Properties

Label 1368.1.ba.b
Level $1368$
Weight $1$
Character orbit 1368.ba
Analytic conductor $0.683$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
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.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.682720937282\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.233928.2
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.14971392.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

Character values

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

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
619.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 1.00000 1.00000 0
1147.1 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.g even 3 1 inner
1368.ba 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.ba.b 2
8.d odd 2 1 CM 1368.1.ba.b 2
9.c even 3 1 1368.1.cj.a yes 2
19.c even 3 1 1368.1.cj.a yes 2
72.p odd 6 1 1368.1.cj.a yes 2
152.k odd 6 1 1368.1.cj.a yes 2
171.g even 3 1 inner 1368.1.ba.b 2
1368.ba odd 6 1 inner 1368.1.ba.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.1.ba.b 2 1.a even 1 1 trivial
1368.1.ba.b 2 8.d odd 2 1 CM
1368.1.ba.b 2 171.g even 3 1 inner
1368.1.ba.b 2 1368.ba odd 6 1 inner
1368.1.cj.a yes 2 9.c even 3 1
1368.1.cj.a yes 2 19.c even 3 1
1368.1.cj.a yes 2 72.p odd 6 1
1368.1.cj.a yes 2 152.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1368, [\chi])\):

\( T_{5} \)
\( T_{11}^{2} + 2 T_{11} + 4 \)

Hecke characteristic polynomials

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