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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,1,Mod(43,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([12, 12, 16, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.43");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
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

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: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
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

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.

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

Display 100 coefficients 10 coefficients

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 43.1
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 \( T_{11}^{8} - 4T_{11}^{7} + 10T_{11}^{6} - 16T_{11}^{5} + 21T_{11}^{4} - 20T_{11}^{3} + 4T_{11}^{2} + 4T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} + 5 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 4 T^{3} + 5 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 2 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
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