Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,6,Mod(37,126)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(126, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.37");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.g (of order \(3\), degree \(2\), 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: | \(20.2083612964\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-1 + \zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−2.00000 | + | 3.46410i | 0 | −8.00000 | − | 13.8564i | −6.00000 | + | 10.3923i | 0 | 59.5000 | + | 115.181i | 64.0000 | 0 | −24.0000 | − | 41.5692i | ||||||||||||||
| 109.1 | −2.00000 | − | 3.46410i | 0 | −8.00000 | + | 13.8564i | −6.00000 | − | 10.3923i | 0 | 59.5000 | − | 115.181i | 64.0000 | 0 | −24.0000 | + | 41.5692i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 126.6.g.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 126.6.g.d | yes | 2 | |
| 7.c | even | 3 | 1 | inner | 126.6.g.a | ✓ | 2 |
| 7.c | even | 3 | 1 | 882.6.a.r | 1 | ||
| 7.d | odd | 6 | 1 | 882.6.a.q | 1 | ||
| 21.g | even | 6 | 1 | 882.6.a.h | 1 | ||
| 21.h | odd | 6 | 1 | 126.6.g.d | yes | 2 | |
| 21.h | odd | 6 | 1 | 882.6.a.d | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.6.g.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 126.6.g.a | ✓ | 2 | 7.c | even | 3 | 1 | inner |
| 126.6.g.d | yes | 2 | 3.b | odd | 2 | 1 | |
| 126.6.g.d | yes | 2 | 21.h | odd | 6 | 1 | |
| 882.6.a.d | 1 | 21.h | odd | 6 | 1 | ||
| 882.6.a.h | 1 | 21.g | even | 6 | 1 | ||
| 882.6.a.q | 1 | 7.d | odd | 6 | 1 | ||
| 882.6.a.r | 1 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 12T_{5} + 144 \)
acting on \(S_{6}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 4T + 16 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 12T + 144 \)
|
| $7$ |
\( T^{2} - 119T + 16807 \)
|
| $11$ |
\( T^{2} + 288T + 82944 \)
|
| $13$ |
\( (T - 737)^{2} \)
|
| $17$ |
\( T^{2} - 156T + 24336 \)
|
| $19$ |
\( T^{2} + 617T + 380689 \)
|
| $23$ |
\( T^{2} + 4596 T + 21123216 \)
|
| $29$ |
\( (T + 5304)^{2} \)
|
| $31$ |
\( T^{2} + 2513 T + 6315169 \)
|
| $37$ |
\( T^{2} + 2375 T + 5640625 \)
|
| $41$ |
\( (T + 14280)^{2} \)
|
| $43$ |
\( (T + 1579)^{2} \)
|
| $47$ |
\( T^{2} + 17268 T + 298183824 \)
|
| $53$ |
\( T^{2} - 18612 T + 346406544 \)
|
| $59$ |
\( T^{2} + 28428 T + 808151184 \)
|
| $61$ |
\( T^{2} + 15566 T + 242300356 \)
|
| $67$ |
\( T^{2} - 8053 T + 64850809 \)
|
| $71$ |
\( (T - 13020)^{2} \)
|
| $73$ |
\( T^{2} + \cdots + 2526369169 \)
|
| $79$ |
\( T^{2} + 30155 T + 909324025 \)
|
| $83$ |
\( (T - 99276)^{2} \)
|
| $89$ |
\( T^{2} + \cdots + 2714826816 \)
|
| $97$ |
\( (T - 116222)^{2} \)
|
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