Properties

Label 1368.1.cj.c
Level $1368$
Weight $1$
Character orbit 1368.cj
Analytic conductor $0.683$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
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.cj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.682720937282\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.1871424.3

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12}^{3} q^{2} - q^{3} - q^{4} -\zeta_{12}^{5} q^{5} + \zeta_{12}^{3} q^{6} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} + q^{9} +O(q^{10})\) \( q -\zeta_{12}^{3} q^{2} - q^{3} - q^{4} -\zeta_{12}^{5} q^{5} + \zeta_{12}^{3} q^{6} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} + q^{9} -\zeta_{12}^{2} q^{10} + \zeta_{12}^{2} q^{11} + q^{12} + \zeta_{12}^{2} q^{14} + \zeta_{12}^{5} q^{15} + q^{16} + \zeta_{12}^{4} q^{17} -\zeta_{12}^{3} q^{18} + q^{19} + \zeta_{12}^{5} q^{20} -\zeta_{12}^{5} q^{21} -\zeta_{12}^{5} q^{22} -\zeta_{12}^{3} q^{24} - q^{27} -\zeta_{12}^{5} q^{28} + \zeta_{12} q^{29} + \zeta_{12}^{2} q^{30} + \zeta_{12} q^{31} -\zeta_{12}^{3} q^{32} -\zeta_{12}^{2} q^{33} + \zeta_{12} q^{34} + \zeta_{12}^{4} q^{35} - q^{36} -2 \zeta_{12}^{3} q^{37} -\zeta_{12}^{3} q^{38} + \zeta_{12}^{2} q^{40} + \zeta_{12}^{2} q^{41} -\zeta_{12}^{2} q^{42} -\zeta_{12}^{2} q^{44} -\zeta_{12}^{5} q^{45} -\zeta_{12} q^{47} - q^{48} -\zeta_{12}^{4} q^{51} -\zeta_{12}^{5} q^{53} + \zeta_{12}^{3} q^{54} + \zeta_{12} q^{55} -\zeta_{12}^{2} q^{56} - q^{57} -\zeta_{12}^{4} q^{58} -\zeta_{12}^{2} q^{59} -\zeta_{12}^{5} q^{60} + \zeta_{12} q^{61} -\zeta_{12}^{4} q^{62} + \zeta_{12}^{5} q^{63} - q^{64} + \zeta_{12}^{5} q^{66} + 2 q^{67} -\zeta_{12}^{4} q^{68} + \zeta_{12} q^{70} -\zeta_{12} q^{71} + \zeta_{12}^{3} q^{72} + \zeta_{12}^{4} q^{73} -2 q^{74} - q^{76} -\zeta_{12} q^{77} -\zeta_{12}^{5} q^{80} + q^{81} -\zeta_{12}^{5} q^{82} + \zeta_{12}^{2} q^{83} + \zeta_{12}^{5} q^{84} + \zeta_{12}^{3} q^{85} -\zeta_{12} q^{87} + \zeta_{12}^{5} q^{88} -\zeta_{12}^{2} q^{89} -\zeta_{12}^{2} q^{90} -\zeta_{12} q^{93} + \zeta_{12}^{4} q^{94} -\zeta_{12}^{5} q^{95} + \zeta_{12}^{3} q^{96} + \zeta_{12}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{27} + 2 q^{30} - 2 q^{33} - 2 q^{35} - 4 q^{36} + 2 q^{40} + 2 q^{41} - 2 q^{42} - 2 q^{44} - 4 q^{48} + 2 q^{51} - 2 q^{56} - 4 q^{57} + 2 q^{58} - 2 q^{59} + 2 q^{62} - 4 q^{64} + 8 q^{67} + 2 q^{68} - 2 q^{73} - 8 q^{74} - 4 q^{76} + 4 q^{81} + 2 q^{83} - 2 q^{89} - 2 q^{90} - 2 q^{94} + 2 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{12}^{2}\) \(\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
691.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
1.00000i −1.00000 −1.00000 −0.866025 0.500000i 1.00000i 0.866025 + 0.500000i 1.00000i 1.00000 −0.500000 + 0.866025i
691.2 1.00000i −1.00000 −1.00000 0.866025 + 0.500000i 1.00000i −0.866025 0.500000i 1.00000i 1.00000 −0.500000 + 0.866025i
1075.1 1.00000i −1.00000 −1.00000 0.866025 0.500000i 1.00000i −0.866025 + 0.500000i 1.00000i 1.00000 −0.500000 0.866025i
1075.2 1.00000i −1.00000 −1.00000 −0.866025 + 0.500000i 1.00000i 0.866025 0.500000i 1.00000i 1.00000 −0.500000 0.866025i
\(n\): 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 inner
171.h even 3 1 inner
1368.cj 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.cj.c yes 4
8.d odd 2 1 inner 1368.1.cj.c yes 4
9.c even 3 1 1368.1.ba.c 4
19.c even 3 1 1368.1.ba.c 4
72.p odd 6 1 1368.1.ba.c 4
152.k odd 6 1 1368.1.ba.c 4
171.h even 3 1 inner 1368.1.cj.c yes 4
1368.cj odd 6 1 inner 1368.1.cj.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.1.ba.c 4 9.c even 3 1
1368.1.ba.c 4 19.c even 3 1
1368.1.ba.c 4 72.p odd 6 1
1368.1.ba.c 4 152.k odd 6 1
1368.1.cj.c yes 4 1.a even 1 1 trivial
1368.1.cj.c yes 4 8.d odd 2 1 inner
1368.1.cj.c yes 4 171.h even 3 1 inner
1368.1.cj.c yes 4 1368.cj odd 6 1 inner

Hecke kernels

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

\( T_{5}^{4} - T_{5}^{2} + 1 \)
\( T_{11}^{2} - T_{11} + 1 \)

Hecke characteristic polynomials

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