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Properties

Label 980.1.p.c
Level $980$
Weight $1$
Character orbit 980.p
Analytic conductor $0.489$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -4
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,1,Mod(79,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.79");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 980.p (of order \(6\), degree \(2\), not 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 \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.137200.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{10} q^{2} - \zeta_{24}^{8} q^{4} - \zeta_{24}^{7} q^{5} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{4} q^{9} - \zeta_{24}^{5} q^{10} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{13} - \zeta_{24}^{4} q^{16} + \cdots + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{9} - 4 q^{16} + 8 q^{36} - 8 q^{50} - 8 q^{64} + 4 q^{65} - 8 q^{74} - 4 q^{81} - 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-\zeta_{24}^{4}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−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
−0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.965926 + 0.258819i 0 0 − 1.00000i 0.500000 − 0.866025i 0.965926 + 0.258819i
79.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.965926 − 0.258819i 0 0 − 1.00000i 0.500000 − 0.866025i −0.965926 − 0.258819i
79.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.258819 − 0.965926i 0 0 1.00000i 0.500000 − 0.866025i 0.258819 − 0.965926i
79.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.258819 + 0.965926i 0 0 1.00000i 0.500000 − 0.866025i −0.258819 + 0.965926i
459.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.965926 − 0.258819i 0 0 1.00000i 0.500000 + 0.866025i 0.965926 − 0.258819i
459.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.965926 + 0.258819i 0 0 1.00000i 0.500000 + 0.866025i −0.965926 + 0.258819i
459.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.258819 + 0.965926i 0 0 − 1.00000i 0.500000 + 0.866025i 0.258819 + 0.965926i
459.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.258819 − 0.965926i 0 0 − 1.00000i 0.500000 + 0.866025i −0.258819 − 0.965926i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.4
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.b even 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner
140.c even 2 1 inner
140.p odd 6 1 inner
140.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.p.c 8
4.b odd 2 1 CM 980.1.p.c 8
5.b even 2 1 inner 980.1.p.c 8
7.b odd 2 1 inner 980.1.p.c 8
7.c even 3 1 980.1.f.e ✓ 4
7.c even 3 1 inner 980.1.p.c 8
7.d odd 6 1 980.1.f.e ✓ 4
7.d odd 6 1 inner 980.1.p.c 8
20.d odd 2 1 inner 980.1.p.c 8
28.d even 2 1 inner 980.1.p.c 8
28.f even 6 1 980.1.f.e ✓ 4
28.f even 6 1 inner 980.1.p.c 8
28.g odd 6 1 980.1.f.e ✓ 4
28.g odd 6 1 inner 980.1.p.c 8
35.c odd 2 1 inner 980.1.p.c 8
35.i odd 6 1 980.1.f.e ✓ 4
35.i odd 6 1 inner 980.1.p.c 8
35.j even 6 1 980.1.f.e ✓ 4
35.j even 6 1 inner 980.1.p.c 8
140.c even 2 1 inner 980.1.p.c 8
140.p odd 6 1 980.1.f.e ✓ 4
140.p odd 6 1 inner 980.1.p.c 8
140.s even 6 1 980.1.f.e ✓ 4
140.s even 6 1 inner 980.1.p.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.f.e ✓ 4 7.c even 3 1
980.1.f.e ✓ 4 7.d odd 6 1
980.1.f.e ✓ 4 28.f even 6 1
980.1.f.e ✓ 4 28.g odd 6 1
980.1.f.e ✓ 4 35.i odd 6 1
980.1.f.e ✓ 4 35.j even 6 1
980.1.f.e ✓ 4 140.p odd 6 1
980.1.f.e ✓ 4 140.s even 6 1
980.1.p.c 8 1.a even 1 1 trivial
980.1.p.c 8 4.b odd 2 1 CM
980.1.p.c 8 5.b even 2 1 inner
980.1.p.c 8 7.b odd 2 1 inner
980.1.p.c 8 7.c even 3 1 inner
980.1.p.c 8 7.d odd 6 1 inner
980.1.p.c 8 20.d odd 2 1 inner
980.1.p.c 8 28.d even 2 1 inner
980.1.p.c 8 28.f even 6 1 inner
980.1.p.c 8 28.g odd 6 1 inner
980.1.p.c 8 35.c odd 2 1 inner
980.1.p.c 8 35.i odd 6 1 inner
980.1.p.c 8 35.j even 6 1 inner
980.1.p.c 8 140.c even 2 1 inner
980.1.p.c 8 140.p odd 6 1 inner
980.1.p.c 8 140.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{1}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
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