Properties

Label 1008.1.u.a
Level $1008$
Weight $1$
Character orbit 1008.u
Analytic conductor $0.503$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,1,Mod(181,1008)]
 
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([0, 1, 0, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.181");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1008.u (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: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.129024.2

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + i q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + i q^{7} - q^{8} + ( - i + 1) q^{11} - i q^{14} + q^{16} + (i - 1) q^{22} + i q^{23} - i q^{25} + i q^{28} + (i + 1) q^{29} - q^{32} + ( - i + 1) q^{37} + (i - 1) q^{43} + ( - i + 1) q^{44} - 2 i q^{46} - q^{49} + i q^{50} + ( - i + 1) q^{53} - i q^{56} + ( - i - 1) q^{58} + q^{64} + (i + 1) q^{67} + (i - 1) q^{74} + (i + 1) q^{77} + ( - i + 1) q^{86} + (i - 1) q^{88} + 2 i q^{92} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{11} + 2 q^{16} - 2 q^{22} + 2 q^{29} - 2 q^{32} + 2 q^{37} - 2 q^{43} + 2 q^{44} - 2 q^{49} + 2 q^{53} - 2 q^{58} + 2 q^{64} + 2 q^{67} - 2 q^{74} + 2 q^{77} + 2 q^{86} - 2 q^{88} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(-i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.00000i
1.00000i
−1.00000 0 1.00000 0 0 1.00000i −1.00000 0 0
685.1 −1.00000 0 1.00000 0 0 1.00000i −1.00000 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 \(\Q(\sqrt{-7}) \)
16.e even 4 1 inner
112.l 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.u.a 2
3.b odd 2 1 1008.1.u.c yes 2
7.b odd 2 1 CM 1008.1.u.a 2
16.e even 4 1 inner 1008.1.u.a 2
21.c even 2 1 1008.1.u.c yes 2
48.i odd 4 1 1008.1.u.c yes 2
112.l odd 4 1 inner 1008.1.u.a 2
336.y even 4 1 1008.1.u.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.1.u.a 2 1.a even 1 1 trivial
1008.1.u.a 2 7.b odd 2 1 CM
1008.1.u.a 2 16.e even 4 1 inner
1008.1.u.a 2 112.l odd 4 1 inner
1008.1.u.c yes 2 3.b odd 2 1
1008.1.u.c yes 2 21.c even 2 1
1008.1.u.c yes 2 48.i odd 4 1
1008.1.u.c yes 2 336.y even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1008, [\chi])\):

\( T_{11}^{2} - 2T_{11} + 2 \) Copy content Toggle raw display
\( T_{23}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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