# Properties

 Label 4030.2.a.f 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: 6.6.4418197.1 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4 \nu^{2} + 3 \nu + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 6 \nu^{3} - \nu^{2} + 8 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 4 \beta_{2} + \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 6 \beta_{3} + \beta_{2} + 16 \beta_{1}$$

## 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
 −1.95223 0.191750 1.67370 −1.59064 −0.467757 2.14519
1.00000 −2.58368 1.00000 −1.00000 −2.58368 −0.667536 1.00000 3.67538 −1.00000
1.2 1.00000 −1.56820 1.00000 −1.00000 −1.56820 −1.49195 1.00000 −0.540753 −1.00000
1.3 1.00000 −1.33263 1.00000 −1.00000 −1.33263 1.60759 1.00000 −1.22409 −1.00000
1.4 1.00000 −0.252625 1.00000 −1.00000 −0.252625 −1.23948 1.00000 −2.93618 −1.00000
1.5 1.00000 0.300926 1.00000 −1.00000 0.300926 3.15038 1.00000 −2.90944 −1.00000
1.6 1.00000 2.43620 1.00000 −1.00000 2.43620 −3.35900 1.00000 2.93509 −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 $$T_{3}^{6} + 3 T_{3}^{5} - 4 T_{3}^{4} - 18 T_{3}^{3} - 12 T_{3}^{2} + 2 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4030))$$.