Properties

Label 1368.1.cz.b
Level $1368$
Weight $1$
Character orbit 1368.cz
Analytic conductor $0.683$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
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.cz (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: \(D_{6}\)
Projective field: Galois closure of 6.2.3119171623488.7

$q$-expansion

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

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
563.1
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
1091.1 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.k even 6 1 inner
1368.cz 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.cz.b yes 2
8.d odd 2 1 CM 1368.1.cz.b yes 2
9.d odd 6 1 1368.1.bt.b 2
19.d odd 6 1 1368.1.bt.b 2
72.l even 6 1 1368.1.bt.b 2
152.o even 6 1 1368.1.bt.b 2
171.k even 6 1 inner 1368.1.cz.b yes 2
1368.cz odd 6 1 inner 1368.1.cz.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.1.bt.b 2 9.d odd 6 1
1368.1.bt.b 2 19.d odd 6 1
1368.1.bt.b 2 72.l even 6 1
1368.1.bt.b 2 152.o even 6 1
1368.1.cz.b yes 2 1.a even 1 1 trivial
1368.1.cz.b yes 2 8.d odd 2 1 CM
1368.1.cz.b yes 2 171.k even 6 1 inner
1368.1.cz.b yes 2 1368.cz odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} \) acting on \(S_{1}^{\mathrm{new}}(1368, [\chi])\).

Hecke characteristic polynomials

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