# Properties

 Label 8036.2.a.l Level 8036 Weight 2 Character orbit 8036.a Self dual Yes Analytic conductor 64.168 Analytic rank 1 Dimension 5 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8036 = 2^{2} \cdot 7^{2} \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 8036.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$64.1677830643$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.470117.1 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( 1 - \beta_{1} - \beta_{4} ) q^{13} + ( -2 - \beta_{3} + \beta_{4} ) q^{15} + ( -2 \beta_{2} - \beta_{3} ) q^{17} + ( 2 \beta_{3} + \beta_{4} ) q^{19} + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{23} + ( -1 + \beta_{3} - \beta_{4} ) q^{25} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{27} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} + ( 1 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -1 - \beta_{1} - \beta_{2} ) q^{33} + ( -4 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{37} + ( -4 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{39} - q^{41} + ( -6 + 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} ) q^{43} + ( \beta_{3} + \beta_{4} ) q^{45} + ( 2 + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{47} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{51} + ( -5 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{53} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{55} + ( -3 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{57} + ( 1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{59} + ( -1 + \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{61} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{65} + ( -5 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} ) q^{67} + ( -1 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{69} + ( -5 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{71} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{73} + ( -3 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{75} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{79} + ( -1 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{81} + ( -3 + 3 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{83} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{85} + ( 7 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{87} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{89} + ( 4 + 3 \beta_{3} - 3 \beta_{4} ) q^{93} + ( 2 + 2 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{95} + ( -2 + \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{97} + ( -7 + \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 2q^{3} + q^{5} + q^{9} + O(q^{10})$$ $$5q + 2q^{3} + q^{5} + q^{9} - 2q^{11} + q^{13} - 9q^{15} + 3q^{17} + 4q^{19} - 8q^{23} - 6q^{25} + 8q^{27} - 9q^{29} + 11q^{31} - 5q^{33} - 11q^{37} - 17q^{39} - 5q^{41} - 27q^{43} + 3q^{45} + 3q^{47} - 3q^{51} - 19q^{53} + 13q^{55} - 11q^{57} + 15q^{59} + 7q^{65} - 21q^{67} - 14q^{69} - 16q^{71} + 10q^{73} - 12q^{75} - 14q^{79} - 7q^{81} + 2q^{83} - 21q^{85} + 36q^{87} - 6q^{89} + 17q^{93} + 9q^{95} - 20q^{97} - 17q^{99} + O(q^{100})$$

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

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

## 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
 −2.05768 −0.475832 −0.189142 1.94177 2.78088
0 −2.05768 0 2.77770 0 0 0 1.23404 0
1.2 0 −0.475832 0 −1.19617 0 0 0 −2.77358 0
1.3 0 −0.189142 0 1.51194 0 0 0 −2.96423 0
1.4 0 1.94177 0 −2.68629 0 0 0 0.770473 0
1.5 0 2.78088 0 0.592821 0 0 0 4.73330 0
 $$n$$: 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$41$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8036))$$:

 $$T_{3}^{5} - 2 T_{3}^{4} - 6 T_{3}^{3} + 8 T_{3}^{2} + 7 T_{3} + 1$$ $$T_{5}^{5} - T_{5}^{4} - 9 T_{5}^{3} + 8 T_{5}^{2} + 12 T_{5} - 8$$ $$T_{11}^{5} + 2 T_{11}^{4} - 29 T_{11}^{3} - 98 T_{11}^{2} - 88 T_{11} - 24$$