# Properties

 Label 2736.1.b.b Level $2736$ Weight $1$ Character orbit 2736.b Analytic conductor $1.365$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ 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: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{5} -\zeta_{24}^{6} q^{7} +O(q^{10})$$ $$q + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{5} -\zeta_{24}^{6} q^{7} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{11} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{17} + \zeta_{24}^{6} q^{19} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{23} + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{10} ) q^{25} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{35} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{43} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{47} + ( -\zeta_{24}^{4} - 2 \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{55} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{61} - q^{73} + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{77} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{83} + ( -\zeta_{24}^{4} + \zeta_{24}^{8} ) q^{85} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{25} - 8q^{73} - 8q^{85} + O(q^{100})$$

## 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
 0.965926 + 0.258819i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 + 0.965926i −0.258819 − 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 − 0.258819i
0 0 0 1.93185i 0 1.00000i 0 0 0
2735.2 0 0 0 1.93185i 0 1.00000i 0 0 0
2735.3 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.4 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.5 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.6 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.7 0 0 0 1.93185i 0 1.00000i 0 0 0
2735.8 0 0 0 1.93185i 0 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2735.8 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.b 8
3.b odd 2 1 inner 2736.1.b.b 8
4.b odd 2 1 inner 2736.1.b.b 8
12.b even 2 1 inner 2736.1.b.b 8
19.b odd 2 1 CM 2736.1.b.b 8
57.d even 2 1 inner 2736.1.b.b 8
76.d even 2 1 inner 2736.1.b.b 8
228.b odd 2 1 inner 2736.1.b.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.1.b.b 8 1.a even 1 1 trivial
2736.1.b.b 8 3.b odd 2 1 inner
2736.1.b.b 8 4.b odd 2 1 inner
2736.1.b.b 8 12.b even 2 1 inner
2736.1.b.b 8 19.b odd 2 1 CM
2736.1.b.b 8 57.d even 2 1 inner
2736.1.b.b 8 76.d even 2 1 inner
2736.1.b.b 8 228.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 4 T_{5}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2736, [\chi])$$.

## Hecke characteristic polynomials

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