# Properties

 Label 2736.1.ga.b Level $2736$ Weight $1$ Character orbit 2736.ga Analytic conductor $1.365$ Analytic rank $0$ Dimension $6$ Projective image $D_{18}$ 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.ga (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{18}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{18} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{18}^{7} - \zeta_{18}^{8} ) q^{7} +O(q^{10})$$ $$q + ( -\zeta_{18}^{7} - \zeta_{18}^{8} ) q^{7} + ( -\zeta_{18}^{2} - \zeta_{18}^{6} ) q^{13} + \zeta_{18}^{3} q^{19} -\zeta_{18}^{4} q^{25} + ( -\zeta_{18}^{2} + \zeta_{18}^{4} ) q^{31} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{37} + ( -\zeta_{18} + \zeta_{18}^{3} ) q^{43} + ( -\zeta_{18}^{5} - \zeta_{18}^{6} - \zeta_{18}^{7} ) q^{49} + ( 1 - \zeta_{18}^{5} ) q^{61} + ( \zeta_{18}^{5} + \zeta_{18}^{6} ) q^{67} + ( \zeta_{18}^{3} + \zeta_{18}^{7} ) q^{73} + ( 1 + \zeta_{18} ) q^{79} + ( -1 - \zeta_{18} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{91} + \zeta_{18}^{8} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + O(q^{10})$$ $$6 q + 3 q^{13} + 3 q^{19} + 3 q^{43} + 3 q^{49} + 6 q^{61} - 3 q^{67} + 3 q^{73} + 6 q^{79} - 6 q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-\zeta_{18}^{7}$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 0.939693 + 0.342020i −0.766044 + 0.642788i −0.766044 − 0.642788i −0.173648 − 0.984808i −0.173648 + 0.984808i 0.939693 − 0.342020i
0 0 0 0 0 1.70574 0.984808i 0 0 0
415.1 0 0 0 0 0 −0.592396 + 0.342020i 0 0 0
1279.1 0 0 0 0 0 −0.592396 0.342020i 0 0 0
1423.1 0 0 0 0 0 −1.11334 + 0.642788i 0 0 0
1567.1 0 0 0 0 0 −1.11334 0.642788i 0 0 0
1999.1 0 0 0 0 0 1.70574 + 0.984808i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1999.1 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.l odd 18 1 inner
228.v even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.ga.b yes 6
3.b odd 2 1 CM 2736.1.ga.b yes 6
4.b odd 2 1 2736.1.ga.a 6
12.b even 2 1 2736.1.ga.a 6
19.e even 9 1 2736.1.ga.a 6
57.l odd 18 1 2736.1.ga.a 6
76.l odd 18 1 inner 2736.1.ga.b yes 6
228.v even 18 1 inner 2736.1.ga.b yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.1.ga.a 6 4.b odd 2 1
2736.1.ga.a 6 12.b even 2 1
2736.1.ga.a 6 19.e even 9 1
2736.1.ga.a 6 57.l odd 18 1
2736.1.ga.b yes 6 1.a even 1 1 trivial
2736.1.ga.b yes 6 3.b odd 2 1 CM
2736.1.ga.b yes 6 76.l odd 18 1 inner
2736.1.ga.b yes 6 228.v even 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} - 3 T_{7}^{4} + 9 T_{7}^{2} + 9 T_{7} + 3$$ acting on $$S_{1}^{\mathrm{new}}(2736, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$3 + 9 T + 9 T^{2} - 3 T^{4} + T^{6}$$
$11$ $$T^{6}$$
$13$ $$1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$17$ $$T^{6}$$
$19$ $$( 1 - T + T^{2} )^{3}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$3 + 9 T + 9 T^{2} - 3 T^{4} + T^{6}$$
$37$ $$( 1 - 3 T + T^{3} )^{2}$$
$41$ $$T^{6}$$
$43$ $$3 - 6 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6}$$
$67$ $$3 + 6 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$71$ $$T^{6}$$
$73$ $$1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$79$ $$3 - 9 T + 18 T^{2} - 21 T^{3} + 15 T^{4} - 6 T^{5} + T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$1 + T^{3} + T^{6}$$