Properties

Label 980.1.f.d
Level $980$
Weight $1$
Character orbit 980.f
Self dual yes
Analytic conductor $0.489$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -20
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,1,Mod(99,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.99");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 980.f (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.489083712380\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.980.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.6722800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} - q^{10} + q^{12} - q^{15} + q^{16} - q^{20} - q^{23} + q^{24} + q^{25} - q^{27} - q^{29} - q^{30} + q^{32} - q^{40} + q^{41} - q^{43} - q^{46} - 2 q^{47} + q^{48} + q^{50} - q^{54} - q^{58} - q^{60} + q^{61} + q^{64} - q^{67} - q^{69} + q^{75} - q^{80} - q^{81} + q^{82} + q^{83} - q^{86} - q^{87} + q^{89} - q^{92} - 2 q^{94} + q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(0\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0
1.00000 1.00000 1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.f.d 1
4.b odd 2 1 980.1.f.a 1
5.b even 2 1 980.1.f.a 1
7.b odd 2 1 980.1.f.c 1
7.c even 3 2 980.1.p.a 2
7.d odd 6 2 140.1.p.a 2
20.d odd 2 1 CM 980.1.f.d 1
21.g even 6 2 1260.1.ci.b 2
28.d even 2 1 980.1.f.b 1
28.f even 6 2 140.1.p.b yes 2
28.g odd 6 2 980.1.p.b 2
35.c odd 2 1 980.1.f.b 1
35.i odd 6 2 140.1.p.b yes 2
35.j even 6 2 980.1.p.b 2
35.k even 12 4 700.1.u.a 4
56.j odd 6 2 2240.1.bt.a 2
56.m even 6 2 2240.1.bt.b 2
84.j odd 6 2 1260.1.ci.a 2
105.p even 6 2 1260.1.ci.a 2
140.c even 2 1 980.1.f.c 1
140.p odd 6 2 980.1.p.a 2
140.s even 6 2 140.1.p.a 2
140.x odd 12 4 700.1.u.a 4
280.ba even 6 2 2240.1.bt.a 2
280.bk odd 6 2 2240.1.bt.b 2
420.be odd 6 2 1260.1.ci.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 7.d odd 6 2
140.1.p.a 2 140.s even 6 2
140.1.p.b yes 2 28.f even 6 2
140.1.p.b yes 2 35.i odd 6 2
700.1.u.a 4 35.k even 12 4
700.1.u.a 4 140.x odd 12 4
980.1.f.a 1 4.b odd 2 1
980.1.f.a 1 5.b even 2 1
980.1.f.b 1 28.d even 2 1
980.1.f.b 1 35.c odd 2 1
980.1.f.c 1 7.b odd 2 1
980.1.f.c 1 140.c even 2 1
980.1.f.d 1 1.a even 1 1 trivial
980.1.f.d 1 20.d odd 2 1 CM
980.1.p.a 2 7.c even 3 2
980.1.p.a 2 140.p odd 6 2
980.1.p.b 2 28.g odd 6 2
980.1.p.b 2 35.j even 6 2
1260.1.ci.a 2 84.j odd 6 2
1260.1.ci.a 2 105.p even 6 2
1260.1.ci.b 2 21.g even 6 2
1260.1.ci.b 2 420.be odd 6 2
2240.1.bt.a 2 56.j odd 6 2
2240.1.bt.a 2 280.ba even 6 2
2240.1.bt.b 2 56.m even 6 2
2240.1.bt.b 2 280.bk odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(980, [\chi])\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{23} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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