Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1064,2,Mod(265,1064)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1064, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("1064.265");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1064 = 2^{3} \cdot 7 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1064.m (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: | \(8.49608277506\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
265.1 | 0 | −2.94661 | 0 | − | 1.44192i | 0 | −0.743572 | − | 2.53911i | 0 | 5.68249 | 0 | |||||||||||||||
265.2 | 0 | −2.94661 | 0 | 1.44192i | 0 | −0.743572 | + | 2.53911i | 0 | 5.68249 | 0 | ||||||||||||||||
265.3 | 0 | −2.88400 | 0 | 0.977713i | 0 | 2.04655 | + | 1.67679i | 0 | 5.31745 | 0 | ||||||||||||||||
265.4 | 0 | −2.88400 | 0 | − | 0.977713i | 0 | 2.04655 | − | 1.67679i | 0 | 5.31745 | 0 | |||||||||||||||
265.5 | 0 | −2.23395 | 0 | − | 4.18039i | 0 | 2.56370 | − | 0.653799i | 0 | 1.99054 | 0 | |||||||||||||||
265.6 | 0 | −2.23395 | 0 | 4.18039i | 0 | 2.56370 | + | 0.653799i | 0 | 1.99054 | 0 | ||||||||||||||||
265.7 | 0 | −1.94588 | 0 | 1.86531i | 0 | 1.44774 | − | 2.21451i | 0 | 0.786449 | 0 | ||||||||||||||||
265.8 | 0 | −1.94588 | 0 | − | 1.86531i | 0 | 1.44774 | + | 2.21451i | 0 | 0.786449 | 0 | |||||||||||||||
265.9 | 0 | −1.42796 | 0 | 1.88393i | 0 | −2.55095 | − | 0.701880i | 0 | −0.960935 | 0 | ||||||||||||||||
265.10 | 0 | −1.42796 | 0 | − | 1.88393i | 0 | −2.55095 | + | 0.701880i | 0 | −0.960935 | 0 | |||||||||||||||
265.11 | 0 | −0.428956 | 0 | 2.73144i | 0 | −0.763462 | + | 2.53320i | 0 | −2.81600 | 0 | ||||||||||||||||
265.12 | 0 | −0.428956 | 0 | − | 2.73144i | 0 | −0.763462 | − | 2.53320i | 0 | −2.81600 | 0 | |||||||||||||||
265.13 | 0 | 0.428956 | 0 | 2.73144i | 0 | −0.763462 | + | 2.53320i | 0 | −2.81600 | 0 | ||||||||||||||||
265.14 | 0 | 0.428956 | 0 | − | 2.73144i | 0 | −0.763462 | − | 2.53320i | 0 | −2.81600 | 0 | |||||||||||||||
265.15 | 0 | 1.42796 | 0 | 1.88393i | 0 | −2.55095 | − | 0.701880i | 0 | −0.960935 | 0 | ||||||||||||||||
265.16 | 0 | 1.42796 | 0 | − | 1.88393i | 0 | −2.55095 | + | 0.701880i | 0 | −0.960935 | 0 | |||||||||||||||
265.17 | 0 | 1.94588 | 0 | 1.86531i | 0 | 1.44774 | − | 2.21451i | 0 | 0.786449 | 0 | ||||||||||||||||
265.18 | 0 | 1.94588 | 0 | − | 1.86531i | 0 | 1.44774 | + | 2.21451i | 0 | 0.786449 | 0 | |||||||||||||||
265.19 | 0 | 2.23395 | 0 | − | 4.18039i | 0 | 2.56370 | − | 0.653799i | 0 | 1.99054 | 0 | |||||||||||||||
265.20 | 0 | 2.23395 | 0 | 4.18039i | 0 | 2.56370 | + | 0.653799i | 0 | 1.99054 | 0 | ||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
133.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1064.2.m.b | ✓ | 24 |
4.b | odd | 2 | 1 | 2128.2.m.h | 24 | ||
7.b | odd | 2 | 1 | inner | 1064.2.m.b | ✓ | 24 |
19.b | odd | 2 | 1 | inner | 1064.2.m.b | ✓ | 24 |
28.d | even | 2 | 1 | 2128.2.m.h | 24 | ||
76.d | even | 2 | 1 | 2128.2.m.h | 24 | ||
133.c | even | 2 | 1 | inner | 1064.2.m.b | ✓ | 24 |
532.b | odd | 2 | 1 | 2128.2.m.h | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1064.2.m.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
1064.2.m.b | ✓ | 24 | 7.b | odd | 2 | 1 | inner |
1064.2.m.b | ✓ | 24 | 19.b | odd | 2 | 1 | inner |
1064.2.m.b | ✓ | 24 | 133.c | even | 2 | 1 | inner |
2128.2.m.h | 24 | 4.b | odd | 2 | 1 | ||
2128.2.m.h | 24 | 28.d | even | 2 | 1 | ||
2128.2.m.h | 24 | 76.d | even | 2 | 1 | ||
2128.2.m.h | 24 | 532.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 28T_{3}^{10} + 298T_{3}^{8} - 1499T_{3}^{6} + 3578T_{3}^{4} - 3392T_{3}^{2} + 512 \)
acting on \(S_{2}^{\mathrm{new}}(1064, [\chi])\).