# Properties

 Label 8036.2.a.n Level 8036 Weight 2 Character orbit 8036.a Self dual Yes Analytic conductor 64.168 Analytic rank 1 Dimension 8 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{5} q^{5} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{5} q^{5} + ( 1 + \beta_{2} ) q^{9} + ( -1 - \beta_{1} + \beta_{3} ) q^{11} + ( 1 - \beta_{4} + \beta_{5} ) q^{13} + ( -\beta_{3} - \beta_{4} ) q^{15} + ( -\beta_{1} + \beta_{4} - \beta_{6} ) q^{17} + ( -1 - \beta_{2} - \beta_{7} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{25} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{27} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{29} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{31} + ( -3 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{33} + ( -4 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{39} - q^{41} + ( -1 - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{43} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{45} + ( -3 - \beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{47} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{51} + ( -\beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{55} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{57} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{61} + ( -3 + \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{65} + ( -\beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{67} + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{69} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{75} + ( 2 + 3 \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + ( -4 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{81} + ( 5 - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{83} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} ) q^{85} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{87} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{89} + ( -4 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{95} + ( 1 - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{97} + ( -1 - 2 \beta_{1} + \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{9} + O(q^{10})$$ $$8q + 4q^{9} - 8q^{11} + 7q^{13} - q^{15} + q^{17} - 4q^{19} - 3q^{23} - 4q^{25} + 12q^{27} - 4q^{29} - 4q^{31} - 23q^{33} - 31q^{37} + 5q^{39} - 8q^{41} - 8q^{43} - q^{45} - 24q^{47} - 23q^{51} - q^{53} - 2q^{55} - 15q^{57} - 4q^{59} + 4q^{61} - 24q^{65} + 21q^{69} + 8q^{71} - 11q^{73} + 15q^{75} + 14q^{79} - 28q^{81} + 42q^{83} - 20q^{85} - 25q^{87} + 11q^{89} - 27q^{93} - 15q^{95} + 16q^{97} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 14 x^{6} - 4 x^{5} + 60 x^{4} + 31 x^{3} - 75 x^{2} - 60 x - 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} + 10 \nu^{5} - 13 \nu^{4} - 34 \nu^{3} + 10 \nu^{2} + 46 \nu + 19$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} - 4 \nu^{6} - 17 \nu^{5} + 26 \nu^{4} + 38 \nu^{3} - 29 \nu^{2} - 26 \nu - 8$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - 16 \nu^{5} - 11 \nu^{4} + 73 \nu^{3} + 35 \nu^{2} - 94 \nu - 40$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{7} - 4 \nu^{6} - 20 \nu^{5} + 29 \nu^{4} + 65 \nu^{3} - 41 \nu^{2} - 80 \nu - 17$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-2 \nu^{7} + \nu^{6} + 26 \nu^{5} - 5 \nu^{4} - 101 \nu^{3} - 7 \nu^{2} + 113 \nu + 44$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} + \beta_{5} + 3 \beta_{3} + 8 \beta_{2} + \beta_{1} + 23$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{7} + 10 \beta_{6} + 10 \beta_{5} + \beta_{4} + 12 \beta_{3} + 13 \beta_{2} + 28 \beta_{1} + 22$$ $$\nu^{6}$$ $$=$$ $$15 \beta_{7} + 23 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} + 36 \beta_{3} + 62 \beta_{2} + 15 \beta_{1} + 149$$ $$\nu^{7}$$ $$=$$ $$83 \beta_{7} + 86 \beta_{6} + 85 \beta_{5} + 14 \beta_{4} + 116 \beta_{3} + 126 \beta_{2} + 173 \beta_{1} + 210$$

## 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
 −2.28881 −2.24904 −1.33338 −0.503623 −0.308117 1.71291 2.10300 2.86706
0 −2.28881 0 −0.350035 0 0 0 2.23863 0
1.2 0 −2.24904 0 1.47097 0 0 0 2.05817 0
1.3 0 −1.33338 0 −1.76687 0 0 0 −1.22209 0
1.4 0 −0.503623 0 −2.23729 0 0 0 −2.74636 0
1.5 0 −0.308117 0 3.30124 0 0 0 −2.90506 0
1.6 0 1.71291 0 −2.15120 0 0 0 −0.0659453 0
1.7 0 2.10300 0 2.93524 0 0 0 1.42260 0
1.8 0 2.86706 0 −1.20206 0 0 0 5.22005 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8036))$$:

 $$T_{3}^{8} - 14 T_{3}^{6} - 4 T_{3}^{5} + 60 T_{3}^{4} + 31 T_{3}^{3} - 75 T_{3}^{2} - 60 T_{3} - 11$$ $$T_{5}^{8} - 18 T_{5}^{6} - 12 T_{5}^{5} + 98 T_{5}^{4} + 117 T_{5}^{3} - 115 T_{5}^{2} - 196 T_{5} - 51$$ $$T_{11}^{8} + \cdots$$