# Properties

 Label 6027.2.a.z Level 6027 Weight 2 Character orbit 6027.a Self dual Yes Analytic conductor 48.126 Analytic rank 1 Dimension 8 CM No Inner twists 1

# Learn more about

## 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} - 8 x^{6} + 14 x^{5} + 18 x^{4} - 24 x^{3} - 10 x^{2} + 10 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 6 \nu^{2} - \nu + 4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 6 \nu^{3} - \nu^{2} + 5 \nu$$ $$\beta_{6}$$ $$=$$ $$\nu^{7} - \nu^{6} - 9 \nu^{5} + 6 \nu^{4} + 24 \nu^{3} - 7 \nu^{2} - 17 \nu + 2$$ $$\beta_{7}$$ $$=$$ $$\nu^{7} - 2 \nu^{6} - 8 \nu^{5} + 13 \nu^{4} + 18 \nu^{3} - 18 \nu^{2} - 9 \nu + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 6 \beta_{2} + \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 6 \beta_{3} + 7 \beta_{2} + 19 \beta_{1} + 9$$ $$\nu^{6}$$ $$=$$ $$-\beta_{7} + \beta_{6} + \beta_{5} + 7 \beta_{4} + 32 \beta_{2} + 10 \beta_{1} + 70$$ $$\nu^{7}$$ $$=$$ $$-\beta_{7} + 2 \beta_{6} + 10 \beta_{5} + \beta_{4} + 30 \beta_{3} + 42 \beta_{2} + 96 \beta_{1} + 62$$

## 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
 2.39346 2.28873 1.17091 0.487949 0.117246 −0.896239 −1.41849 −2.14356
−2.39346 −1.00000 3.72866 −2.51820 2.39346 0 −4.13748 1.00000 6.02722
1.2 −2.28873 −1.00000 3.23826 2.92705 2.28873 0 −2.83405 1.00000 −6.69922
1.3 −1.17091 −1.00000 −0.628975 2.94759 1.17091 0 3.07829 1.00000 −3.45135
1.4 −0.487949 −1.00000 −1.76191 −1.53225 0.487949 0 1.83562 1.00000 0.747657
1.5 −0.117246 −1.00000 −1.98625 −0.562834 0.117246 0 0.467371 1.00000 0.0659898
1.6 0.896239 −1.00000 −1.19676 1.54330 −0.896239 0 −2.86506 1.00000 1.38317
1.7 1.41849 −1.00000 0.0121162 −2.27752 −1.41849 0 −2.81979 1.00000 −3.23064
1.8 2.14356 −1.00000 2.59485 1.47286 −2.14356 0 1.27510 1.00000 3.15717
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6027))$$:

 $$T_{2}^{8} + \cdots$$ $$T_{5}^{8} - \cdots$$ $$T_{13}^{8} - \cdots$$