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

Downloads

Learn more

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:

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

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