Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,4,Mod(127,378)]
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([4, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.127");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 378.f (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: | \(22.3027219822\) |
| 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: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| 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
\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} + 5\nu^{3} - 2\nu^{2} ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} - 11\nu^{3} + 17\nu^{2} - 12\nu + 3 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 16\nu^{3} - 19\nu^{2} + 21\nu - 6 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} + 8\nu^{2} - 7\nu + 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7\nu^{5} - 16\nu^{4} + 62\nu^{3} - 68\nu^{2} + 99\nu - 30 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - 2\beta _1 - 6 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 6\beta_{3} - 4\beta_{2} + \beta _1 - 6 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{5} - 3\beta_{4} - 13\beta_{3} - \beta_{2} + 11\beta _1 + 19 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3\beta_{5} - 6\beta_{4} + 19\beta_{3} + 19\beta_{2} + 11\beta _1 + 37 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(325\) |
| \(\chi(n)\) | \(-1 + \beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 127.1 |
|
−1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −3.21780 | − | 5.57339i | 0 | −3.50000 | + | 6.06218i | 8.00000 | 0 | 12.8712 | ||||||||||||||||||||||||||||
| 127.2 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 3.11273 | + | 5.39140i | 0 | −3.50000 | + | 6.06218i | 8.00000 | 0 | −12.4509 | |||||||||||||||||||||||||||||
| 127.3 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 4.60507 | + | 7.97622i | 0 | −3.50000 | + | 6.06218i | 8.00000 | 0 | −18.4203 | |||||||||||||||||||||||||||||
| 253.1 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −3.21780 | + | 5.57339i | 0 | −3.50000 | − | 6.06218i | 8.00000 | 0 | 12.8712 | |||||||||||||||||||||||||||||
| 253.2 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 3.11273 | − | 5.39140i | 0 | −3.50000 | − | 6.06218i | 8.00000 | 0 | −12.4509 | |||||||||||||||||||||||||||||
| 253.3 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 4.60507 | − | 7.97622i | 0 | −3.50000 | − | 6.06218i | 8.00000 | 0 | −18.4203 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.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.f.a | 6 | |
| 3.b | odd | 2 | 1 | 126.4.f.a | ✓ | 6 | |
| 9.c | even | 3 | 1 | inner | 378.4.f.a | 6 | |
| 9.c | even | 3 | 1 | 1134.4.a.h | 3 | ||
| 9.d | odd | 6 | 1 | 126.4.f.a | ✓ | 6 | |
| 9.d | odd | 6 | 1 | 1134.4.a.g | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.4.f.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 126.4.f.a | ✓ | 6 | 9.d | odd | 6 | 1 | |
| 378.4.f.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 378.4.f.a | 6 | 9.c | even | 3 | 1 | inner | |
| 1134.4.a.g | 3 | 9.d | odd | 6 | 1 | ||
| 1134.4.a.h | 3 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 9T_{5}^{5} + 123T_{5}^{4} - 360T_{5}^{3} + 5085T_{5}^{2} - 15498T_{5} + 136161 \)
acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 9 T^{5} + \cdots + 136161 \)
|
| $7$ |
\( (T^{2} + 7 T + 49)^{3} \)
|
| $11$ |
\( T^{6} - 48 T^{5} + \cdots + 79548561 \)
|
| $13$ |
\( T^{6} + \cdots + 1824400369 \)
|
| $17$ |
\( (T^{3} + 24 T^{2} + \cdots - 157239)^{2} \)
|
| $19$ |
\( (T^{3} + 141 T^{2} + \cdots - 106169)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 104760326889 \)
|
| $29$ |
\( T^{6} + \cdots + 78328896129 \)
|
| $31$ |
\( T^{6} - 165 T^{5} + \cdots + 559369801 \)
|
| $37$ |
\( (T^{3} + 399 T^{2} + \cdots - 18438443)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 264917119401 \)
|
| $43$ |
\( T^{6} + \cdots + 40798917084409 \)
|
| $47$ |
\( T^{6} + \cdots + 57\!\cdots\!61 \)
|
| $53$ |
\( (T^{3} + 522 T^{2} + \cdots - 106645401)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 71\!\cdots\!61 \)
|
| $61$ |
\( T^{6} + \cdots + 837556418281489 \)
|
| $67$ |
\( T^{6} + \cdots + 16\!\cdots\!69 \)
|
| $71$ |
\( (T^{3} + 57 T^{2} + \cdots + 141832269)^{2} \)
|
| $73$ |
\( (T^{3} + 117 T^{2} + \cdots - 225440003)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 32\!\cdots\!89 \)
|
| $83$ |
\( T^{6} + \cdots + 43\!\cdots\!89 \)
|
| $89$ |
\( (T^{3} - 648 T^{2} + \cdots + 21052413)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 66\!\cdots\!09 \)
|
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