Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [138,5,Mod(47,138)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(138, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("138.47");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 138 = 2 \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 138.c (of order \(2\), degree \(1\), 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: | \(14.2650549056\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | − | 2.82843i | −8.97110 | + | 0.720642i | −8.00000 | − | 29.0082i | 2.03828 | + | 25.3741i | 75.1852 | 22.6274i | 79.9613 | − | 12.9299i | −82.0477 | ||||||||||
47.2 | − | 2.82843i | −8.21093 | + | 3.68520i | −8.00000 | 42.4258i | 10.4233 | + | 23.2240i | −2.59538 | 22.6274i | 53.8386 | − | 60.5178i | 119.998 | |||||||||||
47.3 | − | 2.82843i | −7.96199 | + | 4.19604i | −8.00000 | − | 18.2777i | 11.8682 | + | 22.5199i | −45.3951 | 22.6274i | 45.7866 | − | 66.8176i | −51.6971 | ||||||||||
47.4 | − | 2.82843i | −7.90779 | − | 4.29732i | −8.00000 | 10.5993i | −12.1546 | + | 22.3666i | −56.5485 | 22.6274i | 44.0662 | + | 67.9645i | 29.9794 | |||||||||||
47.5 | − | 2.82843i | −6.26270 | − | 6.46363i | −8.00000 | 3.82580i | −18.2819 | + | 17.7136i | 80.3647 | 22.6274i | −2.55712 | + | 80.9596i | 10.8210 | |||||||||||
47.6 | − | 2.82843i | −2.30705 | − | 8.69928i | −8.00000 | − | 39.0683i | −24.6053 | + | 6.52531i | −28.2665 | 22.6274i | −70.3551 | + | 40.1393i | −110.502 | ||||||||||
47.7 | − | 2.82843i | −0.694207 | + | 8.97319i | −8.00000 | 10.5332i | 25.3800 | + | 1.96351i | −61.3754 | 22.6274i | −80.0362 | − | 12.4585i | 29.7925 | |||||||||||
47.8 | − | 2.82843i | 1.63185 | − | 8.85082i | −8.00000 | 5.43082i | −25.0339 | − | 4.61556i | −23.1901 | 22.6274i | −75.6741 | − | 28.8864i | 15.3607 | |||||||||||
47.9 | − | 2.82843i | 2.12978 | − | 8.74437i | −8.00000 | 37.2957i | −24.7328 | − | 6.02393i | 61.4020 | 22.6274i | −71.9281 | − | 37.2472i | 105.488 | |||||||||||
47.10 | − | 2.82843i | 3.06562 | + | 8.46180i | −8.00000 | 31.7511i | 23.9336 | − | 8.67088i | 50.3621 | 22.6274i | −62.2040 | + | 51.8813i | 89.8057 | |||||||||||
47.11 | − | 2.82843i | 6.27270 | + | 6.45393i | −8.00000 | − | 43.2073i | 18.2545 | − | 17.7419i | −83.1420 | 22.6274i | −2.30643 | + | 80.9672i | −122.209 | ||||||||||
47.12 | − | 2.82843i | 7.99491 | + | 4.13297i | −8.00000 | − | 7.26641i | 11.6898 | − | 22.6130i | 48.6143 | 22.6274i | 46.8371 | + | 66.0854i | −20.5525 | ||||||||||
47.13 | − | 2.82843i | 8.55653 | − | 2.79032i | −8.00000 | 33.9236i | −7.89220 | − | 24.2015i | −34.3722 | 22.6274i | 65.4283 | − | 47.7508i | 95.9504 | |||||||||||
47.14 | − | 2.82843i | 8.66438 | − | 2.43487i | −8.00000 | − | 5.01640i | −6.88684 | − | 24.5066i | −33.0433 | 22.6274i | 69.1428 | − | 42.1932i | −14.1885 | ||||||||||
47.15 | 2.82843i | −8.97110 | − | 0.720642i | −8.00000 | 29.0082i | 2.03828 | − | 25.3741i | 75.1852 | − | 22.6274i | 79.9613 | + | 12.9299i | −82.0477 | |||||||||||
47.16 | 2.82843i | −8.21093 | − | 3.68520i | −8.00000 | − | 42.4258i | 10.4233 | − | 23.2240i | −2.59538 | − | 22.6274i | 53.8386 | + | 60.5178i | 119.998 | ||||||||||
47.17 | 2.82843i | −7.96199 | − | 4.19604i | −8.00000 | 18.2777i | 11.8682 | − | 22.5199i | −45.3951 | − | 22.6274i | 45.7866 | + | 66.8176i | −51.6971 | |||||||||||
47.18 | 2.82843i | −7.90779 | + | 4.29732i | −8.00000 | − | 10.5993i | −12.1546 | − | 22.3666i | −56.5485 | − | 22.6274i | 44.0662 | − | 67.9645i | 29.9794 | ||||||||||
47.19 | 2.82843i | −6.26270 | + | 6.46363i | −8.00000 | − | 3.82580i | −18.2819 | − | 17.7136i | 80.3647 | − | 22.6274i | −2.55712 | − | 80.9596i | 10.8210 | ||||||||||
47.20 | 2.82843i | −2.30705 | + | 8.69928i | −8.00000 | 39.0683i | −24.6053 | − | 6.52531i | −28.2665 | − | 22.6274i | −70.3551 | − | 40.1393i | −110.502 | |||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 138.5.c.a | ✓ | 28 |
3.b | odd | 2 | 1 | inner | 138.5.c.a | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
138.5.c.a | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
138.5.c.a | ✓ | 28 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(138, [\chi])\).