Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,2,Mod(37,378)]
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([2, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 378.e (of order \(3\), degree \(2\), not 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: | \(3.01834519640\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.309123.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 126) |
| 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
\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(325\) |
| \(\chi(n)\) | \(-\beta_{4}\) | \(-\beta_{4}\) |
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 |
|
1.00000 | 0 | 1.00000 | −1.59097 | − | 2.75564i | 0 | −2.56238 | − | 0.658939i | 1.00000 | 0 | −1.59097 | − | 2.75564i | ||||||||||||||||||||||||||||||
| 37.2 | 1.00000 | 0 | 1.00000 | 0.296790 | + | 0.514055i | 0 | 2.32383 | + | 1.26483i | 1.00000 | 0 | 0.296790 | + | 0.514055i | |||||||||||||||||||||||||||||||
| 37.3 | 1.00000 | 0 | 1.00000 | 0.794182 | + | 1.37556i | 0 | 1.23855 | − | 2.33795i | 1.00000 | 0 | 0.794182 | + | 1.37556i | |||||||||||||||||||||||||||||||
| 235.1 | 1.00000 | 0 | 1.00000 | −1.59097 | + | 2.75564i | 0 | −2.56238 | + | 0.658939i | 1.00000 | 0 | −1.59097 | + | 2.75564i | |||||||||||||||||||||||||||||||
| 235.2 | 1.00000 | 0 | 1.00000 | 0.296790 | − | 0.514055i | 0 | 2.32383 | − | 1.26483i | 1.00000 | 0 | 0.296790 | − | 0.514055i | |||||||||||||||||||||||||||||||
| 235.3 | 1.00000 | 0 | 1.00000 | 0.794182 | − | 1.37556i | 0 | 1.23855 | + | 2.33795i | 1.00000 | 0 | 0.794182 | − | 1.37556i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.h | even | 3 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + T_{5}^{5} + 7T_{5}^{4} - 12T_{5}^{3} + 33T_{5}^{2} - 18T_{5} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + T^{5} + 7 T^{4} + \cdots + 9 \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots + 343 \)
|
| $11$ |
\( T^{6} - T^{5} + 7 T^{4} + \cdots + 9 \)
|
| $13$ |
\( T^{6} - 8 T^{5} + \cdots + 4761 \)
|
| $17$ |
\( T^{6} - 4 T^{5} + \cdots + 576 \)
|
| $19$ |
\( T^{6} + 3 T^{5} + \cdots + 2401 \)
|
| $23$ |
\( T^{6} - 7 T^{5} + \cdots + 9 \)
|
| $29$ |
\( T^{6} - 5 T^{5} + \cdots + 131769 \)
|
| $31$ |
\( (T^{3} + 20 T^{2} + \cdots + 201)^{2} \)
|
| $37$ |
\( (T^{2} - T + 1)^{3} \)
|
| $41$ |
\( T^{6} + 33 T^{4} + \cdots + 81 \)
|
| $43$ |
\( T^{6} + 6 T^{5} + \cdots + 16129 \)
|
| $47$ |
\( (T^{3} + 9 T^{2} + \cdots - 189)^{2} \)
|
| $53$ |
\( T^{6} + 15 T^{5} + \cdots + 6561 \)
|
| $59$ |
\( (T^{3} + 14 T^{2} + \cdots - 63)^{2} \)
|
| $61$ |
\( (T^{3} + 8 T^{2} - 5 T - 93)^{2} \)
|
| $67$ |
\( (T^{3} + T^{2} - 112 T - 211)^{2} \)
|
| $71$ |
\( (T^{3} + 7 T^{2} + \cdots - 1593)^{2} \)
|
| $73$ |
\( T^{6} - 19 T^{5} + \cdots + 398161 \)
|
| $79$ |
\( (T^{3} + 5 T^{2} + \cdots - 321)^{2} \)
|
| $83$ |
\( T^{6} + 2 T^{5} + \cdots + 21609 \)
|
| $89$ |
\( T^{6} - 9 T^{5} + \cdots + 81 \)
|
| $97$ |
\( T^{6} - 28 T^{5} + \cdots + 61504 \)
|
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