# Properties

 Label 6027.2.a.x Level 6027 Weight 2 Character orbit 6027.a Self dual Yes Analytic conductor 48.126 Analytic rank 0 Dimension 5 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: $$5$$ Coefficient field: 5.5.1197392.1 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 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} + ( 1 + \beta_{1} - \beta_{4} ) q^{5} + ( 1 - \beta_{1} ) q^{6} + ( 3 - 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} + ( 1 + \beta_{1} - \beta_{4} ) q^{5} + ( 1 - \beta_{1} ) q^{6} + ( 3 - 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{8} + q^{9} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{10} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{11} + ( 2 + \beta_{2} ) q^{12} + ( 1 - \beta_{3} + 2 \beta_{4} ) q^{13} + ( 1 + \beta_{1} - \beta_{4} ) q^{15} + ( 6 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{16} + ( 4 + \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{17} + ( 1 - \beta_{1} ) q^{18} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( -3 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{20} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{22} + ( -1 - \beta_{1} - 2 \beta_{4} ) q^{23} + ( 3 - 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{24} + ( 3 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{25} + ( -1 - 2 \beta_{1} + \beta_{3} + 3 \beta_{4} ) q^{26} + q^{27} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{29} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{30} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{31} + ( 6 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{32} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{33} + ( 5 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + 4 \beta_{4} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{37} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{38} + ( 1 - \beta_{3} + 2 \beta_{4} ) q^{39} + ( -9 - 5 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} ) q^{40} + q^{41} + ( 2 - \beta_{1} + \beta_{4} ) q^{43} + ( 3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{44} + ( 1 + \beta_{1} - \beta_{4} ) q^{45} + ( 2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{46} + ( 1 + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{47} + ( 6 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{48} + ( 1 - 7 \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{50} + ( 4 + \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{51} + ( 5 + 4 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} ) q^{52} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{53} + ( 1 - \beta_{1} ) q^{54} + ( 1 + 3 \beta_{1} + 3 \beta_{2} - \beta_{4} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{57} + ( 4 + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{58} + ( -2 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{59} + ( -3 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{60} + ( 5 \beta_{3} + \beta_{4} ) q^{61} + ( -3 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{62} + ( 7 - 3 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{64} + ( -6 + 3 \beta_{1} + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{65} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{66} + ( 5 - 6 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{67} + ( 9 + 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} ) q^{68} + ( -1 - \beta_{1} - 2 \beta_{4} ) q^{69} + ( 6 + 3 \beta_{3} + 3 \beta_{4} ) q^{71} + ( 3 - 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{72} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{73} + ( -7 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{4} ) q^{74} + ( 3 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{75} + ( 6 + 2 \beta_{1} + \beta_{3} + 3 \beta_{4} ) q^{76} + ( -1 - 2 \beta_{1} + \beta_{3} + 3 \beta_{4} ) q^{78} + ( 4 - \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{79} + ( -6 + 6 \beta_{1} + \beta_{2} - 4 \beta_{3} - 8 \beta_{4} ) q^{80} + q^{81} + ( 1 - \beta_{1} ) q^{82} + ( 4 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{83} + ( 1 + 8 \beta_{1} - \beta_{2} + 4 \beta_{3} - 4 \beta_{4} ) q^{85} + ( 5 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{86} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{87} + ( -5 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{88} + ( 7 - 3 \beta_{1} + \beta_{2} + \beta_{4} ) q^{89} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{90} + ( -3 - 7 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - 5 \beta_{4} ) q^{92} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{93} + ( 1 - 10 \beta_{1} - 6 \beta_{3} - 3 \beta_{4} ) q^{94} + ( -4 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{95} + ( 6 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{96} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{97} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 3q^{2} + 5q^{3} + 11q^{4} + 9q^{5} + 3q^{6} + 9q^{8} + 5q^{9} + O(q^{10})$$ $$5q + 3q^{2} + 5q^{3} + 11q^{4} + 9q^{5} + 3q^{6} + 9q^{8} + 5q^{9} - 9q^{10} + 11q^{12} + 3q^{13} + 9q^{15} + 27q^{16} + 16q^{17} + 3q^{18} - 4q^{19} + 7q^{20} - 6q^{22} - 3q^{23} + 9q^{24} + 20q^{25} - 17q^{26} + 5q^{27} + 13q^{29} - 9q^{30} + 4q^{31} + 21q^{32} + 4q^{34} + 11q^{36} + 17q^{37} - 4q^{38} + 3q^{39} - 37q^{40} + 5q^{41} + 6q^{43} + 32q^{44} + 9q^{45} + 27q^{46} + 15q^{47} + 27q^{48} - 14q^{50} + 16q^{51} + 17q^{52} + 11q^{53} + 3q^{54} + 16q^{55} - 4q^{57} + 9q^{58} - 12q^{59} + 7q^{60} - 12q^{61} - 8q^{62} + 19q^{64} - 19q^{65} - 6q^{66} + 11q^{67} + 28q^{68} - 3q^{69} + 18q^{71} + 9q^{72} - 12q^{73} - 27q^{74} + 20q^{75} + 26q^{76} - 17q^{78} + 23q^{79} + 7q^{80} + 5q^{81} + 3q^{82} + 2q^{83} + 20q^{85} + 18q^{86} + 13q^{87} - 22q^{88} + 28q^{89} - 9q^{90} - 7q^{92} + 4q^{93} + 3q^{94} - 10q^{95} + 21q^{96} - 3q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 8 x^{3} + 6 x^{2} + 14 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 3 \nu^{3} - 4 \nu^{2} + 7 \nu + 2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 4 \nu^{3} - \nu^{2} + 11 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} + 3 \beta_{2} + 10 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{4} + 4 \beta_{3} + 13 \beta_{2} + 31 \beta_{1} + 19$$

## 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
 3.48439 1.61299 0.136381 −1.54585 −1.68791
−2.48439 1.00000 4.17219 4.10940 −2.48439 0 −5.39657 1.00000 −10.2093
1.2 −0.612990 1.00000 −1.62424 1.48910 −0.612990 0 2.22162 1.00000 −0.912805
1.3 0.863619 1.00000 −1.25416 3.66459 0.863619 0 −2.81036 1.00000 3.16481
1.4 2.54585 1.00000 4.48134 2.36161 2.54585 0 6.31711 1.00000 6.01230
1.5 2.68791 1.00000 5.22488 −2.62470 2.68791 0 8.66818 1.00000 −7.05495
 $$n$$: 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

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6027))$$:

 $$T_{2}^{5} - 3 T_{2}^{4} - 6 T_{2}^{3} + 20 T_{2}^{2} - T_{2} - 9$$ $$T_{5}^{5} - 9 T_{5}^{4} + 18 T_{5}^{3} + 42 T_{5}^{2} - 171 T_{5} + 139$$ $$T_{13}^{5} - 3 T_{13}^{4} - 56 T_{13}^{3} + 192 T_{13}^{2} + 473 T_{13} - 1681$$