# Properties

 Label 1224.1.cp.a Level $1224$ Weight $1$ Character orbit 1224.cp Analytic conductor $0.611$ Analytic rank $0$ Dimension $8$ Projective image $D_{24}$ 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.cp (of order $$24$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.610855575463$$ Analytic rank: $$0$$ Dimension: $$8$$ 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 $$D_{24}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{24} q^{2} -\zeta_{24} q^{3} + \zeta_{24}^{2} q^{4} -\zeta_{24}^{2} q^{6} + \zeta_{24}^{3} q^{8} + \zeta_{24}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{24} q^{2} -\zeta_{24} q^{3} + \zeta_{24}^{2} q^{4} -\zeta_{24}^{2} q^{6} + \zeta_{24}^{3} q^{8} + \zeta_{24}^{2} q^{9} + ( -\zeta_{24}^{8} + \zeta_{24}^{9} ) q^{11} -\zeta_{24}^{3} q^{12} + \zeta_{24}^{4} q^{16} + \zeta_{24}^{5} q^{17} + \zeta_{24}^{3} q^{18} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{19} + ( -\zeta_{24}^{9} + \zeta_{24}^{10} ) q^{22} -\zeta_{24}^{4} q^{24} + \zeta_{24}^{7} q^{25} -\zeta_{24}^{3} q^{27} + \zeta_{24}^{5} q^{32} + ( \zeta_{24}^{9} - \zeta_{24}^{10} ) q^{33} + \zeta_{24}^{6} q^{34} + \zeta_{24}^{4} q^{36} + ( 1 - \zeta_{24}^{8} ) q^{38} + ( \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{41} + ( 1 - \zeta_{24}^{2} ) q^{43} + ( -\zeta_{24}^{10} + \zeta_{24}^{11} ) q^{44} -\zeta_{24}^{5} q^{48} -\zeta_{24}^{5} q^{49} + \zeta_{24}^{8} q^{50} -\zeta_{24}^{6} q^{51} -\zeta_{24}^{4} q^{54} + ( -1 + \zeta_{24}^{8} ) q^{57} + ( -1 + \zeta_{24}^{10} ) q^{59} + \zeta_{24}^{6} q^{64} + ( \zeta_{24}^{10} - \zeta_{24}^{11} ) q^{66} + ( -\zeta_{24} + \zeta_{24}^{3} ) q^{67} + \zeta_{24}^{7} q^{68} + \zeta_{24}^{5} q^{72} + ( -\zeta_{24}^{5} - \zeta_{24}^{10} ) q^{73} -\zeta_{24}^{8} q^{75} + ( \zeta_{24} - \zeta_{24}^{9} ) q^{76} + \zeta_{24}^{4} q^{81} + ( \zeta_{24}^{7} - \zeta_{24}^{8} ) q^{82} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{83} + ( \zeta_{24} - \zeta_{24}^{3} ) q^{86} + ( -1 - \zeta_{24}^{11} ) q^{88} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{89} -\zeta_{24}^{6} q^{96} + ( -1 + \zeta_{24}^{11} ) q^{97} -\zeta_{24}^{6} q^{98} + ( -\zeta_{24}^{10} + \zeta_{24}^{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 4q^{11} + 4q^{16} - 4q^{24} + 4q^{36} + 12q^{38} + 8q^{43} - 4q^{50} - 4q^{54} - 12q^{57} - 8q^{59} + 4q^{75} + 4q^{81} + 4q^{82} - 4q^{83} - 8q^{88} - 8q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$137$$ $$613$$ $$649$$ $$919$$ $$\chi(n)$$ $$\zeta_{24}^{8}$$ $$-1$$ $$\zeta_{24}^{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −0.258819 − 0.965926i 0.258819 + 0.965926i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 − 0.258819i
−0.258819 0.965926i 0.258819 + 0.965926i −0.866025 + 0.500000i 0 0.866025 0.500000i 0 0.707107 + 0.707107i −0.866025 + 0.500000i 0
331.1 0.258819 + 0.965926i −0.258819 0.965926i −0.866025 + 0.500000i 0 0.866025 0.500000i 0 −0.707107 0.707107i −0.866025 + 0.500000i 0
355.1 0.258819 0.965926i −0.258819 + 0.965926i −0.866025 0.500000i 0 0.866025 + 0.500000i 0 −0.707107 + 0.707107i −0.866025 0.500000i 0
427.1 −0.258819 + 0.965926i 0.258819 0.965926i −0.866025 0.500000i 0 0.866025 + 0.500000i 0 0.707107 0.707107i −0.866025 0.500000i 0
763.1 −0.965926 + 0.258819i 0.965926 0.258819i 0.866025 0.500000i 0 −0.866025 + 0.500000i 0 −0.707107 + 0.707107i 0.866025 0.500000i 0
835.1 0.965926 0.258819i −0.965926 + 0.258819i 0.866025 0.500000i 0 −0.866025 + 0.500000i 0 0.707107 0.707107i 0.866025 0.500000i 0
859.1 0.965926 + 0.258819i −0.965926 0.258819i 0.866025 + 0.500000i 0 −0.866025 0.500000i 0 0.707107 + 0.707107i 0.866025 + 0.500000i 0
1147.1 −0.965926 0.258819i 0.965926 + 0.258819i 0.866025 + 0.500000i 0 −0.866025 0.500000i 0 −0.707107 0.707107i 0.866025 + 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1147.1 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.r even 24 1 inner
1224.cp odd 24 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.cp.a 8
3.b odd 2 1 3672.1.df.a 8
8.d odd 2 1 CM 1224.1.cp.a 8
9.c even 3 1 1224.1.cp.b yes 8
9.d odd 6 1 3672.1.df.b 8
17.d even 8 1 1224.1.cp.b yes 8
24.f even 2 1 3672.1.df.a 8
51.g odd 8 1 3672.1.df.b 8
72.l even 6 1 3672.1.df.b 8
72.p odd 6 1 1224.1.cp.b yes 8
136.p odd 8 1 1224.1.cp.b yes 8
153.q odd 24 1 3672.1.df.a 8
153.r even 24 1 inner 1224.1.cp.a 8
408.bd even 8 1 3672.1.df.b 8
1224.cn even 24 1 3672.1.df.a 8
1224.cp odd 24 1 inner 1224.1.cp.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.cp.a 8 1.a even 1 1 trivial
1224.1.cp.a 8 8.d odd 2 1 CM
1224.1.cp.a 8 153.r even 24 1 inner
1224.1.cp.a 8 1224.cp odd 24 1 inner
1224.1.cp.b yes 8 9.c even 3 1
1224.1.cp.b yes 8 17.d even 8 1
1224.1.cp.b yes 8 72.p odd 6 1
1224.1.cp.b yes 8 136.p odd 8 1
3672.1.df.a 8 3.b odd 2 1
3672.1.df.a 8 24.f even 2 1
3672.1.df.a 8 153.q odd 24 1
3672.1.df.a 8 1224.cn even 24 1
3672.1.df.b 8 9.d odd 6 1
3672.1.df.b 8 51.g odd 8 1
3672.1.df.b 8 72.l even 6 1
3672.1.df.b 8 408.bd even 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{8} - \cdots$$ acting on $$S_{1}^{\mathrm{new}}(1224, [\chi])$$.

## Hecke characteristic polynomials

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