Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [572,2,Mod(49,572)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(572, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([0, 12, 25]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("572.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 572 = 2^{2} \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 572.bq (of order \(30\), 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: | \(4.56744299562\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 | 0 | −0.319495 | + | 3.03979i | 0 | −0.103194 | − | 0.0335297i | 0 | 4.65033 | − | 0.488770i | 0 | −6.20380 | − | 1.31866i | 0 | ||||||||||
49.2 | 0 | −0.292873 | + | 2.78650i | 0 | −1.71678 | − | 0.557815i | 0 | −3.86952 | + | 0.406703i | 0 | −4.74434 | − | 1.00844i | 0 | ||||||||||
49.3 | 0 | −0.228377 | + | 2.17287i | 0 | −1.96627 | − | 0.638880i | 0 | 0.214213 | − | 0.0225147i | 0 | −1.73475 | − | 0.368732i | 0 | ||||||||||
49.4 | 0 | −0.140590 | + | 1.33763i | 0 | 2.78220 | + | 0.903990i | 0 | −3.77335 | + | 0.396595i | 0 | 1.16496 | + | 0.247620i | 0 | ||||||||||
49.5 | 0 | −0.130188 | + | 1.23865i | 0 | 3.26836 | + | 1.06196i | 0 | −0.124860 | + | 0.0131234i | 0 | 1.41713 | + | 0.301220i | 0 | ||||||||||
49.6 | 0 | −0.118461 | + | 1.12708i | 0 | 1.16420 | + | 0.378273i | 0 | 3.19565 | − | 0.335877i | 0 | 1.67816 | + | 0.356704i | 0 | ||||||||||
49.7 | 0 | −0.0476950 | + | 0.453788i | 0 | −3.72881 | − | 1.21156i | 0 | 0.128901 | − | 0.0135480i | 0 | 2.73079 | + | 0.580448i | 0 | ||||||||||
49.8 | 0 | 0.0149789 | − | 0.142515i | 0 | −1.45545 | − | 0.472905i | 0 | −3.21049 | + | 0.337436i | 0 | 2.91436 | + | 0.619466i | 0 | ||||||||||
49.9 | 0 | 0.107949 | − | 1.02707i | 0 | −2.11328 | − | 0.686646i | 0 | 0.696123 | − | 0.0731655i | 0 | 1.89122 | + | 0.401992i | 0 | ||||||||||
49.10 | 0 | 0.136254 | − | 1.29637i | 0 | 2.36621 | + | 0.768828i | 0 | 2.00645 | − | 0.210887i | 0 | 1.27243 | + | 0.270463i | 0 | ||||||||||
49.11 | 0 | 0.188862 | − | 1.79690i | 0 | 0.283508 | + | 0.0921172i | 0 | 0.331103 | − | 0.0348003i | 0 | −0.258739 | − | 0.0549966i | 0 | ||||||||||
49.12 | 0 | 0.201901 | − | 1.92096i | 0 | 1.20386 | + | 0.391159i | 0 | −4.99946 | + | 0.525464i | 0 | −0.714895 | − | 0.151956i | 0 | ||||||||||
49.13 | 0 | 0.287109 | − | 2.73166i | 0 | −3.50557 | − | 1.13903i | 0 | 2.82501 | − | 0.296921i | 0 | −4.44507 | − | 0.944828i | 0 | ||||||||||
49.14 | 0 | 0.340624 | − | 3.24082i | 0 | 3.52100 | + | 1.14404i | 0 | 1.92989 | − | 0.202840i | 0 | −7.45247 | − | 1.58407i | 0 | ||||||||||
69.1 | 0 | −3.10684 | − | 0.660380i | 0 | −2.01179 | + | 2.76900i | 0 | 0.661166 | + | 3.11054i | 0 | 6.47574 | + | 2.88318i | 0 | ||||||||||
69.2 | 0 | −2.87954 | − | 0.612064i | 0 | 0.464082 | − | 0.638753i | 0 | −0.0706912 | − | 0.332576i | 0 | 5.17647 | + | 2.30471i | 0 | ||||||||||
69.3 | 0 | −2.15084 | − | 0.457176i | 0 | 2.38726 | − | 3.28578i | 0 | −1.00599 | − | 4.73282i | 0 | 1.67648 | + | 0.746415i | 0 | ||||||||||
69.4 | 0 | −1.52573 | − | 0.324303i | 0 | −1.60954 | + | 2.21535i | 0 | −0.454255 | − | 2.13710i | 0 | −0.517971 | − | 0.230616i | 0 | ||||||||||
69.5 | 0 | −1.41919 | − | 0.301659i | 0 | 0.650916 | − | 0.895909i | 0 | 0.629373 | + | 2.96097i | 0 | −0.817523 | − | 0.363985i | 0 | ||||||||||
69.6 | 0 | −1.18856 | − | 0.252636i | 0 | 0.729880 | − | 1.00459i | 0 | 0.643353 | + | 3.02674i | 0 | −1.39179 | − | 0.619666i | 0 | ||||||||||
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
13.e | even | 6 | 1 | inner |
143.u | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 572.2.bq.a | ✓ | 112 |
11.c | even | 5 | 1 | inner | 572.2.bq.a | ✓ | 112 |
13.e | even | 6 | 1 | inner | 572.2.bq.a | ✓ | 112 |
143.u | even | 30 | 1 | inner | 572.2.bq.a | ✓ | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
572.2.bq.a | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
572.2.bq.a | ✓ | 112 | 11.c | even | 5 | 1 | inner |
572.2.bq.a | ✓ | 112 | 13.e | even | 6 | 1 | inner |
572.2.bq.a | ✓ | 112 | 143.u | even | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(572, [\chi])\).