# Properties

 Label 8036.2.a.k Level 8036 Weight 2 Character orbit 8036.a Self dual Yes Analytic conductor 64.168 Analytic rank 0 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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.287349.1 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{2} + \beta_{4} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( 2 - \beta_{1} + \beta_{4} ) q^{15} + ( \beta_{3} + \beta_{4} ) q^{17} + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{19} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{23} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{25} + ( -2 - \beta_{1} + \beta_{2} ) q^{27} + ( 2 - 3 \beta_{1} - \beta_{4} ) q^{29} + ( -3 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( -1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{33} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{37} + ( 4 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{39} + q^{41} + ( 6 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{43} + ( -2 + \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{45} + ( -2 - 3 \beta_{4} ) q^{47} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{51} + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{53} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{55} + ( 3 + 5 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{57} + ( -1 + 2 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{59} + ( 3 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{61} + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{65} + ( 3 - 4 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{67} + ( -3 + 8 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{69} + ( 3 + 7 \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{71} + ( 3 - 5 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{73} + ( 3 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{75} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{79} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{81} + ( 1 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{83} + ( 3 - \beta_{2} - \beta_{3} ) q^{85} + ( -1 + 7 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{87} + ( 5 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} ) q^{89} + ( -2 + 9 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{93} + ( 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{95} + ( 6 - 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 4 \beta_{4} ) q^{97} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 2q^{3} + 3q^{5} + q^{9} + O(q^{10})$$ $$5q - 2q^{3} + 3q^{5} + q^{9} + 6q^{11} - 7q^{13} + 9q^{15} - q^{17} - 2q^{19} + 4q^{23} + 2q^{25} - 8q^{27} + 11q^{29} - 13q^{31} - 11q^{33} + 5q^{37} + 23q^{39} + 5q^{41} + 29q^{43} - 11q^{45} - 7q^{47} - 3q^{51} + 21q^{53} - 19q^{55} + 9q^{57} - 3q^{59} + 8q^{61} - 5q^{65} + 3q^{67} - 10q^{69} + 22q^{71} + 16q^{73} + 18q^{75} + 4q^{79} - 15q^{81} + 6q^{83} + 13q^{85} - 6q^{87} + 20q^{89} - 5q^{93} - 7q^{95} + 24q^{97} + 27q^{99} + O(q^{100})$$

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

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

## 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
 2.13797 −2.05679 −1.36870 1.14203 0.145487
0 −2.57090 0 −0.350332 0 0 0 3.60954 0
1.2 0 −2.23037 0 −1.70418 0 0 0 1.97454 0
1.3 0 0.126667 0 4.03743 0 0 0 −2.98396 0
1.4 0 0.695770 0 −1.38288 0 0 0 −2.51590 0
1.5 0 1.97883 0 2.39996 0 0 0 0.915782 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8036))$$:

 $$T_{3}^{5} + 2 T_{3}^{4} - 6 T_{3}^{3} - 8 T_{3}^{2} + 9 T_{3} - 1$$ $$T_{5}^{5} - 3 T_{5}^{4} - 9 T_{5}^{3} + 12 T_{5}^{2} + 28 T_{5} + 8$$ $$T_{11}^{5} - 6 T_{11}^{4} - 7 T_{11}^{3} + 68 T_{11}^{2} - 84 T_{11} + 24$$