# Properties

 Label 2736.2.a.d 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 114) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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.d 1
3.b odd 2 1 912.2.a.k 1
4.b odd 2 1 342.2.a.b 1
12.b even 2 1 114.2.a.b 1
20.d odd 2 1 8550.2.a.ba 1
24.f even 2 1 3648.2.a.x 1
24.h odd 2 1 3648.2.a.c 1
60.h even 2 1 2850.2.a.j 1
60.l odd 4 2 2850.2.d.b 2
76.d even 2 1 6498.2.a.p 1
84.h odd 2 1 5586.2.a.y 1
228.b odd 2 1 2166.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.b 1 12.b even 2 1
342.2.a.b 1 4.b odd 2 1
912.2.a.k 1 3.b odd 2 1
2166.2.a.d 1 228.b odd 2 1
2736.2.a.d 1 1.a even 1 1 trivial
2850.2.a.j 1 60.h even 2 1
2850.2.d.b 2 60.l odd 4 2
3648.2.a.c 1 24.h odd 2 1
3648.2.a.x 1 24.f even 2 1
5586.2.a.y 1 84.h odd 2 1
6498.2.a.p 1 76.d even 2 1
8550.2.a.ba 1 20.d odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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