# Properties

 Label 1368.1.ce.a Level $1368$ Weight $1$ Character orbit 1368.ce Analytic conductor $0.683$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ CM discriminant -152 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.682720937282$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.283680450809856.6

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{6} q^{2} + \zeta_{18}^{4} q^{3} -\zeta_{18}^{3} q^{4} -\zeta_{18} q^{6} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{7} + q^{8} + \zeta_{18}^{8} q^{9} +O(q^{10})$$ $$q + \zeta_{18}^{6} q^{2} + \zeta_{18}^{4} q^{3} -\zeta_{18}^{3} q^{4} -\zeta_{18} q^{6} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{7} + q^{8} + \zeta_{18}^{8} q^{9} -\zeta_{18}^{7} q^{12} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{13} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{14} + \zeta_{18}^{6} q^{16} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{17} -\zeta_{18}^{5} q^{18} + q^{19} + ( -\zeta_{18}^{5} + \zeta_{18}^{6} ) q^{21} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{23} + \zeta_{18}^{4} q^{24} + \zeta_{18}^{6} q^{25} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{26} -\zeta_{18}^{3} q^{27} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{28} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{29} -\zeta_{18}^{3} q^{32} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{34} + \zeta_{18}^{2} q^{36} - q^{37} + \zeta_{18}^{6} q^{38} + ( \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{39} + ( \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{42} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{46} -\zeta_{18}^{6} q^{47} -\zeta_{18} q^{48} + ( \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{49} -\zeta_{18}^{3} q^{50} + ( 1 + \zeta_{18}^{8} ) q^{51} + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{52} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{53} + q^{54} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{56} + \zeta_{18}^{4} q^{57} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{58} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{59} + ( 1 - \zeta_{18} ) q^{63} + q^{64} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{67} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{68} + ( 1 - \zeta_{18}^{5} ) q^{69} + \zeta_{18}^{8} q^{72} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{73} -\zeta_{18}^{6} q^{74} -\zeta_{18} q^{75} -\zeta_{18}^{3} q^{76} + ( -\zeta_{18}^{3} - \zeta_{18}^{5} ) q^{78} -\zeta_{18}^{7} q^{81} + ( 1 + \zeta_{18}^{8} ) q^{84} + ( -\zeta_{18}^{3} + \zeta_{18}^{8} ) q^{87} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} + \zeta_{18}^{6} ) q^{91} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{92} + \zeta_{18}^{3} q^{94} -\zeta_{18}^{7} q^{96} + ( 1 - \zeta_{18} + \zeta_{18}^{8} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} - 3 q^{4} + 6 q^{8} + O(q^{10})$$ $$6 q - 3 q^{2} - 3 q^{4} + 6 q^{8} - 3 q^{16} + 6 q^{19} - 3 q^{21} - 3 q^{25} - 3 q^{27} - 3 q^{32} - 6 q^{37} - 3 q^{38} - 3 q^{39} - 3 q^{42} + 3 q^{47} - 3 q^{49} - 3 q^{50} + 6 q^{51} + 6 q^{54} + 6 q^{63} + 6 q^{64} + 6 q^{69} + 3 q^{74} - 3 q^{76} - 3 q^{78} + 6 q^{84} - 3 q^{87} - 6 q^{91} + 3 q^{94} + 6 q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-\zeta_{18}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
493.1
 −0.766044 + 0.642788i 0.939693 + 0.342020i −0.173648 − 0.984808i −0.766044 − 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i
−0.500000 + 0.866025i −0.939693 0.342020i −0.500000 0.866025i 0 0.766044 0.642788i 0.939693 1.62760i 1.00000 0.766044 + 0.642788i 0
493.2 −0.500000 + 0.866025i 0.173648 + 0.984808i −0.500000 0.866025i 0 −0.939693 0.342020i −0.173648 + 0.300767i 1.00000 −0.939693 + 0.342020i 0
493.3 −0.500000 + 0.866025i 0.766044 0.642788i −0.500000 0.866025i 0 0.173648 + 0.984808i −0.766044 + 1.32683i 1.00000 0.173648 0.984808i 0
949.1 −0.500000 0.866025i −0.939693 + 0.342020i −0.500000 + 0.866025i 0 0.766044 + 0.642788i 0.939693 + 1.62760i 1.00000 0.766044 0.642788i 0
949.2 −0.500000 0.866025i 0.173648 0.984808i −0.500000 + 0.866025i 0 −0.939693 + 0.342020i −0.173648 0.300767i 1.00000 −0.939693 0.342020i 0
949.3 −0.500000 0.866025i 0.766044 + 0.642788i −0.500000 + 0.866025i 0 0.173648 0.984808i −0.766044 1.32683i 1.00000 0.173648 + 0.984808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 949.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
152.g odd 2 1 CM by $$\Q(\sqrt{-38})$$
9.c even 3 1 inner
1368.ce odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.1.ce.a 6
8.b even 2 1 1368.1.ce.b yes 6
9.c even 3 1 inner 1368.1.ce.a 6
19.b odd 2 1 1368.1.ce.b yes 6
72.n even 6 1 1368.1.ce.b yes 6
152.g odd 2 1 CM 1368.1.ce.a 6
171.o odd 6 1 1368.1.ce.b yes 6
1368.ce odd 6 1 inner 1368.1.ce.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.1.ce.a 6 1.a even 1 1 trivial
1368.1.ce.a 6 9.c even 3 1 inner
1368.1.ce.a 6 152.g odd 2 1 CM
1368.1.ce.a 6 1368.ce odd 6 1 inner
1368.1.ce.b yes 6 8.b even 2 1
1368.1.ce.b yes 6 19.b odd 2 1
1368.1.ce.b yes 6 72.n even 6 1
1368.1.ce.b yes 6 171.o odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{6} + 3 T_{13}^{4} + 2 T_{13}^{3} + 9 T_{13}^{2} + 3 T_{13} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1368, [\chi])$$.

## Hecke characteristic polynomials

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