Properties

Label 468.1.bx.a
Level $468$
Weight $1$
Character orbit 468.bx
Analytic conductor $0.234$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [468,1,Mod(71,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.71");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 468.bx (of order \(12\), degree \(4\), 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: \(0.233562425912\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
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: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{5} + \zeta_{24}^{9} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{5} + \zeta_{24}^{9} q^{8} + (\zeta_{24}^{8} - 1) q^{10} - \zeta_{24}^{4} q^{13} + \zeta_{24}^{4} q^{16} + ( - \zeta_{24}^{3} + \zeta_{24}) q^{17} + (\zeta_{24}^{7} + \zeta_{24}^{3}) q^{20} + (\zeta_{24}^{10} + \zeta_{24}^{6} + \zeta_{24}^{2}) q^{25} + \zeta_{24}^{11} q^{26} + ( - \zeta_{24}^{11} - \zeta_{24}^{9}) q^{29} - \zeta_{24}^{11} q^{32} + (\zeta_{24}^{10} - \zeta_{24}^{8}) q^{34} + ( - \zeta_{24}^{8} - \zeta_{24}^{6}) q^{37} + ( - \zeta_{24}^{10} + \zeta_{24}^{2}) q^{40} + \zeta_{24}^{7} q^{41} - \zeta_{24}^{10} q^{49} + ( - \zeta_{24}^{9} + \zeta_{24}^{5} + \zeta_{24}) q^{50} + \zeta_{24}^{6} q^{52} + (\zeta_{24}^{7} + \zeta_{24}^{5}) q^{53} + ( - \zeta_{24}^{6} - \zeta_{24}^{4}) q^{58} + \zeta_{24}^{8} q^{61} - \zeta_{24}^{6} q^{64} + (\zeta_{24}^{9} + \zeta_{24}^{5}) q^{65} + (\zeta_{24}^{5} - \zeta_{24}^{3}) q^{68} + ( - \zeta_{24}^{10} + \zeta_{24}^{8}) q^{73} + ( - \zeta_{24}^{3} - \zeta_{24}) q^{74} + ( - \zeta_{24}^{9} - \zeta_{24}^{5}) q^{80} + \zeta_{24}^{2} q^{82} + (\zeta_{24}^{8} - \zeta_{24}^{6} + \zeta_{24}^{4} - \zeta_{24}^{2}) q^{85} + ( - \zeta_{24}^{8} - \zeta_{24}^{2}) q^{97} - \zeta_{24}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{10} - 4 q^{13} + 4 q^{16} + 4 q^{34} + 4 q^{37} - 4 q^{58} - 4 q^{61} - 4 q^{73} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(\zeta_{24}^{2}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i 0 0.866025 0.500000i 1.22474 + 1.22474i 0 0 −0.707107 + 0.707107i 0 −1.50000 0.866025i
71.2 0.965926 0.258819i 0 0.866025 0.500000i −1.22474 1.22474i 0 0 0.707107 0.707107i 0 −1.50000 0.866025i
215.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 1.22474 1.22474i 0 0 0.707107 + 0.707107i 0 −1.50000 0.866025i
215.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −1.22474 + 1.22474i 0 0 −0.707107 0.707107i 0 −1.50000 0.866025i
323.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 1.22474 1.22474i 0 0 −0.707107 0.707107i 0 −1.50000 + 0.866025i
323.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i −1.22474 + 1.22474i 0 0 0.707107 + 0.707107i 0 −1.50000 + 0.866025i
431.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 1.22474 + 1.22474i 0 0 0.707107 0.707107i 0 −1.50000 + 0.866025i
431.2 0.258819 0.965926i 0 −0.866025 0.500000i −1.22474 1.22474i 0 0 −0.707107 + 0.707107i 0 −1.50000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b even 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner
52.l even 12 1 inner
156.v odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.1.bx.a 8
3.b odd 2 1 inner 468.1.bx.a 8
4.b odd 2 1 CM 468.1.bx.a 8
12.b even 2 1 inner 468.1.bx.a 8
13.f odd 12 1 inner 468.1.bx.a 8
39.k even 12 1 inner 468.1.bx.a 8
52.l even 12 1 inner 468.1.bx.a 8
156.v odd 12 1 inner 468.1.bx.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.1.bx.a 8 1.a even 1 1 trivial
468.1.bx.a 8 3.b odd 2 1 inner
468.1.bx.a 8 4.b odd 2 1 CM
468.1.bx.a 8 12.b even 2 1 inner
468.1.bx.a 8 13.f odd 12 1 inner
468.1.bx.a 8 39.k even 12 1 inner
468.1.bx.a 8 52.l even 12 1 inner
468.1.bx.a 8 156.v odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(468, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{6} + 15 T^{4} - 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + 5 T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
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