# Properties

 Label 2736.1.o.c Level $2736$ Weight $1$ Character orbit 2736.o Self dual yes Analytic conductor $1.365$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 684) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.155952.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{5} - q^{7} +O(q^{10})$$ $$q -\beta q^{5} - q^{7} -\beta q^{11} + \beta q^{17} + q^{19} + 2 q^{25} + \beta q^{35} + q^{43} + \beta q^{47} + 3 q^{55} + q^{61} - q^{73} + \beta q^{77} -3 q^{85} -\beta q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{7} + O(q^{10})$$ $$2q - 2q^{7} + 2q^{19} + 4q^{25} + 2q^{43} + 6q^{55} + 2q^{61} - 2q^{73} - 6q^{85} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 1.73205 −1.73205
0 0 0 −1.73205 0 −1.00000 0 0 0
721.2 0 0 0 1.73205 0 −1.00000 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
3.b odd 2 1 inner
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 2736.1.o.c 2
3.b odd 2 1 inner 2736.1.o.c 2
4.b odd 2 1 684.1.h.b 2
12.b even 2 1 684.1.h.b 2
19.b odd 2 1 CM 2736.1.o.c 2
57.d even 2 1 inner 2736.1.o.c 2
76.d even 2 1 684.1.h.b 2
228.b odd 2 1 684.1.h.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3$$ acting on $$S_{1}^{\mathrm{new}}(2736, [\chi])$$.

## Hecke characteristic polynomials

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