# Properties

 Label 2736.2.f.a Level $2736$ Weight $2$ Character orbit 2736.f Analytic conductor $21.847$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 342) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} -2 q^{7} +O(q^{10})$$ $$q + \beta q^{5} -2 q^{7} -\beta q^{11} + \beta q^{17} + ( -1 + 3 \beta ) q^{19} -\beta q^{23} + 3 q^{25} -6 q^{29} -2 \beta q^{35} -6 \beta q^{37} -6 q^{41} + 4 q^{43} + 5 \beta q^{47} -3 q^{49} -6 q^{53} + 2 q^{55} -4 q^{61} -6 \beta q^{67} -12 q^{71} -10 q^{73} + 2 \beta q^{77} -6 \beta q^{79} + 11 \beta q^{83} -2 q^{85} -6 q^{89} + ( -6 - \beta ) q^{95} + 6 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{7} + O(q^{10})$$ $$2q - 4q^{7} - 2q^{19} + 6q^{25} - 12q^{29} - 12q^{41} + 8q^{43} - 6q^{49} - 12q^{53} + 4q^{55} - 8q^{61} - 24q^{71} - 20q^{73} - 4q^{85} - 12q^{89} - 12q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 − 1.41421i 1.41421i
0 0 0 1.41421i 0 −2.00000 0 0 0
1025.2 0 0 0 1.41421i 0 −2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.f.a 2
3.b odd 2 1 2736.2.f.b 2
4.b odd 2 1 342.2.b.b yes 2
12.b even 2 1 342.2.b.a 2
19.b odd 2 1 2736.2.f.b 2
57.d even 2 1 inner 2736.2.f.a 2
76.d even 2 1 342.2.b.a 2
228.b odd 2 1 342.2.b.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.b.a 2 12.b even 2 1
342.2.b.a 2 76.d even 2 1
342.2.b.b yes 2 4.b odd 2 1
342.2.b.b yes 2 228.b odd 2 1
2736.2.f.a 2 1.a even 1 1 trivial
2736.2.f.a 2 57.d even 2 1 inner
2736.2.f.b 2 3.b odd 2 1
2736.2.f.b 2 19.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 2$$ $$T_{7} + 2$$ $$T_{11}^{2} + 2$$ $$T_{29} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$2 + T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$2 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$2 + T^{2}$$
$19$ $$19 + 2 T + T^{2}$$
$23$ $$2 + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$72 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$50 + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$( 4 + T )^{2}$$
$67$ $$72 + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$72 + T^{2}$$
$83$ $$242 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$72 + T^{2}$$