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Properties

Label 980.1.y.a
Level $980$
Weight $1$
Character orbit 980.y
Analytic conductor $0.489$
Analytic rank $0$
Dimension $16$
Projective image $D_{8}$
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(227,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 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("980.227");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 980.y (of order \(12\), degree \(4\), 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) 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.

\(f(q)\) \(=\) \( q - \zeta_{48}^{2} q^{2} + \zeta_{48}^{4} q^{4} + \zeta_{48}^{5} q^{5} - \zeta_{48}^{6} q^{8} + \zeta_{48}^{20} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{48}^{2} q^{2} + \zeta_{48}^{4} q^{4} + \zeta_{48}^{5} q^{5} - \zeta_{48}^{6} q^{8} + \zeta_{48}^{20} q^{9} - \zeta_{48}^{7} q^{10} + ( - \zeta_{48}^{21} - \zeta_{48}^{15}) q^{13} + \zeta_{48}^{8} q^{16} + ( - \zeta_{48}^{13} + \zeta_{48}^{7}) q^{17} - \zeta_{48}^{22} q^{18} + \zeta_{48}^{9} q^{20} + \zeta_{48}^{10} q^{25} + (\zeta_{48}^{23} + \zeta_{48}^{17}) q^{26} + (\zeta_{48}^{18} + \zeta_{48}^{6}) q^{29} - \zeta_{48}^{10} q^{32} + (\zeta_{48}^{15} - \zeta_{48}^{9}) q^{34} - q^{36} - \zeta_{48}^{11} q^{40} + (\zeta_{48}^{21} + \zeta_{48}^{3}) q^{41} - \zeta_{48} q^{45} - \zeta_{48}^{12} q^{50} + ( - \zeta_{48}^{19} + \zeta_{48}) q^{52} + (\zeta_{48}^{16} - \zeta_{48}^{4}) q^{53} + ( - \zeta_{48}^{20} - \zeta_{48}^{8}) q^{58} + ( - \zeta_{48}^{23} - \zeta_{48}^{17}) q^{61} + \zeta_{48}^{12} q^{64} + ( - \zeta_{48}^{20} + \zeta_{48}^{2}) q^{65} + ( - \zeta_{48}^{17} + \zeta_{48}^{11}) q^{68} + \zeta_{48}^{2} q^{72} + (\zeta_{48}^{19} + \zeta_{48}) q^{73} + \zeta_{48}^{13} q^{80} - \zeta_{48}^{16} q^{81} + ( - \zeta_{48}^{23} - \zeta_{48}^{5}) q^{82} + ( - \zeta_{48}^{18} + \zeta_{48}^{12}) q^{85} + ( - \zeta_{48}^{11} - \zeta_{48}^{5}) q^{89} + \zeta_{48}^{3} q^{90} + ( - \zeta_{48}^{9} - \zeta_{48}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{16} - 16 q^{36} - 8 q^{53} - 8 q^{58} + 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(\zeta_{48}^{8}\) \(-\zeta_{48}^{12}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
227.1
0.793353 − 0.608761i
−0.793353 + 0.608761i
0.608761 + 0.793353i
−0.608761 − 0.793353i
−0.991445 − 0.130526i
0.991445 + 0.130526i
−0.130526 + 0.991445i
0.130526 − 0.991445i
−0.991445 + 0.130526i
0.991445 − 0.130526i
−0.130526 − 0.991445i
0.130526 + 0.991445i
0.793353 + 0.608761i
−0.793353 − 0.608761i
0.608761 − 0.793353i
−0.608761 + 0.793353i
−0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.991445 + 0.130526i 0 0 0.707107 − 0.707107i 0.866025 − 0.500000i 0.130526 − 0.991445i
227.2 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 0.991445 − 0.130526i 0 0 0.707107 − 0.707107i 0.866025 − 0.500000i −0.130526 + 0.991445i
227.3 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −0.130526 − 0.991445i 0 0 −0.707107 + 0.707107i 0.866025 − 0.500000i −0.991445 − 0.130526i
227.4 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0.130526 + 0.991445i 0 0 −0.707107 + 0.707107i 0.866025 − 0.500000i 0.991445 + 0.130526i
423.1 −0.965926 − 0.258819i 0 0.866025 + 0.500000i −0.793353 − 0.608761i 0 0 −0.707107 − 0.707107i −0.866025 + 0.500000i 0.608761 + 0.793353i
423.2 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0.793353 + 0.608761i 0 0 −0.707107 − 0.707107i −0.866025 + 0.500000i −0.608761 − 0.793353i
423.3 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.608761 + 0.793353i 0 0 0.707107 + 0.707107i −0.866025 + 0.500000i −0.793353 + 0.608761i
423.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0.608761 − 0.793353i 0 0 0.707107 + 0.707107i −0.866025 + 0.500000i 0.793353 − 0.608761i
607.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i −0.793353 + 0.608761i 0 0 −0.707107 + 0.707107i −0.866025 − 0.500000i 0.608761 − 0.793353i
607.2 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 0.793353 − 0.608761i 0 0 −0.707107 + 0.707107i −0.866025 − 0.500000i −0.608761 + 0.793353i
607.3 0.965926 − 0.258819i 0 0.866025 − 0.500000i −0.608761 − 0.793353i 0 0 0.707107 − 0.707107i −0.866025 − 0.500000i −0.793353 − 0.608761i
607.4 0.965926 − 0.258819i 0 0.866025 − 0.500000i 0.608761 + 0.793353i 0 0 0.707107 − 0.707107i −0.866025 − 0.500000i 0.793353 + 0.608761i
803.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.991445 − 0.130526i 0 0 0.707107 + 0.707107i 0.866025 + 0.500000i 0.130526 + 0.991445i
803.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 0.991445 + 0.130526i 0 0 0.707107 + 0.707107i 0.866025 + 0.500000i −0.130526 − 0.991445i
803.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.130526 + 0.991445i 0 0 −0.707107 − 0.707107i 0.866025 + 0.500000i −0.991445 + 0.130526i
803.4 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0.130526 − 0.991445i 0 0 −0.707107 − 0.707107i 0.866025 + 0.500000i 0.991445 − 0.130526i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 227.4
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
7.c even 3 1 inner
7.d odd 6 1 inner
20.e even 4 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner
140.j odd 4 1 inner
140.w even 12 1 inner
140.x odd 12 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + 12 T^{4} + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 12 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 12 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 12 T^{4} + 4)^{2} \) Copy content Toggle raw display
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