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Properties

Label 980.2.k.f
Level $980$
Weight $2$
Character orbit 980.k
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
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] = [980,2,Mod(687,980)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.687");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.k (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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{8}^{2} + 1) q^{2} - 2 \zeta_{8}^{2} q^{4} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + ( - 2 \zeta_{8}^{2} - 2) q^{8} + 3 \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{8}^{2} + 1) q^{2} - 2 \zeta_{8}^{2} q^{4} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + ( - 2 \zeta_{8}^{2} - 2) q^{8} + 3 \zeta_{8}^{2} q^{9} + (3 \zeta_{8}^{3} - \zeta_{8}) q^{10} - 6 \zeta_{8} q^{13} - 4 q^{16} + 8 \zeta_{8}^{3} q^{17} + (3 \zeta_{8}^{2} + 3) q^{18} + (4 \zeta_{8}^{3} + 2 \zeta_{8}) q^{20} + (3 \zeta_{8}^{2} + 4) q^{25} + (6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{26} - 10 \zeta_{8}^{2} q^{29} + (4 \zeta_{8}^{2} - 4) q^{32} + (8 \zeta_{8}^{3} + 8 \zeta_{8}) q^{34} + 6 q^{36} + (7 \zeta_{8}^{2} - 7) q^{37} + (2 \zeta_{8}^{3} + 6 \zeta_{8}) q^{40} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{41} + ( - 6 \zeta_{8}^{3} - 3 \zeta_{8}) q^{45} + ( - \zeta_{8}^{2} + 7) q^{50} + 12 \zeta_{8}^{3} q^{52} + ( - 5 \zeta_{8}^{2} - 5) q^{53} + ( - 10 \zeta_{8}^{2} - 10) q^{58} + (\zeta_{8}^{3} - \zeta_{8}) q^{61} + 8 \zeta_{8}^{2} q^{64} + (12 \zeta_{8}^{2} + 6) q^{65} + 16 \zeta_{8} q^{68} + ( - 6 \zeta_{8}^{2} + 6) q^{72} + 6 \zeta_{8} q^{73} + 14 \zeta_{8}^{2} q^{74} + ( - 4 \zeta_{8}^{3} + 8 \zeta_{8}) q^{80} - 9 q^{81} - 2 \zeta_{8}^{3} q^{82} + ( - 8 \zeta_{8}^{2} + 16) q^{85} + ( - 13 \zeta_{8}^{3} - 13 \zeta_{8}) q^{89} + ( - 3 \zeta_{8}^{3} - 9 \zeta_{8}) q^{90} - 8 \zeta_{8}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{8} - 16 q^{16} + 12 q^{18} + 16 q^{25} - 16 q^{32} + 24 q^{36} - 28 q^{37} + 28 q^{50} - 20 q^{53} - 40 q^{58} + 24 q^{65} + 24 q^{72} - 36 q^{81} + 64 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(1\) \(-\zeta_{8}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
687.1
0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 − 0.707107i
1.00000 + 1.00000i 0 2.00000i −2.12132 + 0.707107i 0 0 −2.00000 + 2.00000i − 3.00000i −2.82843 − 1.41421i
687.2 1.00000 + 1.00000i 0 2.00000i 2.12132 − 0.707107i 0 0 −2.00000 + 2.00000i − 3.00000i 2.82843 + 1.41421i
883.1 1.00000 − 1.00000i 0 − 2.00000i −2.12132 − 0.707107i 0 0 −2.00000 − 2.00000i 3.00000i −2.82843 + 1.41421i
883.2 1.00000 − 1.00000i 0 − 2.00000i 2.12132 + 0.707107i 0 0 −2.00000 − 2.00000i 3.00000i 2.82843 − 1.41421i
\(n\): 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.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 980.2.k.f ✓ 4
4.b odd 2 1 CM 980.2.k.f ✓ 4
5.c odd 4 1 inner 980.2.k.f ✓ 4
7.b odd 2 1 inner 980.2.k.f ✓ 4
7.c even 3 2 980.2.x.g 8
7.d odd 6 2 980.2.x.g 8
20.e even 4 1 inner 980.2.k.f ✓ 4
28.d even 2 1 inner 980.2.k.f ✓ 4
28.f even 6 2 980.2.x.g 8
28.g odd 6 2 980.2.x.g 8
35.f even 4 1 inner 980.2.k.f ✓ 4
35.k even 12 2 980.2.x.g 8
35.l odd 12 2 980.2.x.g 8
140.j odd 4 1 inner 980.2.k.f ✓ 4
140.w even 12 2 980.2.x.g 8
140.x odd 12 2 980.2.x.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.k.f ✓ 4 1.a even 1 1 trivial
980.2.k.f ✓ 4 4.b odd 2 1 CM
980.2.k.f ✓ 4 5.c odd 4 1 inner
980.2.k.f ✓ 4 7.b odd 2 1 inner
980.2.k.f ✓ 4 20.e even 4 1 inner
980.2.k.f ✓ 4 28.d even 2 1 inner
980.2.k.f ✓ 4 35.f even 4 1 inner
980.2.k.f ✓ 4 140.j odd 4 1 inner
980.2.x.g 8 7.c even 3 2
980.2.x.g 8 7.d odd 6 2
980.2.x.g 8 28.f even 6 2
980.2.x.g 8 28.g odd 6 2
980.2.x.g 8 35.k even 12 2
980.2.x.g 8 35.l odd 12 2
980.2.x.g 8 140.w even 12 2
980.2.x.g 8 140.x odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{13}^{4} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 1296 \) Copy content Toggle raw display
$17$ \( T^{4} + 4096 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 14 T + 98)^{2} \) 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
$53$ \( (T^{2} + 10 T + 50)^{2} \) 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
$73$ \( T^{4} + 1296 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 338)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 4096 \) Copy content Toggle raw display
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