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Properties

Label 980.2.bf.a
Level $980$
Weight $2$
Character orbit 980.bf
Analytic conductor $7.825$
Analytic rank $0$
Dimension $24$
CM discriminant -20
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(139,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 7, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.139");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.bf (of order \(14\), degree \(6\), 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: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{14}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{4} + 28 q^{6} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{4} + 28 q^{6} - 28 q^{9} + 12 q^{14} - 16 q^{16} - 20 q^{21} + 20 q^{25} + 60 q^{29} - 40 q^{30} - 56 q^{36} - 20 q^{46} - 8 q^{49} + 84 q^{54} + 24 q^{56} - 32 q^{64} + 20 q^{70} - 104 q^{81} - 40 q^{84} + 72 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1 −0.881748 − 1.10568i −0.969951 − 2.01412i −0.445042 + 1.94986i 0.970194 + 2.01463i −1.37172 + 2.84840i −2.01353 + 1.71630i 2.54832 − 1.22721i −1.24542 + 1.56171i 1.37206 − 2.84911i
139.2 −0.881748 − 1.10568i 1.50242 + 3.11980i −0.445042 + 1.94986i −0.970194 − 2.01463i 2.12474 − 4.41207i 1.06946 + 2.41997i 2.54832 − 1.22721i −5.60543 + 7.02899i −1.37206 + 2.84911i
139.3 0.881748 + 1.10568i −1.50242 − 3.11980i −0.445042 + 1.94986i −0.970194 − 2.01463i 2.12474 − 4.41207i −1.06946 − 2.41997i −2.54832 + 1.22721i −5.60543 + 7.02899i 1.37206 − 2.84911i
139.4 0.881748 + 1.10568i 0.969951 + 2.01412i −0.445042 + 1.94986i 0.970194 + 2.01463i −1.37172 + 2.84840i 2.01353 − 1.71630i −2.54832 + 1.22721i −1.24542 + 1.56171i −1.37206 + 2.84911i
279.1 −0.314692 + 1.37876i −2.40595 − 1.91868i −1.80194 − 0.867767i 1.74823 + 1.39417i 3.40253 − 2.71342i 2.59727 − 0.504150i 1.76350 − 2.21135i 1.43969 + 6.30771i −2.47237 + 1.97165i
279.2 −0.314692 + 1.37876i 0.677044 + 0.539924i −1.80194 − 0.867767i −1.74823 − 1.39417i −0.957484 + 0.763568i 1.22521 + 2.34496i 1.76350 − 2.21135i −0.500693 − 2.19368i 2.47237 − 1.97165i
279.3 0.314692 − 1.37876i −0.677044 − 0.539924i −1.80194 − 0.867767i −1.74823 − 1.39417i −0.957484 + 0.763568i −1.22521 − 2.34496i −1.76350 + 2.21135i −0.500693 − 2.19368i −2.47237 + 1.97165i
279.4 0.314692 − 1.37876i 2.40595 + 1.91868i −1.80194 − 0.867767i 1.74823 + 1.39417i 3.40253 − 2.71342i −2.59727 + 0.504150i −1.76350 + 2.21135i 1.43969 + 6.30771i 2.47237 − 1.97165i
419.1 −1.27416 + 0.613604i −2.03022 − 0.463384i 1.24698 − 1.56366i −2.18001 − 0.497572i 2.87116 − 0.655324i −0.0864375 + 2.64434i −0.629384 + 2.75751i 1.20416 + 0.579891i 3.08299 − 0.703673i
419.2 −1.27416 + 0.613604i −0.658158 − 0.150220i 1.24698 − 1.56366i 2.18001 + 0.497572i 0.930775 − 0.212443i −2.55881 + 0.672691i −0.629384 + 2.75751i −2.29230 − 1.10391i −3.08299 + 0.703673i
419.3 1.27416 − 0.613604i 0.658158 + 0.150220i 1.24698 − 1.56366i 2.18001 + 0.497572i 0.930775 − 0.212443i 2.55881 − 0.672691i 0.629384 − 2.75751i −2.29230 − 1.10391i 3.08299 − 0.703673i
419.4 1.27416 − 0.613604i 2.03022 + 0.463384i 1.24698 − 1.56366i −2.18001 − 0.497572i 2.87116 − 0.655324i 0.0864375 − 2.64434i 0.629384 − 2.75751i 1.20416 + 0.579891i −3.08299 + 0.703673i
559.1 −1.27416 − 0.613604i −2.03022 + 0.463384i 1.24698 + 1.56366i −2.18001 + 0.497572i 2.87116 + 0.655324i −0.0864375 − 2.64434i −0.629384 − 2.75751i 1.20416 − 0.579891i 3.08299 + 0.703673i
559.2 −1.27416 − 0.613604i −0.658158 + 0.150220i 1.24698 + 1.56366i 2.18001 − 0.497572i 0.930775 + 0.212443i −2.55881 − 0.672691i −0.629384 − 2.75751i −2.29230 + 1.10391i −3.08299 − 0.703673i
559.3 1.27416 + 0.613604i 0.658158 − 0.150220i 1.24698 + 1.56366i 2.18001 − 0.497572i 0.930775 + 0.212443i 2.55881 + 0.672691i 0.629384 + 2.75751i −2.29230 + 1.10391i 3.08299 + 0.703673i
559.4 1.27416 + 0.613604i 2.03022 − 0.463384i 1.24698 + 1.56366i −2.18001 + 0.497572i 2.87116 + 0.655324i 0.0864375 + 2.64434i 0.629384 + 2.75751i 1.20416 − 0.579891i −3.08299 − 0.703673i
699.1 −0.314692 − 1.37876i −2.40595 + 1.91868i −1.80194 + 0.867767i 1.74823 − 1.39417i 3.40253 + 2.71342i 2.59727 + 0.504150i 1.76350 + 2.21135i 1.43969 − 6.30771i −2.47237 − 1.97165i
699.2 −0.314692 − 1.37876i 0.677044 − 0.539924i −1.80194 + 0.867767i −1.74823 + 1.39417i −0.957484 − 0.763568i 1.22521 − 2.34496i 1.76350 + 2.21135i −0.500693 + 2.19368i 2.47237 + 1.97165i
699.3 0.314692 + 1.37876i −0.677044 + 0.539924i −1.80194 + 0.867767i −1.74823 + 1.39417i −0.957484 − 0.763568i −1.22521 + 2.34496i −1.76350 − 2.21135i −0.500693 + 2.19368i −2.47237 − 1.97165i
699.4 0.314692 + 1.37876i 2.40595 − 1.91868i −1.80194 + 0.867767i 1.74823 − 1.39417i 3.40253 + 2.71342i −2.59727 − 0.504150i −1.76350 − 2.21135i 1.43969 − 6.30771i 2.47237 + 1.97165i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
49.f odd 14 1 inner
196.j even 14 1 inner
245.o odd 14 1 inner
980.bf even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.bf.a ✓ 24
4.b odd 2 1 inner 980.2.bf.a ✓ 24
5.b even 2 1 inner 980.2.bf.a ✓ 24
20.d odd 2 1 CM 980.2.bf.a ✓ 24
49.f odd 14 1 inner 980.2.bf.a ✓ 24
196.j even 14 1 inner 980.2.bf.a ✓ 24
245.o odd 14 1 inner 980.2.bf.a ✓ 24
980.bf even 14 1 inner 980.2.bf.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.bf.a ✓ 24 1.a even 1 1 trivial
980.2.bf.a ✓ 24 4.b odd 2 1 inner
980.2.bf.a ✓ 24 5.b even 2 1 inner
980.2.bf.a ✓ 24 20.d odd 2 1 CM
980.2.bf.a ✓ 24 49.f odd 14 1 inner
980.2.bf.a ✓ 24 196.j even 14 1 inner
980.2.bf.a ✓ 24 245.o odd 14 1 inner
980.2.bf.a ✓ 24 980.bf even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 8 T_{3}^{22} + 139 T_{3}^{20} + 172 T_{3}^{18} + 10289 T_{3}^{16} - 44144 T_{3}^{14} + \cdots + 707281 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display