Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [538,2,Mod(9,538)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(538, base_ring=CyclotomicField(134))
chi = DirichletCharacter(H, H._module([109]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("538.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 538 = 2 \cdot 269 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 538.e (of order \(134\), degree \(66\), 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: | \(4.29595162874\) |
Analytic rank: | \(0\) |
Dimension: | \(1452\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{134})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{134}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | −0.833094 | + | 0.553131i | −1.43549 | + | 2.53183i | 0.388092 | − | 0.921621i | 1.94372 | + | 1.35724i | −0.204537 | − | 2.90327i | 0.225409 | + | 1.59183i | 0.186460 | + | 0.982463i | −2.80913 | − | 4.69454i | −2.37004 | − | 0.0555750i |
9.2 | −0.833094 | + | 0.553131i | −1.05597 | + | 1.86246i | 0.388092 | − | 0.921621i | −1.04785 | − | 0.731677i | −0.150460 | − | 2.13569i | −0.574974 | − | 4.06045i | 0.186460 | + | 0.982463i | −0.813253 | − | 1.35909i | 1.27767 | + | 0.0299601i |
9.3 | −0.833094 | + | 0.553131i | −0.805229 | + | 1.42021i | 0.388092 | − | 0.921621i | −1.22081 | − | 0.852450i | −0.114733 | − | 1.62857i | 0.0206347 | + | 0.145722i | 0.186460 | + | 0.982463i | 0.171807 | + | 0.287119i | 1.48856 | + | 0.0349053i |
9.4 | −0.833094 | + | 0.553131i | −0.552092 | + | 0.973747i | 0.388092 | − | 0.921621i | 1.05205 | + | 0.734610i | −0.0786651 | − | 1.11660i | 0.586003 | + | 4.13834i | 0.186460 | + | 0.982463i | 0.897045 | + | 1.49912i | −1.28279 | − | 0.0300801i |
9.5 | −0.833094 | + | 0.553131i | 0.0651030 | − | 0.114825i | 0.388092 | − | 0.921621i | 2.50671 | + | 1.75036i | 0.00927622 | + | 0.131670i | −0.310567 | − | 2.19321i | 0.186460 | + | 0.982463i | 1.53148 | + | 2.55936i | −3.05651 | − | 0.0716721i |
9.6 | −0.833094 | + | 0.553131i | 0.283306 | − | 0.499678i | 0.388092 | − | 0.921621i | −0.723479 | − | 0.505183i | 0.0403669 | + | 0.572984i | −0.341899 | − | 2.41448i | 0.186460 | + | 0.982463i | 1.37101 | + | 2.29119i | 0.882159 | + | 0.0206858i |
9.7 | −0.833094 | + | 0.553131i | 0.602356 | − | 1.06240i | 0.388092 | − | 0.921621i | −2.46979 | − | 1.72457i | 0.0858269 | + | 1.21826i | 0.527223 | + | 3.72323i | 0.186460 | + | 0.982463i | 0.774562 | + | 1.29443i | 3.01148 | + | 0.0706163i |
9.8 | −0.833094 | + | 0.553131i | 0.865942 | − | 1.52730i | 0.388092 | − | 0.921621i | 0.921477 | + | 0.643438i | 0.123384 | + | 1.75136i | 0.350952 | + | 2.47841i | 0.186460 | + | 0.982463i | −0.0423564 | − | 0.0707850i | −1.12358 | − | 0.0263469i |
9.9 | −0.833094 | + | 0.553131i | 1.30591 | − | 2.30329i | 0.388092 | − | 0.921621i | −3.05509 | − | 2.13327i | 0.186074 | + | 2.64120i | −0.565573 | − | 3.99406i | 0.186460 | + | 0.982463i | −2.05933 | − | 3.44150i | 3.72515 | + | 0.0873512i |
9.10 | −0.833094 | + | 0.553131i | 1.52425 | − | 2.68838i | 0.388092 | − | 0.921621i | −0.943666 | − | 0.658932i | 0.217183 | + | 3.08279i | 0.465524 | + | 3.28751i | 0.186460 | + | 0.982463i | −3.36364 | − | 5.62122i | 1.15064 | + | 0.0269813i |
9.11 | −0.833094 | + | 0.553131i | 1.61279 | − | 2.84454i | 0.388092 | − | 0.921621i | 2.64265 | + | 1.84528i | 0.229799 | + | 3.26186i | −0.192859 | − | 1.36197i | 0.186460 | + | 0.982463i | −3.94990 | − | 6.60098i | −3.22226 | − | 0.0755588i |
9.12 | 0.833094 | − | 0.553131i | −1.66856 | + | 2.94291i | 0.388092 | − | 0.921621i | −0.524262 | − | 0.366075i | 0.237745 | + | 3.37465i | −0.443772 | − | 3.13391i | −0.186460 | − | 0.982463i | −4.33618 | − | 7.24652i | −0.639247 | − | 0.0149897i |
9.13 | 0.833094 | − | 0.553131i | −1.12161 | + | 1.97822i | 0.388092 | − | 0.921621i | 3.58960 | + | 2.50650i | 0.159813 | + | 2.26844i | 0.304181 | + | 2.14812i | −0.186460 | − | 0.982463i | −1.11495 | − | 1.86327i | 4.37690 | + | 0.102634i |
9.14 | 0.833094 | − | 0.553131i | −0.952778 | + | 1.68045i | 0.388092 | − | 0.921621i | 0.475910 | + | 0.332313i | 0.135757 | + | 1.92699i | −0.0793537 | − | 0.560394i | −0.186460 | − | 0.982463i | −0.375712 | − | 0.627880i | 0.580291 | + | 0.0136072i |
9.15 | 0.833094 | − | 0.553131i | −0.745603 | + | 1.31505i | 0.388092 | − | 0.921621i | −3.35521 | − | 2.34284i | 0.106238 | + | 1.50798i | 0.255112 | + | 1.80159i | −0.186460 | − | 0.982463i | 0.366990 | + | 0.613304i | −4.09110 | − | 0.0959323i |
9.16 | 0.833094 | − | 0.553131i | −0.559957 | + | 0.987619i | 0.388092 | − | 0.921621i | −2.31476 | − | 1.61632i | 0.0797857 | + | 1.13251i | −0.0304501 | − | 0.215038i | −0.186460 | − | 0.982463i | 0.878584 | + | 1.46827i | −2.82245 | − | 0.0661836i |
9.17 | 0.833094 | − | 0.553131i | 0.115924 | − | 0.204459i | 0.388092 | − | 0.921621i | −1.66371 | − | 1.16171i | −0.0165174 | − | 0.234455i | −0.662220 | − | 4.67658i | −0.186460 | − | 0.982463i | 1.51206 | + | 2.52691i | −2.02860 | − | 0.0475687i |
9.18 | 0.833094 | − | 0.553131i | 0.261679 | − | 0.461534i | 0.388092 | − | 0.921621i | 1.06696 | + | 0.745027i | −0.0372855 | − | 0.529245i | 0.525099 | + | 3.70823i | −0.186460 | − | 0.982463i | 1.39588 | + | 2.33277i | 1.30098 | + | 0.0305067i |
9.19 | 0.833094 | − | 0.553131i | 0.453043 | − | 0.799051i | 0.388092 | − | 0.921621i | 0.923295 | + | 0.644708i | −0.0645520 | − | 0.916277i | −0.343905 | − | 2.42865i | −0.186460 | − | 0.982463i | 1.10719 | + | 1.85031i | 1.12580 | + | 0.0263989i |
9.20 | 0.833094 | − | 0.553131i | 1.10088 | − | 1.94167i | 0.388092 | − | 0.921621i | −0.0185458 | − | 0.0129499i | −0.156860 | − | 2.22653i | −0.0718859 | − | 0.507656i | −0.186460 | − | 0.982463i | −1.01772 | − | 1.70078i | −0.0226134 | 0.000530262i | |
See next 80 embeddings (of 1452 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
269.e | even | 134 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 538.2.e.a | ✓ | 1452 |
269.e | even | 134 | 1 | inner | 538.2.e.a | ✓ | 1452 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
538.2.e.a | ✓ | 1452 | 1.a | even | 1 | 1 | trivial |
538.2.e.a | ✓ | 1452 | 269.e | even | 134 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(538, [\chi])\).