Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(944,945)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.944");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 945.g (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: | \(7.54586299101\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
944.1 | −2.59590 | 0 | 4.73872 | −0.178464 | − | 2.22893i | 0 | 1.33752 | + | 2.28277i | −7.10944 | 0 | 0.463274 | + | 5.78610i | ||||||||||||
944.2 | −2.59590 | 0 | 4.73872 | −0.178464 | + | 2.22893i | 0 | 1.33752 | − | 2.28277i | −7.10944 | 0 | 0.463274 | − | 5.78610i | ||||||||||||
944.3 | −2.59590 | 0 | 4.73872 | 0.178464 | − | 2.22893i | 0 | −1.33752 | − | 2.28277i | −7.10944 | 0 | −0.463274 | + | 5.78610i | ||||||||||||
944.4 | −2.59590 | 0 | 4.73872 | 0.178464 | + | 2.22893i | 0 | −1.33752 | + | 2.28277i | −7.10944 | 0 | −0.463274 | − | 5.78610i | ||||||||||||
944.5 | −1.85125 | 0 | 1.42712 | −2.19051 | − | 0.449085i | 0 | −1.13986 | + | 2.38762i | 1.06054 | 0 | 4.05517 | + | 0.831368i | ||||||||||||
944.6 | −1.85125 | 0 | 1.42712 | −2.19051 | + | 0.449085i | 0 | −1.13986 | − | 2.38762i | 1.06054 | 0 | 4.05517 | − | 0.831368i | ||||||||||||
944.7 | −1.85125 | 0 | 1.42712 | 2.19051 | − | 0.449085i | 0 | 1.13986 | − | 2.38762i | 1.06054 | 0 | −4.05517 | + | 0.831368i | ||||||||||||
944.8 | −1.85125 | 0 | 1.42712 | 2.19051 | + | 0.449085i | 0 | 1.13986 | + | 2.38762i | 1.06054 | 0 | −4.05517 | − | 0.831368i | ||||||||||||
944.9 | −1.29594 | 0 | −0.320531 | −1.01211 | − | 1.99390i | 0 | −2.26270 | + | 1.37120i | 3.00728 | 0 | 1.31163 | + | 2.58398i | ||||||||||||
944.10 | −1.29594 | 0 | −0.320531 | −1.01211 | + | 1.99390i | 0 | −2.26270 | − | 1.37120i | 3.00728 | 0 | 1.31163 | − | 2.58398i | ||||||||||||
944.11 | −1.29594 | 0 | −0.320531 | 1.01211 | − | 1.99390i | 0 | 2.26270 | − | 1.37120i | 3.00728 | 0 | −1.31163 | + | 2.58398i | ||||||||||||
944.12 | −1.29594 | 0 | −0.320531 | 1.01211 | + | 1.99390i | 0 | 2.26270 | + | 1.37120i | 3.00728 | 0 | −1.31163 | − | 2.58398i | ||||||||||||
944.13 | −0.393311 | 0 | −1.84531 | −1.54773 | − | 1.61386i | 0 | 1.59434 | − | 2.11142i | 1.51240 | 0 | 0.608740 | + | 0.634748i | ||||||||||||
944.14 | −0.393311 | 0 | −1.84531 | −1.54773 | + | 1.61386i | 0 | 1.59434 | + | 2.11142i | 1.51240 | 0 | 0.608740 | − | 0.634748i | ||||||||||||
944.15 | −0.393311 | 0 | −1.84531 | 1.54773 | − | 1.61386i | 0 | −1.59434 | + | 2.11142i | 1.51240 | 0 | −0.608740 | + | 0.634748i | ||||||||||||
944.16 | −0.393311 | 0 | −1.84531 | 1.54773 | + | 1.61386i | 0 | −1.59434 | − | 2.11142i | 1.51240 | 0 | −0.608740 | − | 0.634748i | ||||||||||||
944.17 | 0.393311 | 0 | −1.84531 | −1.54773 | − | 1.61386i | 0 | −1.59434 | − | 2.11142i | −1.51240 | 0 | −0.608740 | − | 0.634748i | ||||||||||||
944.18 | 0.393311 | 0 | −1.84531 | −1.54773 | + | 1.61386i | 0 | −1.59434 | + | 2.11142i | −1.51240 | 0 | −0.608740 | + | 0.634748i | ||||||||||||
944.19 | 0.393311 | 0 | −1.84531 | 1.54773 | − | 1.61386i | 0 | 1.59434 | + | 2.11142i | −1.51240 | 0 | 0.608740 | − | 0.634748i | ||||||||||||
944.20 | 0.393311 | 0 | −1.84531 | 1.54773 | + | 1.61386i | 0 | 1.59434 | − | 2.11142i | −1.51240 | 0 | 0.608740 | + | 0.634748i | ||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
105.g | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 945.2.g.a | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 945.2.g.a | ✓ | 32 |
5.b | even | 2 | 1 | inner | 945.2.g.a | ✓ | 32 |
7.b | odd | 2 | 1 | inner | 945.2.g.a | ✓ | 32 |
15.d | odd | 2 | 1 | inner | 945.2.g.a | ✓ | 32 |
21.c | even | 2 | 1 | inner | 945.2.g.a | ✓ | 32 |
35.c | odd | 2 | 1 | inner | 945.2.g.a | ✓ | 32 |
105.g | even | 2 | 1 | inner | 945.2.g.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
945.2.g.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
945.2.g.a | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
945.2.g.a | ✓ | 32 | 5.b | even | 2 | 1 | inner |
945.2.g.a | ✓ | 32 | 7.b | odd | 2 | 1 | inner |
945.2.g.a | ✓ | 32 | 15.d | odd | 2 | 1 | inner |
945.2.g.a | ✓ | 32 | 21.c | even | 2 | 1 | inner |
945.2.g.a | ✓ | 32 | 35.c | odd | 2 | 1 | inner |
945.2.g.a | ✓ | 32 | 105.g | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 12T_{2}^{6} + 42T_{2}^{4} - 45T_{2}^{2} + 6 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\).