# Properties

 Label 1368.1.dk.b Level $1368$ Weight $1$ Character orbit 1368.dk Analytic conductor $0.683$ Analytic rank $0$ Dimension $6$ Projective image $D_{18}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1368.dk (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.682720937282$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{18} q^{2} -\zeta_{18}^{6} q^{3} + \zeta_{18}^{2} q^{4} + \zeta_{18}^{7} q^{6} -\zeta_{18}^{3} q^{8} -\zeta_{18}^{3} q^{9} +O(q^{10})$$ $$q -\zeta_{18} q^{2} -\zeta_{18}^{6} q^{3} + \zeta_{18}^{2} q^{4} + \zeta_{18}^{7} q^{6} -\zeta_{18}^{3} q^{8} -\zeta_{18}^{3} q^{9} + ( -\zeta_{18} - \zeta_{18}^{2} ) q^{11} -\zeta_{18}^{8} q^{12} + \zeta_{18}^{4} q^{16} + \zeta_{18}^{4} q^{18} -\zeta_{18}^{2} q^{19} + ( \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{22} - q^{24} -\zeta_{18}^{2} q^{25} - q^{27} -\zeta_{18}^{5} q^{32} + ( \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{33} -\zeta_{18}^{5} q^{36} + \zeta_{18}^{3} q^{38} + ( -\zeta_{18}^{6} - \zeta_{18}^{8} ) q^{41} -\zeta_{18}^{7} q^{43} + ( -\zeta_{18}^{3} - \zeta_{18}^{4} ) q^{44} + \zeta_{18} q^{48} + q^{49} + \zeta_{18}^{3} q^{50} + \zeta_{18} q^{54} + \zeta_{18}^{8} q^{57} + ( \zeta_{18}^{2} + \zeta_{18}^{6} ) q^{59} + \zeta_{18}^{6} q^{64} + ( 1 - \zeta_{18}^{8} ) q^{66} + ( -1 - \zeta_{18}^{7} ) q^{67} + \zeta_{18}^{6} q^{72} + ( \zeta_{18}^{5} - \zeta_{18}^{6} ) q^{73} + \zeta_{18}^{8} q^{75} -\zeta_{18}^{4} q^{76} + \zeta_{18}^{6} q^{81} + ( -1 + \zeta_{18}^{7} ) q^{82} + ( -\zeta_{18}^{4} - \zeta_{18}^{5} ) q^{83} + \zeta_{18}^{8} q^{86} + ( \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{88} -\zeta_{18}^{8} q^{89} -\zeta_{18}^{2} q^{96} + ( \zeta_{18}^{3} + \zeta_{18}^{8} ) q^{97} -\zeta_{18} q^{98} + ( \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{3} - 3 q^{8} - 3 q^{9} + O(q^{10})$$ $$6 q + 3 q^{3} - 3 q^{8} - 3 q^{9} + 3 q^{22} - 6 q^{24} - 6 q^{27} + 3 q^{38} + 3 q^{41} - 3 q^{44} + 6 q^{49} + 3 q^{50} - 3 q^{59} - 3 q^{64} + 6 q^{66} - 6 q^{67} - 3 q^{72} + 3 q^{73} - 3 q^{81} - 6 q^{82} + 3 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$$.

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{18}^{5}$$ $$-\zeta_{18}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
155.1
 0.939693 − 0.342020i 0.939693 + 0.342020i −0.173648 + 0.984808i −0.766044 − 0.642788i −0.173648 − 0.984808i −0.766044 + 0.642788i
−0.939693 + 0.342020i 0.500000 + 0.866025i 0.766044 0.642788i 0 −0.766044 0.642788i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0
203.1 −0.939693 0.342020i 0.500000 0.866025i 0.766044 + 0.642788i 0 −0.766044 + 0.642788i 0 −0.500000 0.866025i −0.500000 0.866025i 0
299.1 0.173648 0.984808i 0.500000 + 0.866025i −0.939693 0.342020i 0 0.939693 0.342020i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0
515.1 0.766044 + 0.642788i 0.500000 + 0.866025i 0.173648 + 0.984808i 0 −0.173648 + 0.984808i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0
851.1 0.173648 + 0.984808i 0.500000 0.866025i −0.939693 + 0.342020i 0 0.939693 + 0.342020i 0 −0.500000 0.866025i −0.500000 0.866025i 0
1283.1 0.766044 0.642788i 0.500000 0.866025i 0.173648 0.984808i 0 −0.173648 0.984808i 0 −0.500000 0.866025i −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1283.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
171.bd even 18 1 inner
1368.dk odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.1.dk.b 6
8.d odd 2 1 CM 1368.1.dk.b 6
9.d odd 6 1 1368.1.ee.b yes 6
19.f odd 18 1 1368.1.ee.b yes 6
72.l even 6 1 1368.1.ee.b yes 6
152.v even 18 1 1368.1.ee.b yes 6
171.bd even 18 1 inner 1368.1.dk.b 6
1368.dk odd 18 1 inner 1368.1.dk.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.1.dk.b 6 1.a even 1 1 trivial
1368.1.dk.b 6 8.d odd 2 1 CM
1368.1.dk.b 6 171.bd even 18 1 inner
1368.1.dk.b 6 1368.dk odd 18 1 inner
1368.1.ee.b yes 6 9.d odd 6 1
1368.1.ee.b yes 6 19.f odd 18 1
1368.1.ee.b yes 6 72.l even 6 1
1368.1.ee.b yes 6 152.v even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{6} - 3 T_{11}^{4} + 9 T_{11}^{2} - 9 T_{11} + 3$$ acting on $$S_{1}^{\mathrm{new}}(1368, [\chi])$$.

## Hecke characteristic polynomials

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