# Properties

 Label 2736.2.a.x Level $2736$ Weight $2$ Character orbit 2736.a Self dual yes Analytic conductor $21.847$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1368) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{5} + O(q^{10})$$ $$q + 4q^{5} + 6q^{11} + 2q^{13} + 4q^{17} + q^{19} - 6q^{23} + 11q^{25} + 10q^{29} - 8q^{31} - 10q^{37} - 6q^{41} + 4q^{43} - 6q^{47} - 7q^{49} + 2q^{53} + 24q^{55} - 4q^{59} + 10q^{61} + 8q^{65} + 12q^{67} - 12q^{71} - 6q^{73} + 4q^{79} - 14q^{83} + 16q^{85} - 6q^{89} + 4q^{95} - 2q^{97} + O(q^{100})$$

## 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 0 0 4.00000 0 0 0 0 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
$$2$$ $$1$$
$$3$$ $$1$$
$$19$$ $$-1$$

## 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 2736.2.a.x 1
3.b odd 2 1 2736.2.a.b 1
4.b odd 2 1 1368.2.a.j yes 1
12.b even 2 1 1368.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.a.a 1 12.b even 2 1
1368.2.a.j yes 1 4.b odd 2 1
2736.2.a.b 1 3.b odd 2 1
2736.2.a.x 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2736))$$:

 $$T_{5} - 4$$ $$T_{7}$$ $$T_{11} - 6$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$T$$
$11$ $$-6 + T$$
$13$ $$-2 + T$$
$17$ $$-4 + T$$
$19$ $$-1 + T$$
$23$ $$6 + T$$
$29$ $$-10 + T$$
$31$ $$8 + T$$
$37$ $$10 + T$$
$41$ $$6 + T$$
$43$ $$-4 + T$$
$47$ $$6 + T$$
$53$ $$-2 + T$$
$59$ $$4 + T$$
$61$ $$-10 + T$$
$67$ $$-12 + T$$
$71$ $$12 + T$$
$73$ $$6 + T$$
$79$ $$-4 + T$$
$83$ $$14 + T$$
$89$ $$6 + T$$
$97$ $$2 + T$$