# Properties

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{7} + O(q^{10})$$ $$q - 4q^{7} + 4q^{11} + 2q^{17} - q^{19} - 2q^{23} - 5q^{25} + 6q^{29} - 6q^{31} - 8q^{37} - 10q^{41} + 12q^{43} + 10q^{47} + 9q^{49} - 2q^{53} + 4q^{59} - 10q^{61} - 16q^{71} - 2q^{73} - 16q^{77} - 10q^{79} - 16q^{83} + 2q^{89} - 10q^{97} + O(q^{100})$$

## 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 0 0 −4.00000 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.j 1
3.b odd 2 1 912.2.a.h 1
4.b odd 2 1 342.2.a.f 1
12.b even 2 1 114.2.a.a 1
20.d odd 2 1 8550.2.a.a 1
24.f even 2 1 3648.2.a.bb 1
24.h odd 2 1 3648.2.a.j 1
60.h even 2 1 2850.2.a.x 1
60.l odd 4 2 2850.2.d.s 2
76.d even 2 1 6498.2.a.h 1
84.h odd 2 1 5586.2.a.p 1
228.b odd 2 1 2166.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.a 1 12.b even 2 1
342.2.a.f 1 4.b odd 2 1
912.2.a.h 1 3.b odd 2 1
2166.2.a.i 1 228.b odd 2 1
2736.2.a.j 1 1.a even 1 1 trivial
2850.2.a.x 1 60.h even 2 1
2850.2.d.s 2 60.l odd 4 2
3648.2.a.j 1 24.h odd 2 1
3648.2.a.bb 1 24.f even 2 1
5586.2.a.p 1 84.h odd 2 1
6498.2.a.h 1 76.d even 2 1
8550.2.a.a 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2736))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$-4 + T$$
$13$ $$T$$
$17$ $$-2 + T$$
$19$ $$1 + T$$
$23$ $$2 + T$$
$29$ $$-6 + T$$
$31$ $$6 + T$$
$37$ $$8 + T$$
$41$ $$10 + T$$
$43$ $$-12 + T$$
$47$ $$-10 + T$$
$53$ $$2 + T$$
$59$ $$-4 + T$$
$61$ $$10 + T$$
$67$ $$T$$
$71$ $$16 + T$$
$73$ $$2 + T$$
$79$ $$10 + T$$
$83$ $$16 + T$$
$89$ $$-2 + T$$
$97$ $$10 + T$$