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 $4$

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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:

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

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