( 3 - 3 \beta ) q^{73} -9 \beta q^{74} + ( -3 + 3 \beta ) q^{75} + ( 12 + 6 \beta ) q^{76} + ( -4 - \beta ) q^{78} + ( 2 - 5 \beta ) q^{79} + ( -12 + 3 \beta ) q^{80} + q^{81} + \beta q^{82} -6 q^{83} + ( 10 - 4 \beta ) q^{85} + ( 4 + 6 \beta ) q^{86} + 9 q^{87} + ( 8 + 6 \beta ) q^{88} + ( 2 - 2 \beta ) q^{89} + ( 4 - \beta ) q^{90} + ( -12 - 9 \beta ) q^{92} + ( 3 + \beta ) q^{93} + ( -8 - 5 \beta ) q^{94} + ( 12 - 6 \beta ) q^{95} + ( 4 + \beta ) q^{96} + ( 14 - 3 \beta ) q^{97} + ( -1 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - 2q^{3} + 5q^{4} + 3q^{5} + q^{6} - 9q^{8} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} - 2q^{3} + 5q^{4} + 3q^{5} + q^{6} - 9q^{8} + 2q^{9} + 7q^{10} - 3q^{11} - 5q^{12} - q^{13} - 3q^{15} + 3q^{16} + 5q^{17} - q^{18} + 12q^{19} - q^{20} + 10q^{22} - 3q^{23} + 9q^{24} + 3q^{25} + 9q^{26} - 2q^{27} - 18q^{29} - 7q^{30} - 7q^{31} - 9q^{32} + 3q^{33} + 6q^{34} + 5q^{36} + 18q^{37} - 6q^{38} + q^{39} - 5q^{40} - 2q^{41} - 11q^{43} - 16q^{44} + 3q^{45} + 27q^{46} + 8q^{47} - 3q^{48} + 24q^{50} - 5q^{51} - 11q^{52} - 7q^{53} + q^{54} + 4q^{55} - 12q^{57} + 9q^{58} - 12q^{59} + q^{60} - 7q^{61} + 12q^{62} + 7q^{64} + 7q^{65} - 10q^{66} - 5q^{67} + 4q^{68} + 3q^{69} - 7q^{71} - 9q^{72} + 3q^{73} - 9q^{74} - 3q^{75} + 30q^{76} - 9q^{78} - q^{79} - 21q^{80} + 2q^{81} + q^{82} - 12q^{83} + 16q^{85} + 14q^{86} + 18q^{87} + 22q^{88} + 2q^{89} + 7q^{90} - 33q^{92} + 7q^{93} - 21q^{94} + 18q^{95} + 9q^{96} + 25q^{97} - 3q^{99} + O(q^{100})$$

## 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
 2.56155 −1.56155
−2.56155 −1.00000 4.56155 −0.561553 2.56155 0 −6.56155 1.00000 1.43845
1.2 1.56155 −1.00000 0.438447 3.56155 −1.56155 0 −2.43845 1.00000 5.56155
 $$n$$: 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.

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$
$$41$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6027))$$:

 $$T_{2}^{2} + T_{2} - 4$$ $$T_{5}^{2} - 3 T_{5} - 2$$ $$T_{13}^{2} + T_{13} - 4$$