# Properties

 Label 8036.2.a.d Level $8036$ Weight $2$ Character orbit 8036.a Self dual yes Analytic conductor $64.168$ Analytic rank $0$ 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: $$0$$ 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

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

## 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 −3.00000 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.d 1
7.b odd 2 1 8036.2.a.c 1
7.c even 3 2 1148.2.i.a 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$3 + T$$
$7$ $$T$$
$11$ $$-3 + T$$
$13$ $$4 + T$$
$17$ $$T$$
$19$ $$7 + T$$
$23$ $$-6 + T$$
$29$ $$-6 + T$$
$31$ $$10 + T$$
$37$ $$-2 + T$$
$41$ $$1 + T$$
$43$ $$4 + T$$
$47$ $$-12 + T$$
$53$ $$6 + T$$
$59$ $$-6 + T$$
$61$ $$13 + T$$
$67$ $$4 + T$$
$71$ $$9 + T$$
$73$ $$-14 + T$$
$79$ $$1 + T$$
$83$ $$-12 + T$$
$89$ $$12 + T$$
$97$ $$-2 + T$$