# Properties

 Label 1368.1.dh.a Level $1368$ Weight $1$ Character orbit 1368.dh Analytic conductor $0.683$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ 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.dh (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: $$D_{9}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{9} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{4} q^{2} + \zeta_{18}^{6} q^{3} + \zeta_{18}^{8} q^{4} -\zeta_{18} q^{6} -\zeta_{18}^{3} q^{8} -\zeta_{18}^{3} q^{9} +O(q^{10})$$ $$q + \zeta_{18}^{4} q^{2} + \zeta_{18}^{6} q^{3} + \zeta_{18}^{8} q^{4} -\zeta_{18} q^{6} -\zeta_{18}^{3} q^{8} -\zeta_{18}^{3} q^{9} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{11} -\zeta_{18}^{5} q^{12} -\zeta_{18}^{7} q^{16} -2 \zeta_{18} q^{17} -\zeta_{18}^{7} q^{18} + \zeta_{18}^{8} q^{19} + ( -\zeta_{18}^{3} + \zeta_{18}^{8} ) q^{22} + q^{24} + \zeta_{18}^{8} q^{25} + q^{27} + \zeta_{18}^{2} q^{32} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{33} -2 \zeta_{18}^{5} q^{34} + \zeta_{18}^{2} q^{36} -\zeta_{18}^{3} q^{38} + ( -\zeta_{18}^{5} + \zeta_{18}^{6} ) q^{41} + \zeta_{18} q^{43} + ( -\zeta_{18}^{3} - \zeta_{18}^{7} ) q^{44} + \zeta_{18}^{4} q^{48} + q^{49} -\zeta_{18}^{3} q^{50} -2 \zeta_{18}^{7} q^{51} + \zeta_{18}^{4} q^{54} -\zeta_{18}^{5} q^{57} + ( \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{59} + \zeta_{18}^{6} q^{64} + ( 1 - \zeta_{18}^{5} ) q^{66} + ( 1 - \zeta_{18} ) q^{67} + 2 q^{68} + \zeta_{18}^{6} q^{72} + ( \zeta_{18}^{2} + \zeta_{18}^{6} ) q^{73} -\zeta_{18}^{5} q^{75} -\zeta_{18}^{7} q^{76} + \zeta_{18}^{6} q^{81} + ( 1 - \zeta_{18} ) q^{82} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{83} + \zeta_{18}^{5} q^{86} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{88} + \zeta_{18}^{5} q^{89} + \zeta_{18}^{8} q^{96} + ( -\zeta_{18}^{3} - \zeta_{18}^{5} ) q^{97} + \zeta_{18}^{4} q^{98} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{3} - 3 q^{8} - 3 q^{9} + O(q^{10})$$ $$6 q - 3 q^{3} - 3 q^{8} - 3 q^{9} - 3 q^{22} + 6 q^{24} + 6 q^{27} - 3 q^{38} - 3 q^{41} - 3 q^{44} + 6 q^{49} - 3 q^{50} - 3 q^{59} - 3 q^{64} + 6 q^{66} + 6 q^{67} + 12 q^{68} - 3 q^{72} - 3 q^{73} - 3 q^{81} + 6 q^{82} - 3 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{18}^{2}$$ $$\zeta_{18}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 0.939693 − 0.342020i −0.766044 + 0.642788i −0.766044 − 0.642788i −0.173648 − 0.984808i 0.939693 + 0.342020i −0.173648 + 0.984808i
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
139.1 −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
187.1 −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
427.1 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
859.1 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
1195.1 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 1195.1 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.w even 9 1 inner
1368.dh 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.dh.a 6
8.d odd 2 1 CM 1368.1.dh.a 6
9.c even 3 1 1368.1.en.a yes 6
19.e even 9 1 1368.1.en.a yes 6
72.p odd 6 1 1368.1.en.a yes 6
152.u odd 18 1 1368.1.en.a yes 6
171.w even 9 1 inner 1368.1.dh.a 6
1368.dh odd 18 1 inner 1368.1.dh.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.1.dh.a 6 1.a even 1 1 trivial
1368.1.dh.a 6 8.d odd 2 1 CM
1368.1.dh.a 6 171.w even 9 1 inner
1368.1.dh.a 6 1368.dh odd 18 1 inner
1368.1.en.a yes 6 9.c even 3 1
1368.1.en.a yes 6 19.e even 9 1
1368.1.en.a yes 6 72.p odd 6 1
1368.1.en.a yes 6 152.u odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{6} + 8 T_{17}^{3} + 64$$ acting on $$S_{1}^{\mathrm{new}}(1368, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{3} + T^{6}$$
$3$ $$( 1 + T + T^{2} )^{3}$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$64 + 8 T^{3} + T^{6}$$
$19$ $$1 + T^{3} + T^{6}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6}$$
$37$ $$T^{6}$$
$41$ $$1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$43$ $$1 - T^{3} + T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$61$ $$T^{6}$$
$67$ $$1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6}$$
$71$ $$T^{6}$$
$73$ $$1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$79$ $$T^{6}$$
$83$ $$( 1 - 3 T + T^{3} )^{2}$$
$89$ $$1 - T^{3} + T^{6}$$
$97$ $$1 - 3 T + 3 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$