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Properties

Label 980.1.f.e
Level $980$
Weight $1$
Character orbit 980.f
Analytic conductor $0.489$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
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(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: no
Analytic conductor: \(0.489083712380\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{2} - q^{4} - \zeta_{8} q^{5} - \zeta_{8}^{2} q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{2} q^{2} - q^{4} - \zeta_{8} q^{5} - \zeta_{8}^{2} q^{8} - q^{9} - \zeta_{8}^{3} q^{10} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{13} + q^{16} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{17} - \zeta_{8}^{2} q^{18} + \zeta_{8} q^{20} + \zeta_{8}^{2} q^{25} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{26} + \zeta_{8}^{2} q^{32} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{34} + q^{36} - \zeta_{8}^{2} q^{37} + \zeta_{8}^{3} q^{40} + (\zeta_{8}^{3} - \zeta_{8}) q^{41} + \zeta_{8} q^{45} - q^{50} + (\zeta_{8}^{3} + \zeta_{8}) q^{52} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{61} - q^{64} + (\zeta_{8}^{2} - 1) q^{65} + (\zeta_{8}^{3} + \zeta_{8}) q^{68} + \zeta_{8}^{2} q^{72} + (\zeta_{8}^{3} + \zeta_{8}) q^{73} + 2 q^{74} - \zeta_{8} q^{80} + q^{81} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{82} + (\zeta_{8}^{2} - 1) q^{85} + (\zeta_{8}^{3} - \zeta_{8}) q^{89} + \zeta_{8}^{3} q^{90} + (\zeta_{8}^{3} + \zeta_{8}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{9} + 4 q^{16} + 4 q^{36} - 4 q^{50} - 4 q^{64} - 4 q^{65} + 8 q^{74} + 4 q^{81} - 4 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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
0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 − 0.707107i
− 1.00000i 0 −1.00000 −0.707107 + 0.707107i 0 0 1.00000i −1.00000 0.707107 + 0.707107i
99.2 − 1.00000i 0 −1.00000 0.707107 − 0.707107i 0 0 1.00000i −1.00000 −0.707107 − 0.707107i
99.3 1.00000i 0 −1.00000 −0.707107 − 0.707107i 0 0 − 1.00000i −1.00000 0.707107 − 0.707107i
99.4 1.00000i 0 −1.00000 0.707107 + 0.707107i 0 0 − 1.00000i −1.00000 −0.707107 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
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
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

Twists

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

Hecke kernels

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

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

Hecke characteristic polynomials

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