Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [460,2,Mod(17,460)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(44))
chi = DirichletCharacter(H, H._module([0, 11, 14]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("460.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 460 = 2^{2} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 460.x (of order \(44\), degree \(20\), 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: | \(3.67311849298\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{44})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{44}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −2.71854 | + | 1.48444i | 0 | 1.97074 | + | 1.05650i | 0 | 2.65480 | + | 3.54640i | 0 | 3.56500 | − | 5.54725i | 0 | ||||||||||
17.2 | 0 | −1.95329 | + | 1.06658i | 0 | 0.617840 | − | 2.14902i | 0 | 0.0518807 | + | 0.0693045i | 0 | 1.05584 | − | 1.64292i | 0 | ||||||||||
17.3 | 0 | −1.92230 | + | 1.04965i | 0 | 0.766931 | + | 2.10043i | 0 | −2.30217 | − | 3.07534i | 0 | 0.971534 | − | 1.51174i | 0 | ||||||||||
17.4 | 0 | −1.47141 | + | 0.803448i | 0 | −1.48903 | − | 1.66817i | 0 | 0.330540 | + | 0.441549i | 0 | −0.102418 | + | 0.159366i | 0 | ||||||||||
17.5 | 0 | −1.02467 | + | 0.559510i | 0 | 2.07680 | − | 0.828788i | 0 | −2.22248 | − | 2.96889i | 0 | −0.885033 | + | 1.37714i | 0 | ||||||||||
17.6 | 0 | 0.302476 | − | 0.165164i | 0 | 0.209636 | + | 2.22622i | 0 | 1.78504 | + | 2.38454i | 0 | −1.55771 | + | 2.42384i | 0 | ||||||||||
17.7 | 0 | 0.416910 | − | 0.227650i | 0 | −2.14737 | − | 0.623543i | 0 | −0.887718 | − | 1.18585i | 0 | −1.49993 | + | 2.33394i | 0 | ||||||||||
17.8 | 0 | 0.483512 | − | 0.264018i | 0 | −0.935821 | + | 2.03082i | 0 | −1.13426 | − | 1.51520i | 0 | −1.45784 | + | 2.26845i | 0 | ||||||||||
17.9 | 0 | 0.707493 | − | 0.386321i | 0 | 1.31269 | − | 1.81021i | 0 | 2.84997 | + | 3.80711i | 0 | −1.27062 | + | 1.97712i | 0 | ||||||||||
17.10 | 0 | 1.92117 | − | 1.04904i | 0 | −0.336864 | − | 2.21055i | 0 | −2.02919 | − | 2.71068i | 0 | 0.968479 | − | 1.50698i | 0 | ||||||||||
17.11 | 0 | 2.20798 | − | 1.20565i | 0 | 1.99649 | + | 1.00699i | 0 | −0.375619 | − | 0.501768i | 0 | 1.79966 | − | 2.80032i | 0 | ||||||||||
17.12 | 0 | 2.69738 | − | 1.47288i | 0 | −2.20694 | + | 0.359751i | 0 | 2.19560 | + | 2.93298i | 0 | 3.48456 | − | 5.42208i | 0 | ||||||||||
33.1 | 0 | −0.632093 | − | 2.90569i | 0 | −1.58894 | − | 1.57330i | 0 | 0.867718 | + | 1.58911i | 0 | −5.31457 | + | 2.42708i | 0 | ||||||||||
33.2 | 0 | −0.486808 | − | 2.23782i | 0 | 0.0638162 | + | 2.23516i | 0 | 1.23306 | + | 2.25818i | 0 | −2.04197 | + | 0.932535i | 0 | ||||||||||
33.3 | 0 | −0.409167 | − | 1.88091i | 0 | 2.14867 | + | 0.619038i | 0 | −1.26125 | − | 2.30981i | 0 | −0.641504 | + | 0.292965i | 0 | ||||||||||
33.4 | 0 | −0.299812 | − | 1.37822i | 0 | 0.675503 | − | 2.13159i | 0 | −0.759843 | − | 1.39155i | 0 | 0.919306 | − | 0.419833i | 0 | ||||||||||
33.5 | 0 | −0.0813911 | − | 0.374149i | 0 | −2.18808 | − | 0.460771i | 0 | 0.364123 | + | 0.666841i | 0 | 2.59553 | − | 1.18534i | 0 | ||||||||||
33.6 | 0 | 0.0980316 | + | 0.450644i | 0 | 2.20605 | − | 0.365176i | 0 | 1.83859 | + | 3.36712i | 0 | 2.53543 | − | 1.15789i | 0 | ||||||||||
33.7 | 0 | 0.132785 | + | 0.610405i | 0 | −1.61522 | + | 1.54631i | 0 | −2.27438 | − | 4.16522i | 0 | 2.37393 | − | 1.08414i | 0 | ||||||||||
33.8 | 0 | 0.195370 | + | 0.898100i | 0 | −2.15927 | − | 0.580987i | 0 | 1.50914 | + | 2.76379i | 0 | 1.96048 | − | 0.895322i | 0 | ||||||||||
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
23.d | odd | 22 | 1 | inner |
115.l | even | 44 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 460.2.x.a | ✓ | 240 |
5.c | odd | 4 | 1 | inner | 460.2.x.a | ✓ | 240 |
23.d | odd | 22 | 1 | inner | 460.2.x.a | ✓ | 240 |
115.l | even | 44 | 1 | inner | 460.2.x.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
460.2.x.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
460.2.x.a | ✓ | 240 | 5.c | odd | 4 | 1 | inner |
460.2.x.a | ✓ | 240 | 23.d | odd | 22 | 1 | inner |
460.2.x.a | ✓ | 240 | 115.l | even | 44 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(460, [\chi])\).