# Properties

 Label 8036.2.a.a Level $8036$ Weight $2$ Character orbit 8036.a Self dual yes Analytic conductor $64.168$ Analytic rank $2$ Dimension $1$ 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: $$2$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1148) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
0 −1.00000 0 −1.00000 0 0 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$
$$41$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8036.2.a.a 1
7.b odd 2 1 8036.2.a.e 1
7.c even 3 2 1148.2.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.b 2 7.c even 3 2
8036.2.a.a 1 1.a even 1 1 trivial
8036.2.a.e 1 7.b odd 2 1

## Hecke kernels

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

 $$T_{3} + 1$$ $$T_{5} + 1$$ $$T_{11} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$5 + T$$
$13$ $$2 + T$$
$17$ $$3 + T$$
$19$ $$3 + T$$
$23$ $$1 + T$$
$29$ $$10 + T$$
$31$ $$11 + T$$
$37$ $$-7 + T$$
$41$ $$1 + T$$
$43$ $$-8 + T$$
$47$ $$7 + T$$
$53$ $$11 + T$$
$59$ $$7 + T$$
$61$ $$1 + T$$
$67$ $$-7 + T$$
$71$ $$12 + T$$
$73$ $$5 + T$$
$79$ $$-9 + T$$
$83$ $$T$$
$89$ $$3 + T$$
$97$ $$-2 + T$$