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Properties

Label 980.2.bd.a
Level $980$
Weight $2$
Character orbit 980.bd
Analytic conductor $7.825$
Analytic rank $0$
Dimension $168$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(29,980)]
 
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([0, 7, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.bd (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: \(168\)
Relative dimension: \(28\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 168 q + 2 q^{5} + 18 q^{9} + 6 q^{11} + 16 q^{15} + 20 q^{19} - 6 q^{21} + 6 q^{25} - 14 q^{29} + 20 q^{31} + 2 q^{35} + 40 q^{39} - 14 q^{41} + 96 q^{45} + 32 q^{49} + 4 q^{51} - 30 q^{55} - 28 q^{59}+ \cdots + 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 0 −3.08880 + 0.704999i 0 2.21123 + 0.332343i 0 −2.59041 − 0.538307i 0 6.34078 − 3.05356i 0
29.2 0 −2.98304 + 0.680860i 0 0.808094 + 2.08494i 0 2.38737 + 1.14038i 0 5.73208 − 2.76042i 0
29.3 0 −2.74305 + 0.626084i 0 −2.22364 + 0.235416i 0 0.0771841 − 2.64463i 0 4.42946 − 2.13311i 0
29.4 0 −2.55828 + 0.583912i 0 −1.96921 − 1.05934i 0 −1.86517 + 1.87647i 0 3.50096 − 1.68597i 0
29.5 0 −2.19846 + 0.501785i 0 −1.31265 + 1.81023i 0 0.228420 + 2.63587i 0 1.87855 − 0.904663i 0
29.6 0 −1.89281 + 0.432022i 0 −0.697623 − 2.12446i 0 −2.41857 + 1.07263i 0 0.693184 − 0.333820i 0
29.7 0 −1.87433 + 0.427803i 0 1.93896 − 1.11375i 0 1.31675 + 2.29481i 0 0.627182 − 0.302035i 0
29.8 0 −1.72524 + 0.393776i 0 1.68714 − 1.46750i 0 1.73556 − 1.99695i 0 0.118499 − 0.0570660i 0
29.9 0 −1.42811 + 0.325956i 0 1.41538 + 1.73110i 0 1.54156 − 2.15025i 0 −0.769660 + 0.370649i 0
29.10 0 −0.984698 + 0.224751i 0 −1.10153 + 1.94593i 0 −2.10132 − 1.60762i 0 −1.78379 + 0.859028i 0
29.11 0 −0.631234 + 0.144075i 0 −2.01504 − 0.969345i 0 1.64972 − 2.06843i 0 −2.32521 + 1.11976i 0
29.12 0 −0.622220 + 0.142018i 0 1.29465 + 1.82315i 0 −2.58484 + 0.564461i 0 −2.33592 + 1.12492i 0
29.13 0 −0.441575 + 0.100787i 0 −0.605716 − 2.15247i 0 1.86214 + 1.87948i 0 −2.51808 + 1.21264i 0
29.14 0 −0.332725 + 0.0759423i 0 −2.22696 + 0.201644i 0 2.62035 + 0.365762i 0 −2.59797 + 1.25112i 0
29.15 0 0.332725 − 0.0759423i 0 2.09391 − 0.784566i 0 −2.62035 − 0.365762i 0 −2.59797 + 1.25112i 0
29.16 0 0.441575 − 0.100787i 0 −0.388188 − 2.20211i 0 −1.86214 − 1.87948i 0 −2.51808 + 1.21264i 0
29.17 0 0.622220 − 0.142018i 0 −0.375399 + 2.20433i 0 2.58484 − 0.564461i 0 −2.33592 + 1.12492i 0
29.18 0 0.631234 − 0.144075i 0 1.39490 − 1.74764i 0 −1.64972 + 2.06843i 0 −2.32521 + 1.11976i 0
29.19 0 0.984698 − 0.224751i 0 1.83675 + 1.27529i 0 2.10132 + 1.60762i 0 −1.78379 + 0.859028i 0
29.20 0 1.42811 − 0.325956i 0 −0.524117 + 2.17378i 0 −1.54156 + 2.15025i 0 −0.769660 + 0.370649i 0
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
49.e even 7 1 inner
245.p 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.bd.a ✓ 168
5.b even 2 1 inner 980.2.bd.a ✓ 168
49.e even 7 1 inner 980.2.bd.a ✓ 168
245.p even 14 1 inner 980.2.bd.a ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.bd.a ✓ 168 1.a even 1 1 trivial
980.2.bd.a ✓ 168 5.b even 2 1 inner
980.2.bd.a ✓ 168 49.e even 7 1 inner
980.2.bd.a ✓ 168 245.p even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(980, [\chi])\).