# Properties

 Label 2736.2.a.r Level $2736$ Weight $2$ Character orbit 2736.a Self dual yes Analytic conductor $21.847$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 342) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{5} + O(q^{10})$$ $$q + 2q^{5} - 2q^{11} - 4q^{13} + q^{19} - 8q^{23} - q^{25} - 2q^{29} + 2q^{31} - 8q^{37} - 2q^{41} - 4q^{43} + 4q^{47} - 7q^{49} + 2q^{53} - 4q^{55} - 10q^{61} - 8q^{65} + 16q^{71} + 6q^{73} - 14q^{79} + 6q^{83} - 18q^{89} + 2q^{95} + 10q^{97} + O(q^{100})$$

## 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 2.00000 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.r 1
3.b odd 2 1 2736.2.a.f 1
4.b odd 2 1 342.2.a.g yes 1
12.b even 2 1 342.2.a.a 1
20.d odd 2 1 8550.2.a.i 1
60.h even 2 1 8550.2.a.y 1
76.d even 2 1 6498.2.a.i 1
228.b odd 2 1 6498.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.a.a 1 12.b even 2 1
342.2.a.g yes 1 4.b odd 2 1
2736.2.a.f 1 3.b odd 2 1
2736.2.a.r 1 1.a even 1 1 trivial
6498.2.a.i 1 76.d even 2 1
6498.2.a.o 1 228.b odd 2 1
8550.2.a.i 1 20.d odd 2 1
8550.2.a.y 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2736))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-2 + T$$
$7$ $$T$$
$11$ $$2 + T$$
$13$ $$4 + T$$
$17$ $$T$$
$19$ $$-1 + T$$
$23$ $$8 + T$$
$29$ $$2 + T$$
$31$ $$-2 + T$$
$37$ $$8 + T$$
$41$ $$2 + T$$
$43$ $$4 + T$$
$47$ $$-4 + T$$
$53$ $$-2 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$T$$
$71$ $$-16 + T$$
$73$ $$-6 + T$$
$79$ $$14 + T$$
$83$ $$-6 + T$$
$89$ $$18 + T$$
$97$ $$-10 + T$$