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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,1,Mod(11,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([24, 24, 8, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.11");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.610855575463\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 11.1
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 \( T_{11}^{16} + 8 T_{11}^{14} + 28 T_{11}^{12} - 8 T_{11}^{11} + 56 T_{11}^{10} - 24 T_{11}^{9} + 69 T_{11}^{8} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(1224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$19$ \( T^{16} + 194T^{8} + 1 \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} - 2 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{8} - 4 T^{7} + 10 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{8} - 2 T^{6} + 4 T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} - 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 8 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{8} - 4 T^{7} + 10 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 12 T^{4} + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
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