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 $4$

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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:

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

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