Properties

Label 1368.1.bw.a
Level $1368$
Weight $1$
Character orbit 1368.bw
Analytic conductor $0.683$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1368.bw (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.682720937282\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.623808.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} - q^{7} + \zeta_{24}^{3} q^{8} +O(q^{10})\) \( q -\zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} - q^{7} + \zeta_{24}^{3} q^{8} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{11} + \zeta_{24}^{10} q^{13} + \zeta_{24}^{5} q^{14} -\zeta_{24}^{8} q^{16} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{17} -\zeta_{24}^{6} q^{19} + ( -\zeta_{24}^{2} - \zeta_{24}^{8} ) q^{22} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{23} + \zeta_{24}^{4} q^{25} + \zeta_{24}^{3} q^{26} -\zeta_{24}^{10} q^{28} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{29} + q^{31} -\zeta_{24} q^{32} + ( -\zeta_{24}^{4} - \zeta_{24}^{10} ) q^{34} + \zeta_{24}^{6} q^{37} + \zeta_{24}^{11} q^{38} -\zeta_{24}^{2} q^{43} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{44} + ( -1 - \zeta_{24}^{6} ) q^{46} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{47} -\zeta_{24}^{9} q^{50} -\zeta_{24}^{8} q^{52} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{53} -\zeta_{24}^{3} q^{56} + ( -1 + \zeta_{24}^{6} ) q^{58} -\zeta_{24}^{10} q^{61} -\zeta_{24}^{5} q^{62} + \zeta_{24}^{6} q^{64} -\zeta_{24}^{10} q^{67} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{68} -\zeta_{24}^{8} q^{73} -\zeta_{24}^{11} q^{74} + \zeta_{24}^{4} q^{76} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{77} + \zeta_{24}^{8} q^{79} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{83} + \zeta_{24}^{7} q^{86} + ( 1 + \zeta_{24}^{6} ) q^{88} -\zeta_{24}^{10} q^{91} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{92} + ( -1 - \zeta_{24}^{6} ) q^{94} + 2 \zeta_{24}^{8} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7} + O(q^{10}) \) \( 8 q - 8 q^{7} + 4 q^{16} + 4 q^{22} + 4 q^{25} + 8 q^{31} - 4 q^{34} - 8 q^{46} + 4 q^{52} - 8 q^{58} + 4 q^{73} + 4 q^{76} - 4 q^{79} + 8 q^{88} - 8 q^{94} - 8 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{24}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 −1.00000 −0.707107 + 0.707107i 0 0
125.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 −1.00000 0.707107 + 0.707107i 0 0
125.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 −1.00000 −0.707107 0.707107i 0 0
125.4 0.965926 0.258819i 0 0.866025 0.500000i 0 0 −1.00000 0.707107 0.707107i 0 0
197.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 −1.00000 −0.707107 0.707107i 0 0
197.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 −1.00000 0.707107 0.707107i 0 0
197.3 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 −1.00000 −0.707107 + 0.707107i 0 0
197.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 −1.00000 0.707107 + 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
19.c even 3 1 inner
24.h odd 2 1 inner
57.h odd 6 1 inner
152.p even 6 1 inner
456.x 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.bw.a 8
3.b odd 2 1 inner 1368.1.bw.a 8
8.b even 2 1 inner 1368.1.bw.a 8
19.c even 3 1 inner 1368.1.bw.a 8
24.h odd 2 1 inner 1368.1.bw.a 8
57.h odd 6 1 inner 1368.1.bw.a 8
152.p even 6 1 inner 1368.1.bw.a 8
456.x odd 6 1 inner 1368.1.bw.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.1.bw.a 8 1.a even 1 1 trivial
1368.1.bw.a 8 3.b odd 2 1 inner
1368.1.bw.a 8 8.b even 2 1 inner
1368.1.bw.a 8 19.c even 3 1 inner
1368.1.bw.a 8 24.h odd 2 1 inner
1368.1.bw.a 8 57.h odd 6 1 inner
1368.1.bw.a 8 152.p even 6 1 inner
1368.1.bw.a 8 456.x odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1368, [\chi])\).

Hecke characteristic polynomials

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