# Properties

 Label 1224.1.cx.a Level $1224$ Weight $1$ Character orbit 1224.cx Analytic conductor $0.611$ Analytic rank $0$ Dimension $16$ Projective image $D_{48}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1224 = 2^{3} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1224.cx (of order $$48$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.610855575463$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{48}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{48} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{48}^{5} q^{2} + \zeta_{48}^{14} q^{3} + \zeta_{48}^{10} q^{4} + \zeta_{48}^{19} q^{6} + \zeta_{48}^{15} q^{8} -\zeta_{48}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{48}^{5} q^{2} + \zeta_{48}^{14} q^{3} + \zeta_{48}^{10} q^{4} + \zeta_{48}^{19} q^{6} + \zeta_{48}^{15} q^{8} -\zeta_{48}^{4} q^{9} + ( -\zeta_{48}^{12} + \zeta_{48}^{13} ) q^{11} - q^{12} + \zeta_{48}^{20} q^{16} + \zeta_{48}^{17} q^{17} -\zeta_{48}^{9} q^{18} + ( \zeta_{48}^{7} + \zeta_{48}^{11} ) q^{19} + ( -\zeta_{48}^{17} + \zeta_{48}^{18} ) q^{22} -\zeta_{48}^{5} q^{24} -\zeta_{48}^{11} q^{25} -\zeta_{48}^{18} q^{27} -\zeta_{48} q^{32} + ( \zeta_{48}^{2} - \zeta_{48}^{3} ) q^{33} + \zeta_{48}^{22} q^{34} -\zeta_{48}^{14} q^{36} + ( \zeta_{48}^{12} + \zeta_{48}^{16} ) q^{38} + ( -\zeta_{48}^{14} - \zeta_{48}^{15} ) q^{41} + ( \zeta_{48}^{6} - \zeta_{48}^{16} ) q^{43} + ( -\zeta_{48}^{22} + \zeta_{48}^{23} ) q^{44} -\zeta_{48}^{10} q^{48} -\zeta_{48}^{13} q^{49} -\zeta_{48}^{16} q^{50} -\zeta_{48}^{7} q^{51} -\zeta_{48}^{23} q^{54} + ( -\zeta_{48} + \zeta_{48}^{21} ) q^{57} + ( \zeta_{48}^{18} - \zeta_{48}^{20} ) q^{59} -\zeta_{48}^{6} q^{64} + ( \zeta_{48}^{7} - \zeta_{48}^{8} ) q^{66} + ( \zeta_{48}^{9} - \zeta_{48}^{23} ) q^{67} -\zeta_{48}^{3} q^{68} -\zeta_{48}^{19} q^{72} + ( -\zeta_{48}^{5} - \zeta_{48}^{22} ) q^{73} + \zeta_{48} q^{75} + ( \zeta_{48}^{17} + \zeta_{48}^{21} ) q^{76} + \zeta_{48}^{8} q^{81} + ( -\zeta_{48}^{19} - \zeta_{48}^{20} ) q^{82} + ( -\zeta_{48}^{2} + \zeta_{48}^{8} ) q^{83} + ( \zeta_{48}^{11} - \zeta_{48}^{21} ) q^{86} + ( \zeta_{48}^{3} - \zeta_{48}^{4} ) q^{88} + ( \zeta_{48}^{3} + \zeta_{48}^{9} ) q^{89} -\zeta_{48}^{15} q^{96} + ( \zeta_{48}^{4} + \zeta_{48}^{15} ) q^{97} -\zeta_{48}^{18} q^{98} + ( \zeta_{48}^{16} - \zeta_{48}^{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{12} - 8q^{38} + 8q^{43} + 8q^{50} - 8q^{66} + 8q^{81} + 8q^{83} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$$.

 $$n$$ $$137$$ $$613$$ $$649$$ $$919$$ $$\chi(n)$$ $$-\zeta_{48}^{16}$$ $$-1$$ $$\zeta_{48}^{15}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.130526 + 0.991445i −0.991445 − 0.130526i −0.130526 − 0.991445i −0.130526 + 0.991445i −0.991445 + 0.130526i −0.608761 − 0.793353i 0.793353 + 0.608761i 0.793353 − 0.608761i −0.608761 + 0.793353i −0.793353 + 0.608761i 0.608761 + 0.793353i 0.130526 − 0.991445i 0.991445 + 0.130526i 0.608761 − 0.793353i −0.793353 − 0.608761i 0.991445 − 0.130526i
0.608761 + 0.793353i 0.258819 + 0.965926i −0.258819 + 0.965926i 0 −0.608761 + 0.793353i 0 −0.923880 + 0.382683i −0.866025 + 0.500000i 0
131.1 −0.793353 0.608761i −0.258819 + 0.965926i 0.258819 + 0.965926i 0 0.793353 0.608761i 0 0.382683 0.923880i −0.866025 0.500000i 0
227.1 −0.608761 0.793353i 0.258819 + 0.965926i −0.258819 + 0.965926i 0 0.608761 0.793353i 0 0.923880 0.382683i −0.866025 + 0.500000i 0
275.1 −0.608761 + 0.793353i 0.258819 0.965926i −0.258819 0.965926i 0 0.608761 + 0.793353i 0 0.923880 + 0.382683i −0.866025 0.500000i 0
299.1 −0.793353 + 0.608761i −0.258819 0.965926i 0.258819 0.965926i 0 0.793353 + 0.608761i 0 0.382683 + 0.923880i −0.866025 + 0.500000i 0
347.1 0.130526 + 0.991445i 0.965926 + 0.258819i −0.965926 + 0.258819i 0 −0.130526 + 0.991445i 0 −0.382683 0.923880i 0.866025 + 0.500000i 0
371.1 −0.991445 0.130526i −0.965926 + 0.258819i 0.965926 + 0.258819i 0 0.991445 0.130526i 0 −0.923880 0.382683i 0.866025 0.500000i 0
419.1 −0.991445 + 0.130526i −0.965926 0.258819i 0.965926 0.258819i 0 0.991445 + 0.130526i 0 −0.923880 + 0.382683i 0.866025 + 0.500000i 0
515.1 0.130526 0.991445i 0.965926 0.258819i −0.965926 0.258819i 0 −0.130526 0.991445i 0 −0.382683 + 0.923880i 0.866025 0.500000i 0
635.1 0.991445 0.130526i −0.965926 0.258819i 0.965926 0.258819i 0 −0.991445 0.130526i 0 0.923880 0.382683i 0.866025 + 0.500000i 0
707.1 −0.130526 0.991445i 0.965926 + 0.258819i −0.965926 + 0.258819i 0 0.130526 0.991445i 0 0.382683 + 0.923880i 0.866025 + 0.500000i 0
779.1 0.608761 0.793353i 0.258819 0.965926i −0.258819 0.965926i 0 −0.608761 0.793353i 0 −0.923880 0.382683i −0.866025 0.500000i 0
923.1 0.793353 + 0.608761i −0.258819 + 0.965926i 0.258819 + 0.965926i 0 −0.793353 + 0.608761i 0 −0.382683 + 0.923880i −0.866025 0.500000i 0
947.1 −0.130526 + 0.991445i 0.965926 0.258819i −0.965926 0.258819i 0 0.130526 + 0.991445i 0 0.382683 0.923880i 0.866025 0.500000i 0
1091.1 0.991445 + 0.130526i −0.965926 + 0.258819i 0.965926 + 0.258819i 0 −0.991445 + 0.130526i 0 0.923880 + 0.382683i 0.866025 0.500000i 0
1163.1 0.793353 0.608761i −0.258819 0.965926i 0.258819 0.965926i 0 −0.793353 0.608761i 0 −0.382683 0.923880i −0.866025 + 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1163.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
153.s even 48 1 inner
1224.cx odd 48 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.cx.a 16
3.b odd 2 1 3672.1.dv.a 16
8.d odd 2 1 CM 1224.1.cx.a 16
9.c even 3 1 3672.1.dv.b 16
9.d odd 6 1 1224.1.cx.b yes 16
17.e odd 16 1 1224.1.cx.b yes 16
24.f even 2 1 3672.1.dv.a 16
51.i even 16 1 3672.1.dv.b 16
72.l even 6 1 1224.1.cx.b yes 16
72.p odd 6 1 3672.1.dv.b 16
136.s even 16 1 1224.1.cx.b yes 16
153.s even 48 1 inner 1224.1.cx.a 16
153.t odd 48 1 3672.1.dv.a 16
408.bg odd 16 1 3672.1.dv.b 16
1224.cx odd 48 1 inner 1224.1.cx.a 16
1224.da even 48 1 3672.1.dv.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.cx.a 16 1.a even 1 1 trivial
1224.1.cx.a 16 8.d odd 2 1 CM
1224.1.cx.a 16 153.s even 48 1 inner
1224.1.cx.a 16 1224.cx odd 48 1 inner
1224.1.cx.b yes 16 9.d odd 6 1
1224.1.cx.b yes 16 17.e odd 16 1
1224.1.cx.b yes 16 72.l even 6 1
1224.1.cx.b yes 16 136.s even 16 1
3672.1.dv.a 16 3.b odd 2 1
3672.1.dv.a 16 24.f even 2 1
3672.1.dv.a 16 153.t odd 48 1
3672.1.dv.a 16 1224.da even 48 1
3672.1.dv.b 16 9.c even 3 1
3672.1.dv.b 16 51.i even 16 1
3672.1.dv.b 16 72.p odd 6 1
3672.1.dv.b 16 408.bg odd 16 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{16} + \cdots$$ acting on $$S_{1}^{\mathrm{new}}(1224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{8} + T^{16}$$
$3$ $$( 1 - T^{4} + T^{8} )^{2}$$
$5$ $$1 - T^{16} + T^{32}$$
$7$ $$1 - T^{16} + T^{32}$$
$11$ $$( 1 + T^{2} )^{8}( 1 - T^{8} + T^{16} )$$
$13$ $$( 1 - T^{4} + T^{8} )^{4}$$
$17$ $$1 - T^{8} + T^{16}$$
$19$ $$( 1 - T^{8} + T^{16} )^{2}$$
$23$ $$1 - T^{16} + T^{32}$$
$29$ $$1 - T^{16} + T^{32}$$
$31$ $$1 - T^{16} + T^{32}$$
$37$ $$( 1 + T^{16} )^{2}$$
$41$ $$( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2}$$
$43$ $$( 1 - T + T^{2} )^{8}( 1 + T^{4} )^{4}$$
$47$ $$( 1 - T^{4} + T^{8} )^{4}$$
$53$ $$( 1 + T^{8} )^{4}$$
$59$ $$( 1 - T^{2} + T^{4} )^{4}( 1 + T^{4} )^{4}$$
$61$ $$1 - T^{16} + T^{32}$$
$67$ $$( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} )$$
$71$ $$( 1 + T^{16} )^{2}$$
$73$ $$( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} )$$
$79$ $$1 - T^{16} + T^{32}$$
$83$ $$( 1 - T + T^{2} )^{8}( 1 - T^{4} + T^{8} )^{2}$$
$89$ $$( 1 + T^{8} )^{4}$$
$97$ $$( 1 - T^{2} + T^{4} )^{4}( 1 + T^{8} )^{2}$$