Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [138,4,Mod(137,138)]
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, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("138.137");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 138 = 2 \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 138.d (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: | \(8.14226358079\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
137.1 | − | 2.00000i | −4.96925 | − | 1.51872i | −4.00000 | −17.2734 | −3.03745 | + | 9.93851i | − | 11.5675i | 8.00000i | 22.3870 | + | 15.0938i | 34.5468i | ||||||||||
137.2 | − | 2.00000i | −4.96925 | − | 1.51872i | −4.00000 | 17.2734 | −3.03745 | + | 9.93851i | 11.5675i | 8.00000i | 22.3870 | + | 15.0938i | − | 34.5468i | ||||||||||
137.3 | − | 2.00000i | −3.26555 | + | 4.04180i | −4.00000 | −6.82316 | 8.08360 | + | 6.53111i | − | 3.09939i | 8.00000i | −5.67231 | − | 26.3974i | 13.6463i | ||||||||||
137.4 | − | 2.00000i | −3.26555 | + | 4.04180i | −4.00000 | 6.82316 | 8.08360 | + | 6.53111i | 3.09939i | 8.00000i | −5.67231 | − | 26.3974i | − | 13.6463i | ||||||||||
137.5 | − | 2.00000i | −1.34717 | − | 5.01848i | −4.00000 | −4.58818 | −10.0370 | + | 2.69433i | 31.2911i | 8.00000i | −23.3703 | + | 13.5215i | 9.17636i | |||||||||||
137.6 | − | 2.00000i | −1.34717 | − | 5.01848i | −4.00000 | 4.58818 | −10.0370 | + | 2.69433i | − | 31.2911i | 8.00000i | −23.3703 | + | 13.5215i | − | 9.17636i | |||||||||
137.7 | − | 2.00000i | 2.34341 | + | 4.63772i | −4.00000 | −10.8839 | 9.27544 | − | 4.68682i | − | 21.8179i | 8.00000i | −16.0169 | + | 21.7361i | 21.7678i | ||||||||||
137.8 | − | 2.00000i | 2.34341 | + | 4.63772i | −4.00000 | 10.8839 | 9.27544 | − | 4.68682i | 21.8179i | 8.00000i | −16.0169 | + | 21.7361i | − | 21.7678i | ||||||||||
137.9 | − | 2.00000i | 4.33582 | − | 2.86368i | −4.00000 | −13.9937 | −5.72736 | − | 8.67164i | − | 0.843170i | 8.00000i | 10.5987 | − | 24.8328i | 27.9874i | ||||||||||
137.10 | − | 2.00000i | 4.33582 | − | 2.86368i | −4.00000 | 13.9937 | −5.72736 | − | 8.67164i | 0.843170i | 8.00000i | 10.5987 | − | 24.8328i | − | 27.9874i | ||||||||||
137.11 | − | 2.00000i | 4.90275 | + | 1.72136i | −4.00000 | −10.2829 | 3.44272 | − | 9.80549i | 24.1811i | 8.00000i | 21.0738 | + | 16.8788i | 20.5659i | |||||||||||
137.12 | − | 2.00000i | 4.90275 | + | 1.72136i | −4.00000 | 10.2829 | 3.44272 | − | 9.80549i | − | 24.1811i | 8.00000i | 21.0738 | + | 16.8788i | − | 20.5659i | |||||||||
137.13 | 2.00000i | −4.96925 | + | 1.51872i | −4.00000 | −17.2734 | −3.03745 | − | 9.93851i | 11.5675i | − | 8.00000i | 22.3870 | − | 15.0938i | − | 34.5468i | ||||||||||
137.14 | 2.00000i | −4.96925 | + | 1.51872i | −4.00000 | 17.2734 | −3.03745 | − | 9.93851i | − | 11.5675i | − | 8.00000i | 22.3870 | − | 15.0938i | 34.5468i | ||||||||||
137.15 | 2.00000i | −3.26555 | − | 4.04180i | −4.00000 | −6.82316 | 8.08360 | − | 6.53111i | 3.09939i | − | 8.00000i | −5.67231 | + | 26.3974i | − | 13.6463i | ||||||||||
137.16 | 2.00000i | −3.26555 | − | 4.04180i | −4.00000 | 6.82316 | 8.08360 | − | 6.53111i | − | 3.09939i | − | 8.00000i | −5.67231 | + | 26.3974i | 13.6463i | ||||||||||
137.17 | 2.00000i | −1.34717 | + | 5.01848i | −4.00000 | −4.58818 | −10.0370 | − | 2.69433i | − | 31.2911i | − | 8.00000i | −23.3703 | − | 13.5215i | − | 9.17636i | |||||||||
137.18 | 2.00000i | −1.34717 | + | 5.01848i | −4.00000 | 4.58818 | −10.0370 | − | 2.69433i | 31.2911i | − | 8.00000i | −23.3703 | − | 13.5215i | 9.17636i | |||||||||||
137.19 | 2.00000i | 2.34341 | − | 4.63772i | −4.00000 | −10.8839 | 9.27544 | + | 4.68682i | 21.8179i | − | 8.00000i | −16.0169 | − | 21.7361i | − | 21.7678i | ||||||||||
137.20 | 2.00000i | 2.34341 | − | 4.63772i | −4.00000 | 10.8839 | 9.27544 | + | 4.68682i | − | 21.8179i | − | 8.00000i | −16.0169 | − | 21.7361i | 21.7678i | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
69.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 138.4.d.a | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 138.4.d.a | ✓ | 24 |
23.b | odd | 2 | 1 | inner | 138.4.d.a | ✓ | 24 |
69.c | even | 2 | 1 | inner | 138.4.d.a | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
138.4.d.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
138.4.d.a | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
138.4.d.a | ✓ | 24 | 23.b | odd | 2 | 1 | inner |
138.4.d.a | ✓ | 24 | 69.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(138, [\chi])\).