# Properties

 Label 2736.1.bk.a Level $2736$ Weight $1$ Character orbit 2736.bk Analytic conductor $1.365$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2736.bk (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.225194688.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{7} +O(q^{10})$$ $$q + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{7} -\zeta_{6} q^{13} - q^{19} + \zeta_{6} q^{25} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} - q^{37} + ( -1 - \zeta_{6} ) q^{43} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{49} + \zeta_{6} q^{61} + ( 1 - \zeta_{6}^{2} ) q^{67} + \zeta_{6}^{2} q^{73} + ( 1 + \zeta_{6} ) q^{79} + ( -1 + \zeta_{6}^{2} ) q^{91} -2 \zeta_{6}^{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - q^{13} - 2q^{19} + q^{25} - 2q^{37} - 3q^{43} - 4q^{49} + q^{61} + 3q^{67} - q^{73} + 3q^{79} - 3q^{91} + 2q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
847.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 1.73205i 0 0 0
2287.1 0 0 0 0 0 1.73205i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
76.g odd 6 1 inner
228.m even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.bk.a 2
3.b odd 2 1 CM 2736.1.bk.a 2
4.b odd 2 1 2736.1.bk.b yes 2
12.b even 2 1 2736.1.bk.b yes 2
19.c even 3 1 2736.1.bk.b yes 2
57.h odd 6 1 2736.1.bk.b yes 2
76.g odd 6 1 inner 2736.1.bk.a 2
228.m even 6 1 inner 2736.1.bk.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.1.bk.a 2 1.a even 1 1 trivial
2736.1.bk.a 2 3.b odd 2 1 CM
2736.1.bk.a 2 76.g odd 6 1 inner
2736.1.bk.a 2 228.m even 6 1 inner
2736.1.bk.b yes 2 4.b odd 2 1
2736.1.bk.b yes 2 12.b even 2 1
2736.1.bk.b yes 2 19.c even 3 1
2736.1.bk.b yes 2 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{43}^{2} + 3 T_{43} + 3$$ acting on $$S_{1}^{\mathrm{new}}(2736, [\chi])$$.

## Hecke characteristic polynomials

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