Properties

Label 1008.1.y.a
Level $1008$
Weight $1$
Character orbit 1008.y
Analytic conductor $0.503$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,1,Mod(251,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.251");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1008.y (of order \(4\), degree \(2\), 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: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.2709504.2

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - q^{7} - \zeta_{8}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - q^{7} - \zeta_{8}^{3} q^{8} - 2 \zeta_{8} q^{11} + \zeta_{8} q^{14} - q^{16} + 2 \zeta_{8}^{2} q^{22} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{23} - \zeta_{8}^{2} q^{25} - \zeta_{8}^{2} q^{28} + \zeta_{8} q^{32} + ( - \zeta_{8}^{2} - 1) q^{37} + ( - \zeta_{8}^{2} + 1) q^{43} - 2 \zeta_{8}^{3} q^{44} + (\zeta_{8}^{2} - 1) q^{46} + q^{49} + \zeta_{8}^{3} q^{50} + 2 \zeta_{8}^{3} q^{53} + \zeta_{8}^{3} q^{56} - \zeta_{8}^{2} q^{64} + ( - \zeta_{8}^{2} - 1) q^{67} + (\zeta_{8}^{3} + \zeta_{8}) q^{71} + (\zeta_{8}^{3} + \zeta_{8}) q^{74} + 2 \zeta_{8} q^{77} + 2 \zeta_{8}^{2} q^{79} + (\zeta_{8}^{3} - \zeta_{8}) q^{86} - 2 q^{88} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{92} - \zeta_{8} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} - 4 q^{16} - 4 q^{37} + 4 q^{43} - 4 q^{46} + 4 q^{49} - 4 q^{67} - 8 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{8}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i 0 1.00000i 0 0 −1.00000 0.707107 0.707107i 0 0
251.2 0.707107 + 0.707107i 0 1.00000i 0 0 −1.00000 −0.707107 + 0.707107i 0 0
755.1 −0.707107 + 0.707107i 0 1.00000i 0 0 −1.00000 0.707107 + 0.707107i 0 0
755.2 0.707107 0.707107i 0 1.00000i 0 0 −1.00000 −0.707107 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
16.f odd 4 1 inner
21.c even 2 1 inner
48.k even 4 1 inner
112.j even 4 1 inner
336.v odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.1.y.a 4
3.b odd 2 1 inner 1008.1.y.a 4
7.b odd 2 1 CM 1008.1.y.a 4
16.f odd 4 1 inner 1008.1.y.a 4
21.c even 2 1 inner 1008.1.y.a 4
48.k even 4 1 inner 1008.1.y.a 4
112.j even 4 1 inner 1008.1.y.a 4
336.v odd 4 1 inner 1008.1.y.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.1.y.a 4 1.a even 1 1 trivial
1008.1.y.a 4 3.b odd 2 1 inner
1008.1.y.a 4 7.b odd 2 1 CM
1008.1.y.a 4 16.f odd 4 1 inner
1008.1.y.a 4 21.c even 2 1 inner
1008.1.y.a 4 48.k even 4 1 inner
1008.1.y.a 4 112.j even 4 1 inner
1008.1.y.a 4 336.v odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 16 \) acting on \(S_{1}^{\mathrm{new}}(1008, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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