# Properties

 Label 4030.2.a.g Level 4030 Weight 2 Character orbit 4030.a Self dual Yes Analytic conductor 32.180 Analytic rank 1 Dimension 6 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4030 = 2 \cdot 5 \cdot 13 \cdot 31$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4030.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$32.1797120146$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.10369693.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( -1 + \beta_{1} ) q^{3} + q^{4} + q^{5} + ( -1 + \beta_{1} ) q^{6} + ( -2 - \beta_{3} ) q^{7} + q^{8} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( -1 + \beta_{1} ) q^{3} + q^{4} + q^{5} + ( -1 + \beta_{1} ) q^{6} + ( -2 - \beta_{3} ) q^{7} + q^{8} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + q^{10} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{11} + ( -1 + \beta_{1} ) q^{12} - q^{13} + ( -2 - \beta_{3} ) q^{14} + ( -1 + \beta_{1} ) q^{15} + q^{16} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{18} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{19} + q^{20} + ( 2 - 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{21} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{22} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{23} + ( -1 + \beta_{1} ) q^{24} + q^{25} - q^{26} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{27} + ( -2 - \beta_{3} ) q^{28} + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{29} + ( -1 + \beta_{1} ) q^{30} - q^{31} + q^{32} + ( -\beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{33} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{34} + ( -2 - \beta_{3} ) q^{35} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{36} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{38} + ( 1 - \beta_{1} ) q^{39} + q^{40} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{41} + ( 2 - 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{42} + ( -2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{43} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{44} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{45} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{46} + ( -4 + 4 \beta_{1} + 3 \beta_{4} + \beta_{5} ) q^{47} + ( -1 + \beta_{1} ) q^{48} + ( 1 - \beta_{1} + 4 \beta_{3} - \beta_{5} ) q^{49} + q^{50} + ( -2 - \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{51} - q^{52} + ( 3 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{54} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{55} + ( -2 - \beta_{3} ) q^{56} + ( -6 + \beta_{1} - \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{57} + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{58} + ( -9 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{59} + ( -1 + \beta_{1} ) q^{60} + ( 5 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{61} - q^{62} + ( -4 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{63} + q^{64} - q^{65} + ( -\beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{66} + ( -2 + 4 \beta_{2} + \beta_{4} + \beta_{5} ) q^{67} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{68} + ( -2 \beta_{1} + \beta_{2} - 4 \beta_{4} + 2 \beta_{5} ) q^{69} + ( -2 - \beta_{3} ) q^{70} + ( -7 + 4 \beta_{1} + 4 \beta_{2} ) q^{71} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{72} + ( -\beta_{1} + \beta_{2} + 4 \beta_{3} - 4 \beta_{4} ) q^{73} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{74} + ( -1 + \beta_{1} ) q^{75} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{76} + 3 \beta_{1} q^{77} + ( 1 - \beta_{1} ) q^{78} + ( \beta_{1} - 3 \beta_{2} + 3 \beta_{4} ) q^{79} + q^{80} + ( -5 - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{81} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{82} + ( -8 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{83} + ( 2 - 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{84} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{85} + ( -2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{86} + ( 2 - 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{87} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{88} + ( 3 - 2 \beta_{1} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{90} + ( 2 + \beta_{3} ) q^{91} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{92} + ( 1 - \beta_{1} ) q^{93} + ( -4 + 4 \beta_{1} + 3 \beta_{4} + \beta_{5} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{95} + ( -1 + \beta_{1} ) q^{96} + ( -6 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{97} + ( 1 - \beta_{1} + 4 \beta_{3} - \beta_{5} ) q^{98} + ( 1 - \beta_{2} + \beta_{3} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{2} - 3q^{3} + 6q^{4} + 6q^{5} - 3q^{6} - 10q^{7} + 6q^{8} - q^{9} + O(q^{10})$$ $$6q + 6q^{2} - 3q^{3} + 6q^{4} + 6q^{5} - 3q^{6} - 10q^{7} + 6q^{8} - q^{9} + 6q^{10} - 10q^{11} - 3q^{12} - 6q^{13} - 10q^{14} - 3q^{15} + 6q^{16} - 4q^{17} - q^{18} - 7q^{19} + 6q^{20} + q^{21} - 10q^{22} - 3q^{23} - 3q^{24} + 6q^{25} - 6q^{26} - 3q^{27} - 10q^{28} - 4q^{29} - 3q^{30} - 6q^{31} + 6q^{32} - 4q^{33} - 4q^{34} - 10q^{35} - q^{36} - 2q^{37} - 7q^{38} + 3q^{39} + 6q^{40} - 10q^{41} + q^{42} - 15q^{43} - 10q^{44} - q^{45} - 3q^{46} - 24q^{47} - 3q^{48} - 2q^{49} + 6q^{50} - 3q^{51} - 6q^{52} + 2q^{53} - 3q^{54} - 10q^{55} - 10q^{56} - 23q^{57} - 4q^{58} - 25q^{59} - 3q^{60} + 9q^{61} - 6q^{62} - 9q^{63} + 6q^{64} - 6q^{65} - 4q^{66} - 26q^{67} - 4q^{68} - 2q^{69} - 10q^{70} - 38q^{71} - q^{72} - q^{73} - 2q^{74} - 3q^{75} - 7q^{76} + 9q^{77} + 3q^{78} + 6q^{80} - 18q^{81} - 10q^{82} - 18q^{83} + q^{84} - 4q^{85} - 15q^{86} - 7q^{87} - 10q^{88} + 4q^{89} - q^{90} + 10q^{91} - 3q^{92} + 3q^{93} - 24q^{94} - 7q^{95} - 3q^{96} - 21q^{97} - 2q^{98} + 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} - 4 x^{4} + 12 x^{3} + 5 x^{2} - 7 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2 \nu^{2} + 3 \nu - 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - 3 \nu^{4} - 4 \nu^{3} + 11 \nu^{2} + 6 \nu - 4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 6 \nu^{3} + 8 \nu^{2} + 10 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} - \beta_{4} + 5 \beta_{3} + 7 \beta_{2} + 9 \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{5} - 2 \beta_{4} + 8 \beta_{3} + 18 \beta_{2} + 30 \beta_{1} + 28$$

## 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
 −1.66311 −1.06862 0.172245 0.517376 2.31687 2.72524
1.00000 −2.66311 1.00000 1.00000 −2.66311 −4.14258 1.00000 4.09215 1.00000
1.2 1.00000 −2.06862 1.00000 1.00000 −2.06862 0.701675 1.00000 1.27917 1.00000
1.3 1.00000 −0.827755 1.00000 1.00000 −0.827755 0.429039 1.00000 −2.31482 1.00000
1.4 1.00000 −0.482624 1.00000 1.00000 −0.482624 −0.948993 1.00000 −2.76707 1.00000
1.5 1.00000 1.31687 1.00000 1.00000 1.31687 −4.24970 1.00000 −1.26586 1.00000
1.6 1.00000 1.72524 1.00000 1.00000 1.72524 −1.78945 1.00000 −0.0235611 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$
$$31$$ $$1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{6} + 3 T_{3}^{5} - 4 T_{3}^{4} - 14 T_{3}^{3} + 2 T_{3}^{2} + 14 T_{3} + 5$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4030))$$.