# Properties

 Label 2736.1.b.a Level $2736$ Weight $1$ Character orbit 2736.b Analytic conductor $1.365$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -19 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.155952.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{5} + 2 \zeta_{8}^{2} q^{7} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{5} + 2 \zeta_{8}^{2} q^{7} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{11} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17} + \zeta_{8}^{2} q^{19} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{23} - q^{25} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{35} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{47} -3 q^{49} -2 \zeta_{8}^{2} q^{55} + 2 q^{73} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{77} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{83} + 2 q^{85} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{25} - 12q^{49} + 8q^{73} + 8q^{85} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2735.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 0 0 1.41421i 0 2.00000i 0 0 0
2735.2 0 0 0 1.41421i 0 2.00000i 0 0 0
2735.3 0 0 0 1.41421i 0 2.00000i 0 0 0
2735.4 0 0 0 1.41421i 0 2.00000i 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
4.b odd 2 1 inner
12.b even 2 1 inner
57.d even 2 1 inner
76.d even 2 1 inner
228.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.b.a 4
3.b odd 2 1 inner 2736.1.b.a 4
4.b odd 2 1 inner 2736.1.b.a 4
12.b even 2 1 inner 2736.1.b.a 4
19.b odd 2 1 CM 2736.1.b.a 4
57.d even 2 1 inner 2736.1.b.a 4
76.d even 2 1 inner 2736.1.b.a 4
228.b odd 2 1 inner 2736.1.b.a 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

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