Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [348,2,Mod(77,348)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(348, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([0, 14, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("348.77");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 348.u (of order \(28\), degree \(12\), 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: | \(2.77879399034\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{28})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{28}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
77.1 | 0 | −1.61032 | − | 0.637874i | 0 | −0.257187 | + | 1.12681i | 0 | 1.28537 | − | 0.619002i | 0 | 2.18623 | + | 2.05436i | 0 | ||||||||||
77.2 | 0 | −1.43419 | + | 0.971134i | 0 | −0.315574 | + | 1.38262i | 0 | −4.35058 | + | 2.09513i | 0 | 1.11380 | − | 2.78558i | 0 | ||||||||||
77.3 | 0 | −0.910998 | + | 1.47312i | 0 | −0.668669 | + | 2.92963i | 0 | 2.92243 | − | 1.40737i | 0 | −1.34017 | − | 2.68402i | 0 | ||||||||||
77.4 | 0 | −0.564571 | − | 1.63746i | 0 | 0.446237 | − | 1.95509i | 0 | 1.32620 | − | 0.638664i | 0 | −2.36252 | + | 1.84892i | 0 | ||||||||||
77.5 | 0 | −0.206229 | + | 1.71973i | 0 | 0.668669 | − | 2.92963i | 0 | 2.92243 | − | 1.40737i | 0 | −2.91494 | − | 0.709316i | 0 | ||||||||||
77.6 | 0 | 0.0554564 | − | 1.73116i | 0 | −0.717531 | + | 3.14371i | 0 | −2.94019 | + | 1.41592i | 0 | −2.99385 | − | 0.192008i | 0 | ||||||||||
77.7 | 0 | 0.515802 | + | 1.65347i | 0 | 0.315574 | − | 1.38262i | 0 | −4.35058 | + | 2.09513i | 0 | −2.46790 | + | 1.70572i | 0 | ||||||||||
77.8 | 0 | 1.03600 | − | 1.38805i | 0 | 0.717531 | − | 3.14371i | 0 | −2.94019 | + | 1.41592i | 0 | −0.853388 | − | 2.87606i | 0 | ||||||||||
77.9 | 0 | 1.46234 | − | 0.928210i | 0 | −0.446237 | + | 1.95509i | 0 | 1.32620 | − | 0.638664i | 0 | 1.27685 | − | 2.71471i | 0 | ||||||||||
77.10 | 0 | 1.65670 | + | 0.505306i | 0 | 0.257187 | − | 1.12681i | 0 | 1.28537 | − | 0.619002i | 0 | 2.48933 | + | 1.67428i | 0 | ||||||||||
89.1 | 0 | −1.73198 | − | 0.0158657i | 0 | 1.59205 | + | 1.99637i | 0 | −0.403835 | − | 1.76932i | 0 | 2.99950 | + | 0.0549580i | 0 | ||||||||||
89.2 | 0 | −1.73192 | − | 0.0214617i | 0 | −0.983380 | − | 1.23312i | 0 | 0.714491 | + | 3.13039i | 0 | 2.99908 | + | 0.0743397i | 0 | ||||||||||
89.3 | 0 | −0.826331 | + | 1.52223i | 0 | −0.603757 | − | 0.757088i | 0 | −0.729341 | − | 3.19545i | 0 | −1.63435 | − | 2.51573i | 0 | ||||||||||
89.4 | 0 | −0.770787 | − | 1.55109i | 0 | 0.983380 | + | 1.23312i | 0 | 0.714491 | + | 3.13039i | 0 | −1.81177 | + | 2.39112i | 0 | ||||||||||
89.5 | 0 | −0.765772 | − | 1.55357i | 0 | −1.59205 | − | 1.99637i | 0 | −0.403835 | − | 1.76932i | 0 | −1.82719 | + | 2.37937i | 0 | ||||||||||
89.6 | 0 | 0.360287 | + | 1.69416i | 0 | −0.868530 | − | 1.08910i | 0 | 0.633399 | + | 2.77510i | 0 | −2.74039 | + | 1.22077i | 0 | ||||||||||
89.7 | 0 | 1.01295 | − | 1.40497i | 0 | 0.603757 | + | 0.757088i | 0 | −0.729341 | − | 3.19545i | 0 | −0.947872 | − | 2.84632i | 0 | ||||||||||
89.8 | 0 | 1.08314 | + | 1.35160i | 0 | 2.49932 | + | 3.13404i | 0 | −0.562661 | − | 2.46518i | 0 | −0.653631 | + | 2.92793i | 0 | ||||||||||
89.9 | 0 | 1.68271 | − | 0.410463i | 0 | 0.868530 | + | 1.08910i | 0 | 0.633399 | + | 2.77510i | 0 | 2.66304 | − | 1.38138i | 0 | ||||||||||
89.10 | 0 | 1.68770 | + | 0.389436i | 0 | −2.49932 | − | 3.13404i | 0 | −0.562661 | − | 2.46518i | 0 | 2.69668 | + | 1.31450i | 0 | ||||||||||
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
29.f | odd | 28 | 1 | inner |
87.k | even | 28 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 348.2.u.a | ✓ | 120 |
3.b | odd | 2 | 1 | inner | 348.2.u.a | ✓ | 120 |
29.f | odd | 28 | 1 | inner | 348.2.u.a | ✓ | 120 |
87.k | even | 28 | 1 | inner | 348.2.u.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
348.2.u.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
348.2.u.a | ✓ | 120 | 3.b | odd | 2 | 1 | inner |
348.2.u.a | ✓ | 120 | 29.f | odd | 28 | 1 | inner |
348.2.u.a | ✓ | 120 | 87.k | even | 28 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(348, [\chi])\).