Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,4,Mod(109,378)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(378, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.109");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 378.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: | \(22.3027219822\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.11184604443.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 43x^{4} - 210x^{3} + 1849x^{2} - 4515x + 11025 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} + 43x^{4} - 210x^{3} + 1849x^{2} - 4515x + 11025 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 43\nu^{3} - 105\nu^{2} + 1849\nu ) / 4515 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 4\nu^{3} + 43\nu^{2} - 62\nu + 827 ) / 43 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 3\nu^{3} + 43\nu^{2} - 148\nu + 932 ) / 43 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -29\nu^{5} - 827\nu^{3} + 7560\nu^{2} - 40076\nu + 86835 ) / 4515 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -29\nu^{5} - 932\nu^{3} + 7560\nu^{2} - 35561\nu + 97860 ) / 4515 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{5} - 2\beta_{4} + 7\beta_{3} - 7\beta_{2} + 87\beta _1 - 87 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -86\beta_{5} + 86\beta_{4} - 43\beta_{3} + 43\beta_{2} + 315 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 191\beta_{5} - 320\beta_{4} - 191\beta_{3} + 320\beta_{2} - 3741\beta_1 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2374\beta_{5} - 2059\beta_{4} + 4433\beta_{3} - 4433\beta_{2} + 22680\beta _1 - 22680 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(325\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
−1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −6.70319 | − | 11.6103i | 0 | 5.32531 | − | 17.7381i | 8.00000 | 0 | −13.4064 | + | 23.2205i | ||||||||||||||||||||||||||
| 109.2 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 0.301070 | + | 0.521469i | 0 | −17.6665 | + | 5.55825i | 8.00000 | 0 | 0.602141 | − | 1.04294i | |||||||||||||||||||||||||||
| 109.3 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 3.90212 | + | 6.75867i | 0 | 16.3412 | + | 8.71578i | 8.00000 | 0 | 7.80425 | − | 13.5173i | |||||||||||||||||||||||||||
| 163.1 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −6.70319 | + | 11.6103i | 0 | 5.32531 | + | 17.7381i | 8.00000 | 0 | −13.4064 | − | 23.2205i | |||||||||||||||||||||||||||
| 163.2 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 0.301070 | − | 0.521469i | 0 | −17.6665 | − | 5.55825i | 8.00000 | 0 | 0.602141 | + | 1.04294i | |||||||||||||||||||||||||||
| 163.3 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 3.90212 | − | 6.75867i | 0 | 16.3412 | − | 8.71578i | 8.00000 | 0 | 7.80425 | + | 13.5173i | |||||||||||||||||||||||||||
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 | 378.4.g.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 378.4.g.b | yes | 6 | |
| 7.c | even | 3 | 1 | inner | 378.4.g.a | ✓ | 6 |
| 21.h | odd | 6 | 1 | 378.4.g.b | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 378.4.g.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 378.4.g.a | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 378.4.g.b | yes | 6 | 3.b | odd | 2 | 1 | |
| 378.4.g.b | yes | 6 | 21.h | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 5T_{5}^{5} + 133T_{5}^{4} - 666T_{5}^{3} + 11349T_{5}^{2} - 6804T_{5} + 3969 \)
acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 5 T^{5} + \cdots + 3969 \)
|
| $7$ |
\( T^{6} - 8 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} - 29 T^{5} + \cdots + 4862025 \)
|
| $13$ |
\( (T^{3} - 10 T^{2} + \cdots + 603)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 2114252361 \)
|
| $19$ |
\( T^{6} + \cdots + 19329618961 \)
|
| $23$ |
\( T^{6} + \cdots + 148635694089 \)
|
| $29$ |
\( (T^{3} + 181 T^{2} + \cdots + 862407)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 27799667506521 \)
|
| $37$ |
\( T^{6} + \cdots + 1957974001 \)
|
| $41$ |
\( (T^{3} - 216 T^{2} + \cdots - 164025)^{2} \)
|
| $43$ |
\( (T^{3} + 363 T^{2} + \cdots - 6611597)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 2568361041 \)
|
| $53$ |
\( T^{6} + \cdots + 13\!\cdots\!61 \)
|
| $59$ |
\( T^{6} + \cdots + 15\!\cdots\!69 \)
|
| $61$ |
\( T^{6} + \cdots + 1548419366025 \)
|
| $67$ |
\( T^{6} + \cdots + 70\!\cdots\!25 \)
|
| $71$ |
\( (T^{3} + 395 T^{2} + \cdots - 115887969)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 202747156022761 \)
|
| $79$ |
\( T^{6} + \cdots + 21738803675121 \)
|
| $83$ |
\( (T^{3} + 77 T^{2} + \cdots + 172613511)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 37\!\cdots\!69 \)
|
| $97$ |
\( (T^{3} - 590 T^{2} + \cdots + 67884200)^{2} \)
|
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