# Properties

 Label 4030.2.a.e 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: 6.6.6550837.1 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 8 x^{4} + 6 x^{3} + 17 x^{2} - 9 x - 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5 \nu^{2} + 3 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} + 4 \nu^{2} - 7 \nu - 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 7 \nu^{2} + \nu - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2 \beta_{3} + 6 \beta_{2} + \beta_{1} + 13$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 6 \beta_{4} + 8 \beta_{3} + 9 \beta_{2} + 17 \beta_{1} + 10$$

## 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
 2.39285 1.80051 0.920227 −0.357882 −1.82857 −1.92714
1.00000 −3.39285 1.00000 1.00000 −3.39285 −3.19544 1.00000 8.51145 1.00000
1.2 1.00000 −2.80051 1.00000 1.00000 −2.80051 3.22197 1.00000 4.84285 1.00000
1.3 1.00000 −1.92023 1.00000 1.00000 −1.92023 −3.60391 1.00000 0.687271 1.00000
1.4 1.00000 −0.642118 1.00000 1.00000 −0.642118 1.07525 1.00000 −2.58768 1.00000
1.5 1.00000 0.828569 1.00000 1.00000 0.828569 −0.496108 1.00000 −2.31347 1.00000
1.6 1.00000 0.927138 1.00000 1.00000 0.927138 −5.00176 1.00000 −2.14042 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 $$T_{3}^{6} + 7 T_{3}^{5} + 12 T_{3}^{4} - 8 T_{3}^{3} - 24 T_{3}^{2} + 4 T_{3} + 9$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4030))$$.