This is the first CM form with weight at least $2$, when ordered by analytic conductor.
Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,3,Mod(6,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.6");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.b (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: | yes |
| Analytic conductor: | \(0.190736185052\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Expression as an eta quotient
\(f(z) = \eta(z)^{3}\eta(7z)^{3}=q\prod_{n=1}^\infty(1 - q^{n})^{3}(1 - q^{7n})^{3}\)
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6.1 |
|
−3.00000 | 0 | 5.00000 | 0 | 0 | −7.00000 | −3.00000 | 9.00000 | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7.3.b.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | 63.3.d.a | 1 | ||
| 4.b | odd | 2 | 1 | 112.3.c.a | 1 | ||
| 5.b | even | 2 | 1 | 175.3.d.a | 1 | ||
| 5.c | odd | 4 | 2 | 175.3.c.a | 2 | ||
| 7.b | odd | 2 | 1 | CM | 7.3.b.a | ✓ | 1 |
| 7.c | even | 3 | 2 | 49.3.d.a | 2 | ||
| 7.d | odd | 6 | 2 | 49.3.d.a | 2 | ||
| 8.b | even | 2 | 1 | 448.3.c.a | 1 | ||
| 8.d | odd | 2 | 1 | 448.3.c.b | 1 | ||
| 12.b | even | 2 | 1 | 1008.3.f.a | 1 | ||
| 21.c | even | 2 | 1 | 63.3.d.a | 1 | ||
| 21.g | even | 6 | 2 | 441.3.m.a | 2 | ||
| 21.h | odd | 6 | 2 | 441.3.m.a | 2 | ||
| 28.d | even | 2 | 1 | 112.3.c.a | 1 | ||
| 28.f | even | 6 | 2 | 784.3.s.a | 2 | ||
| 28.g | odd | 6 | 2 | 784.3.s.a | 2 | ||
| 35.c | odd | 2 | 1 | 175.3.d.a | 1 | ||
| 35.f | even | 4 | 2 | 175.3.c.a | 2 | ||
| 56.e | even | 2 | 1 | 448.3.c.b | 1 | ||
| 56.h | odd | 2 | 1 | 448.3.c.a | 1 | ||
| 84.h | odd | 2 | 1 | 1008.3.f.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.3.b.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 7.3.b.a | ✓ | 1 | 7.b | odd | 2 | 1 | CM |
| 49.3.d.a | 2 | 7.c | even | 3 | 2 | ||
| 49.3.d.a | 2 | 7.d | odd | 6 | 2 | ||
| 63.3.d.a | 1 | 3.b | odd | 2 | 1 | ||
| 63.3.d.a | 1 | 21.c | even | 2 | 1 | ||
| 112.3.c.a | 1 | 4.b | odd | 2 | 1 | ||
| 112.3.c.a | 1 | 28.d | even | 2 | 1 | ||
| 175.3.c.a | 2 | 5.c | odd | 4 | 2 | ||
| 175.3.c.a | 2 | 35.f | even | 4 | 2 | ||
| 175.3.d.a | 1 | 5.b | even | 2 | 1 | ||
| 175.3.d.a | 1 | 35.c | odd | 2 | 1 | ||
| 441.3.m.a | 2 | 21.g | even | 6 | 2 | ||
| 441.3.m.a | 2 | 21.h | odd | 6 | 2 | ||
| 448.3.c.a | 1 | 8.b | even | 2 | 1 | ||
| 448.3.c.a | 1 | 56.h | odd | 2 | 1 | ||
| 448.3.c.b | 1 | 8.d | odd | 2 | 1 | ||
| 448.3.c.b | 1 | 56.e | even | 2 | 1 | ||
| 784.3.s.a | 2 | 28.f | even | 6 | 2 | ||
| 784.3.s.a | 2 | 28.g | odd | 6 | 2 | ||
| 1008.3.f.a | 1 | 12.b | even | 2 | 1 | ||
| 1008.3.f.a | 1 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(7, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 3 \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T + 7 \)
|
| $11$ |
\( T + 6 \)
|
| $13$ |
\( T \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T \)
|
| $23$ |
\( T - 18 \)
|
| $29$ |
\( T + 54 \)
|
| $31$ |
\( T \)
|
| $37$ |
\( T + 38 \)
|
| $41$ |
\( T \)
|
| $43$ |
\( T - 58 \)
|
| $47$ |
\( T \)
|
| $53$ |
\( T + 6 \)
|
| $59$ |
\( T \)
|
| $61$ |
\( T \)
|
| $67$ |
\( T + 118 \)
|
| $71$ |
\( T - 114 \)
|
| $73$ |
\( T \)
|
| $79$ |
\( T + 94 \)
|
| $83$ |
\( T \)
|
| $89$ |
\( T \)
|
| $97$ |
\( T \)
|
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