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 $4$

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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:

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

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