# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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