# Properties

 Label 6027.2.a.i Level 6027 Weight 2 Character orbit 6027.a Self dual Yes Analytic conductor 48.126 Analytic rank 1 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−2.41421 −1.00000 3.82843 1.00000 2.41421 0 −4.41421 1.00000 −2.41421
1.2 0.414214 −1.00000 −1.82843 1.00000 −0.414214 0 −1.58579 1.00000 0.414214
 $$n$$: 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6027))$$:

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{5} - 1$$ $$T_{13}^{2} - 6 T_{13} + 7$$