Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [720,6,Mod(719,720)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(720, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("720.719");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 720.o (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: | \(115.476350265\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
719.1 | 0 | 0 | 0 | −55.8906 | − | 1.11466i | 0 | −188.577 | 0 | 0 | 0 | ||||||||||||||||
719.2 | 0 | 0 | 0 | −55.8906 | − | 1.11466i | 0 | 188.577 | 0 | 0 | 0 | ||||||||||||||||
719.3 | 0 | 0 | 0 | −55.8906 | + | 1.11466i | 0 | −188.577 | 0 | 0 | 0 | ||||||||||||||||
719.4 | 0 | 0 | 0 | −55.8906 | + | 1.11466i | 0 | 188.577 | 0 | 0 | 0 | ||||||||||||||||
719.5 | 0 | 0 | 0 | −48.8519 | − | 27.1752i | 0 | −21.9879 | 0 | 0 | 0 | ||||||||||||||||
719.6 | 0 | 0 | 0 | −48.8519 | − | 27.1752i | 0 | 21.9879 | 0 | 0 | 0 | ||||||||||||||||
719.7 | 0 | 0 | 0 | −48.8519 | + | 27.1752i | 0 | −21.9879 | 0 | 0 | 0 | ||||||||||||||||
719.8 | 0 | 0 | 0 | −48.8519 | + | 27.1752i | 0 | 21.9879 | 0 | 0 | 0 | ||||||||||||||||
719.9 | 0 | 0 | 0 | −42.7952 | − | 35.9662i | 0 | −7.21390 | 0 | 0 | 0 | ||||||||||||||||
719.10 | 0 | 0 | 0 | −42.7952 | − | 35.9662i | 0 | 7.21390 | 0 | 0 | 0 | ||||||||||||||||
719.11 | 0 | 0 | 0 | −42.7952 | + | 35.9662i | 0 | −7.21390 | 0 | 0 | 0 | ||||||||||||||||
719.12 | 0 | 0 | 0 | −42.7952 | + | 35.9662i | 0 | 7.21390 | 0 | 0 | 0 | ||||||||||||||||
719.13 | 0 | 0 | 0 | −25.8892 | − | 49.5454i | 0 | −154.886 | 0 | 0 | 0 | ||||||||||||||||
719.14 | 0 | 0 | 0 | −25.8892 | − | 49.5454i | 0 | 154.886 | 0 | 0 | 0 | ||||||||||||||||
719.15 | 0 | 0 | 0 | −25.8892 | + | 49.5454i | 0 | −154.886 | 0 | 0 | 0 | ||||||||||||||||
719.16 | 0 | 0 | 0 | −25.8892 | + | 49.5454i | 0 | 154.886 | 0 | 0 | 0 | ||||||||||||||||
719.17 | 0 | 0 | 0 | −21.5882 | − | 51.5650i | 0 | −168.759 | 0 | 0 | 0 | ||||||||||||||||
719.18 | 0 | 0 | 0 | −21.5882 | − | 51.5650i | 0 | 168.759 | 0 | 0 | 0 | ||||||||||||||||
719.19 | 0 | 0 | 0 | −21.5882 | + | 51.5650i | 0 | −168.759 | 0 | 0 | 0 | ||||||||||||||||
719.20 | 0 | 0 | 0 | −21.5882 | + | 51.5650i | 0 | 168.759 | 0 | 0 | 0 | ||||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
60.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 720.6.o.d | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 720.6.o.d | ✓ | 40 |
4.b | odd | 2 | 1 | inner | 720.6.o.d | ✓ | 40 |
5.b | even | 2 | 1 | inner | 720.6.o.d | ✓ | 40 |
12.b | even | 2 | 1 | inner | 720.6.o.d | ✓ | 40 |
15.d | odd | 2 | 1 | inner | 720.6.o.d | ✓ | 40 |
20.d | odd | 2 | 1 | inner | 720.6.o.d | ✓ | 40 |
60.h | even | 2 | 1 | inner | 720.6.o.d | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
720.6.o.d | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
720.6.o.d | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
720.6.o.d | ✓ | 40 | 4.b | odd | 2 | 1 | inner |
720.6.o.d | ✓ | 40 | 5.b | even | 2 | 1 | inner |
720.6.o.d | ✓ | 40 | 12.b | even | 2 | 1 | inner |
720.6.o.d | ✓ | 40 | 15.d | odd | 2 | 1 | inner |
720.6.o.d | ✓ | 40 | 20.d | odd | 2 | 1 | inner |
720.6.o.d | ✓ | 40 | 60.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{10} - 88566 T_{7}^{8} + 2596254156 T_{7}^{6} - 25663288906440 T_{7}^{4} + \cdots - 61\!\cdots\!52 \)
acting on \(S_{6}^{\mathrm{new}}(720, [\chi])\).