Properties

Label 980.1.j.a
Level $980$
Weight $1$
Character orbit 980.j
Analytic conductor $0.489$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
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] = [980,1,Mod(587,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.587");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 980.j (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: \(0.489083712380\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.823543000000.2

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{16}^{2} q^{2} + \zeta_{16}^{4} q^{4} + \zeta_{16}^{5} q^{5} - \zeta_{16}^{6} q^{8} + \zeta_{16}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{16}^{2} q^{2} + \zeta_{16}^{4} q^{4} + \zeta_{16}^{5} q^{5} - \zeta_{16}^{6} q^{8} + \zeta_{16}^{4} q^{9} - \zeta_{16}^{7} q^{10} + (\zeta_{16}^{7} - \zeta_{16}^{5}) q^{13} - q^{16} + (\zeta_{16}^{7} + \zeta_{16}^{5}) q^{17} - \zeta_{16}^{6} q^{18} - \zeta_{16} q^{20} - \zeta_{16}^{2} q^{25} + (\zeta_{16}^{7} + \zeta_{16}) q^{26} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{29} + \zeta_{16}^{2} q^{32} + ( - \zeta_{16}^{7} + \zeta_{16}) q^{34} - q^{36} + \zeta_{16}^{3} q^{40} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{41} - \zeta_{16} q^{45} + \zeta_{16}^{4} q^{50} + ( - \zeta_{16}^{3} + \zeta_{16}) q^{52} + ( - \zeta_{16}^{4} + 1) q^{53} + ( - \zeta_{16}^{4} + 1) q^{58} + ( - \zeta_{16}^{7} - \zeta_{16}) q^{61} - \zeta_{16}^{4} q^{64} + ( - \zeta_{16}^{4} + \zeta_{16}^{2}) q^{65} + ( - \zeta_{16}^{3} - \zeta_{16}) q^{68} + \zeta_{16}^{2} q^{72} + (\zeta_{16}^{3} + \zeta_{16}) q^{73} - \zeta_{16}^{5} q^{80} - q^{81} + ( - \zeta_{16}^{7} - \zeta_{16}^{5}) q^{82} + ( - \zeta_{16}^{4} - \zeta_{16}^{2}) q^{85} + ( - \zeta_{16}^{5} + \zeta_{16}^{3}) q^{89} + \zeta_{16}^{3} q^{90} + ( - \zeta_{16}^{3} + \zeta_{16}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{16} - 8 q^{36} + 8 q^{53} + 8 q^{58} - 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-1\) \(\zeta_{16}^{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
587.1
0.923880 + 0.382683i
−0.923880 0.382683i
−0.382683 + 0.923880i
0.382683 0.923880i
0.923880 0.382683i
−0.923880 + 0.382683i
−0.382683 0.923880i
0.382683 + 0.923880i
−0.707107 0.707107i 0 1.00000i −0.382683 + 0.923880i 0 0 0.707107 0.707107i 1.00000i 0.923880 0.382683i
587.2 −0.707107 0.707107i 0 1.00000i 0.382683 0.923880i 0 0 0.707107 0.707107i 1.00000i −0.923880 + 0.382683i
587.3 0.707107 + 0.707107i 0 1.00000i −0.923880 0.382683i 0 0 −0.707107 + 0.707107i 1.00000i −0.382683 0.923880i
587.4 0.707107 + 0.707107i 0 1.00000i 0.923880 + 0.382683i 0 0 −0.707107 + 0.707107i 1.00000i 0.382683 + 0.923880i
783.1 −0.707107 + 0.707107i 0 1.00000i −0.382683 0.923880i 0 0 0.707107 + 0.707107i 1.00000i 0.923880 + 0.382683i
783.2 −0.707107 + 0.707107i 0 1.00000i 0.382683 + 0.923880i 0 0 0.707107 + 0.707107i 1.00000i −0.923880 0.382683i
783.3 0.707107 0.707107i 0 1.00000i −0.923880 + 0.382683i 0 0 −0.707107 0.707107i 1.00000i −0.382683 + 0.923880i
783.4 0.707107 0.707107i 0 1.00000i 0.923880 0.382683i 0 0 −0.707107 0.707107i 1.00000i 0.382683 0.923880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 587.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.c odd 4 1 inner
7.b odd 2 1 inner
20.e even 4 1 inner
28.d even 2 1 inner
35.f even 4 1 inner
140.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.j.a 8
4.b odd 2 1 CM 980.1.j.a 8
5.c odd 4 1 inner 980.1.j.a 8
7.b odd 2 1 inner 980.1.j.a 8
7.c even 3 2 980.1.y.a 16
7.d odd 6 2 980.1.y.a 16
20.e even 4 1 inner 980.1.j.a 8
28.d even 2 1 inner 980.1.j.a 8
28.f even 6 2 980.1.y.a 16
28.g odd 6 2 980.1.y.a 16
35.f even 4 1 inner 980.1.j.a 8
35.k even 12 2 980.1.y.a 16
35.l odd 12 2 980.1.y.a 16
140.j odd 4 1 inner 980.1.j.a 8
140.w even 12 2 980.1.y.a 16
140.x odd 12 2 980.1.y.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.j.a 8 1.a even 1 1 trivial
980.1.j.a 8 4.b odd 2 1 CM
980.1.j.a 8 5.c odd 4 1 inner
980.1.j.a 8 7.b odd 2 1 inner
980.1.j.a 8 20.e even 4 1 inner
980.1.j.a 8 28.d even 2 1 inner
980.1.j.a 8 35.f even 4 1 inner
980.1.j.a 8 140.j odd 4 1 inner
980.1.y.a 16 7.c even 3 2
980.1.y.a 16 7.d odd 6 2
980.1.y.a 16 28.f even 6 2
980.1.y.a 16 28.g odd 6 2
980.1.y.a 16 35.k even 12 2
980.1.y.a 16 35.l odd 12 2
980.1.y.a 16 140.w even 12 2
980.1.y.a 16 140.x odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(980, [\chi])\).

Hecke characteristic polynomials

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