Properties

Label 1944.1.j.c
Level $1944$
Weight $1$
Character orbit 1944.j
Analytic conductor $0.970$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1944 = 2^{3} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1944.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.970182384559\)
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 4.2.15552.2

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{5} -\zeta_{24}^{4} q^{7} + \zeta_{24}^{9} q^{8} +O(q^{10})\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{5} -\zeta_{24}^{4} q^{7} + \zeta_{24}^{9} q^{8} + ( -1 - \zeta_{24}^{6} ) q^{10} -\zeta_{24}^{2} q^{13} + \zeta_{24}^{11} q^{14} + \zeta_{24}^{4} q^{16} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{17} -\zeta_{24}^{6} q^{19} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{20} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{23} -\zeta_{24}^{4} q^{25} + \zeta_{24}^{9} q^{26} + \zeta_{24}^{6} q^{28} -\zeta_{24}^{8} q^{31} -\zeta_{24}^{11} q^{32} + ( \zeta_{24}^{4} - \zeta_{24}^{10} ) q^{34} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{35} + \zeta_{24}^{6} q^{37} -\zeta_{24} q^{38} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{40} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{41} -\zeta_{24}^{10} q^{43} + ( -1 + \zeta_{24}^{6} ) q^{46} + \zeta_{24}^{11} q^{50} + \zeta_{24}^{4} q^{52} + \zeta_{24} q^{56} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{59} -\zeta_{24}^{3} q^{62} -\zeta_{24}^{6} q^{64} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{65} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{68} + ( \zeta_{24}^{4} + \zeta_{24}^{10} ) q^{70} + \zeta_{24} q^{74} + \zeta_{24}^{8} q^{76} -\zeta_{24}^{4} q^{79} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{80} + ( 1 - \zeta_{24}^{6} ) q^{82} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{83} + 2 \zeta_{24}^{2} q^{85} -\zeta_{24}^{5} q^{86} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{89} + \zeta_{24}^{6} q^{91} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{92} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{95} -\zeta_{24}^{4} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7} + O(q^{10}) \) \( 8 q - 4 q^{7} - 8 q^{10} + 4 q^{16} - 4 q^{25} + 4 q^{31} + 4 q^{34} - 4 q^{40} - 8 q^{46} + 4 q^{52} + 4 q^{70} - 4 q^{76} - 4 q^{79} + 8 q^{82} - 4 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(973\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{24}^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1133.1
−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 0.707107 1.22474i 0 −0.500000 0.866025i −0.707107 0.707107i 0 −1.00000 + 1.00000i
1133.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i −0.707107 + 1.22474i 0 −0.500000 0.866025i 0.707107 0.707107i 0 −1.00000 1.00000i
1133.3 0.258819 0.965926i 0 −0.866025 0.500000i 0.707107 1.22474i 0 −0.500000 0.866025i −0.707107 + 0.707107i 0 −1.00000 1.00000i
1133.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.707107 + 1.22474i 0 −0.500000 0.866025i 0.707107 + 0.707107i 0 −1.00000 + 1.00000i
1781.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0.707107 + 1.22474i 0 −0.500000 + 0.866025i −0.707107 + 0.707107i 0 −1.00000 1.00000i
1781.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i −0.707107 1.22474i 0 −0.500000 + 0.866025i 0.707107 + 0.707107i 0 −1.00000 + 1.00000i
1781.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0.707107 + 1.22474i 0 −0.500000 + 0.866025i −0.707107 0.707107i 0 −1.00000 + 1.00000i
1781.4 0.965926 0.258819i 0 0.866025 0.500000i −0.707107 1.22474i 0 −0.500000 + 0.866025i 0.707107 0.707107i 0 −1.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1781.4
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
9.c even 3 1 inner
9.d odd 6 1 inner
24.h odd 2 1 inner
72.j odd 6 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1944.1.j.c 8
3.b odd 2 1 inner 1944.1.j.c 8
8.b even 2 1 inner 1944.1.j.c 8
9.c even 3 1 1944.1.h.c 4
9.c even 3 1 inner 1944.1.j.c 8
9.d odd 6 1 1944.1.h.c 4
9.d odd 6 1 inner 1944.1.j.c 8
24.h odd 2 1 inner 1944.1.j.c 8
72.j odd 6 1 1944.1.h.c 4
72.j odd 6 1 inner 1944.1.j.c 8
72.n even 6 1 1944.1.h.c 4
72.n even 6 1 inner 1944.1.j.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1944.1.h.c 4 9.c even 3 1
1944.1.h.c 4 9.d odd 6 1
1944.1.h.c 4 72.j odd 6 1
1944.1.h.c 4 72.n even 6 1
1944.1.j.c 8 1.a even 1 1 trivial
1944.1.j.c 8 3.b odd 2 1 inner
1944.1.j.c 8 8.b even 2 1 inner
1944.1.j.c 8 9.c even 3 1 inner
1944.1.j.c 8 9.d odd 6 1 inner
1944.1.j.c 8 24.h odd 2 1 inner
1944.1.j.c 8 72.j odd 6 1 inner
1944.1.j.c 8 72.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2 T_{5}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1944, [\chi])\).

Hecke characteristic polynomials

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