Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [574,2,Mod(43,574)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(574, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("574.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 574 = 2 \cdot 7 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 574.u (of order \(20\), degree \(8\), 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: | \(4.58341307602\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 | 0.951057 | − | 0.309017i | −2.33712 | + | 2.33712i | 0.809017 | − | 0.587785i | −1.70120 | − | 2.34151i | −1.50053 | + | 2.94495i | 0.453990 | + | 0.891007i | 0.587785 | − | 0.809017i | − | 7.92429i | −2.34151 | − | 1.70120i | |
43.2 | 0.951057 | − | 0.309017i | −1.62445 | + | 1.62445i | 0.809017 | − | 0.587785i | −0.764320 | − | 1.05200i | −1.04296 | + | 2.04693i | −0.453990 | − | 0.891007i | 0.587785 | − | 0.809017i | − | 2.27768i | −1.05200 | − | 0.764320i | |
43.3 | 0.951057 | − | 0.309017i | −1.27895 | + | 1.27895i | 0.809017 | − | 0.587785i | −1.21136 | − | 1.66730i | −0.821136 | + | 1.61157i | −0.453990 | − | 0.891007i | 0.587785 | − | 0.809017i | − | 0.271421i | −1.66730 | − | 1.21136i | |
43.4 | 0.951057 | − | 0.309017i | −0.506337 | + | 0.506337i | 0.809017 | − | 0.587785i | 1.31445 | + | 1.80919i | −0.325088 | + | 0.638022i | 0.453990 | + | 0.891007i | 0.587785 | − | 0.809017i | 2.48725i | 1.80919 | + | 1.31445i | ||
43.5 | 0.951057 | − | 0.309017i | 0.0482483 | − | 0.0482483i | 0.809017 | − | 0.587785i | −0.234147 | − | 0.322276i | 0.0309773 | − | 0.0607964i | 0.453990 | + | 0.891007i | 0.587785 | − | 0.809017i | 2.99534i | −0.322276 | − | 0.234147i | ||
43.6 | 0.951057 | − | 0.309017i | 0.398829 | − | 0.398829i | 0.809017 | − | 0.587785i | 1.92408 | + | 2.64827i | 0.256064 | − | 0.502554i | −0.453990 | − | 0.891007i | 0.587785 | − | 0.809017i | 2.68187i | 2.64827 | + | 1.92408i | ||
43.7 | 0.951057 | − | 0.309017i | 1.34719 | − | 1.34719i | 0.809017 | − | 0.587785i | −0.232029 | − | 0.319360i | 0.864951 | − | 1.69756i | −0.453990 | − | 0.891007i | 0.587785 | − | 0.809017i | − | 0.629857i | −0.319360 | − | 0.232029i | |
43.8 | 0.951057 | − | 0.309017i | 1.54408 | − | 1.54408i | 0.809017 | − | 0.587785i | −1.84191 | − | 2.53517i | 0.991360 | − | 1.94565i | 0.453990 | + | 0.891007i | 0.587785 | − | 0.809017i | − | 1.76836i | −2.53517 | − | 1.84191i | |
43.9 | 0.951057 | − | 0.309017i | 1.79942 | − | 1.79942i | 0.809017 | − | 0.587785i | −1.05059 | − | 1.44602i | 1.15530 | − | 2.26740i | −0.453990 | − | 0.891007i | 0.587785 | − | 0.809017i | − | 3.47580i | −1.44602 | − | 1.05059i | |
43.10 | 0.951057 | − | 0.309017i | 1.89317 | − | 1.89317i | 0.809017 | − | 0.587785i | 1.56096 | + | 2.14848i | 1.21549 | − | 2.38554i | 0.453990 | + | 0.891007i | 0.587785 | − | 0.809017i | − | 4.16820i | 2.14848 | + | 1.56096i | |
169.1 | −0.587785 | − | 0.809017i | −2.34137 | − | 2.34137i | −0.309017 | + | 0.951057i | −0.332119 | − | 0.107912i | −0.517986 | + | 3.27043i | 0.156434 | + | 0.987688i | 0.951057 | − | 0.309017i | 7.96404i | 0.107912 | + | 0.332119i | ||
169.2 | −0.587785 | − | 0.809017i | −1.38661 | − | 1.38661i | −0.309017 | + | 0.951057i | 1.51658 | + | 0.492765i | −0.306761 | + | 1.93682i | 0.156434 | + | 0.987688i | 0.951057 | − | 0.309017i | 0.845358i | −0.492765 | − | 1.51658i | ||
169.3 | −0.587785 | − | 0.809017i | −1.37500 | − | 1.37500i | −0.309017 | + | 0.951057i | 2.70783 | + | 0.879827i | −0.304193 | + | 1.92060i | −0.156434 | − | 0.987688i | 0.951057 | − | 0.309017i | 0.781229i | −0.879827 | − | 2.70783i | ||
169.4 | −0.587785 | − | 0.809017i | −0.887212 | − | 0.887212i | −0.309017 | + | 0.951057i | −0.911575 | − | 0.296189i | −0.196279 | + | 1.23926i | −0.156434 | − | 0.987688i | 0.951057 | − | 0.309017i | − | 1.42571i | 0.296189 | + | 0.911575i | |
169.5 | −0.587785 | − | 0.809017i | −0.364190 | − | 0.364190i | −0.309017 | + | 0.951057i | −2.56002 | − | 0.831800i | −0.0805703 | + | 0.508701i | −0.156434 | − | 0.987688i | 0.951057 | − | 0.309017i | − | 2.73473i | 0.831800 | + | 2.56002i | |
169.6 | −0.587785 | − | 0.809017i | 0.350958 | + | 0.350958i | −0.309017 | + | 0.951057i | −0.961282 | − | 0.312339i | 0.0776431 | − | 0.490219i | 0.156434 | + | 0.987688i | 0.951057 | − | 0.309017i | − | 2.75366i | 0.312339 | + | 0.961282i | |
169.7 | −0.587785 | − | 0.809017i | 0.756754 | + | 0.756754i | −0.309017 | + | 0.951057i | 3.33364 | + | 1.08317i | 0.167418 | − | 1.05704i | −0.156434 | − | 0.987688i | 0.951057 | − | 0.309017i | − | 1.85465i | −1.08317 | − | 3.33364i | |
169.8 | −0.587785 | − | 0.809017i | 1.52921 | + | 1.52921i | −0.309017 | + | 0.951057i | −3.74452 | − | 1.21667i | 0.338311 | − | 2.13601i | 0.156434 | + | 0.987688i | 0.951057 | − | 0.309017i | 1.67699i | 1.21667 | + | 3.74452i | ||
169.9 | −0.587785 | − | 0.809017i | 2.06904 | + | 2.06904i | −0.309017 | + | 0.951057i | 1.41569 | + | 0.459984i | 0.457737 | − | 2.89004i | 0.156434 | + | 0.987688i | 0.951057 | − | 0.309017i | 5.56184i | −0.459984 | − | 1.41569i | ||
169.10 | −0.587785 | − | 0.809017i | 2.09088 | + | 2.09088i | −0.309017 | + | 0.951057i | 1.77185 | + | 0.575708i | 0.462568 | − | 2.92054i | −0.156434 | − | 0.987688i | 0.951057 | − | 0.309017i | 5.74352i | −0.575708 | − | 1.77185i | ||
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
41.g | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 574.2.u.a | ✓ | 80 |
41.g | even | 20 | 1 | inner | 574.2.u.a | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
574.2.u.a | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
574.2.u.a | ✓ | 80 | 41.g | even | 20 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} - 4 T_{3}^{79} + 8 T_{3}^{78} - 8 T_{3}^{77} + 566 T_{3}^{76} - 2296 T_{3}^{75} + \cdots + 430336 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\).