Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,6,Mod(7,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.7");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 38.c (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: | \(6.09458515289\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 133x^{4} - 60x^{3} + 17689x^{2} - 3990x + 900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{6} + 133x^{4} - 60x^{3} + 17689x^{2} - 3990x + 900 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 60 ) / 133 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 133\nu^{3} - 30\nu^{2} + 17689\nu ) / 3990 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 133\nu^{2} - 30\nu + 11837 ) / 133 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -89\nu^{5} - 11837\nu^{3} + 6660\nu^{2} - 1574321\nu + 355110 ) / 3990 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 89\beta_{3} - 89 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 133\beta_{2} + 60 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -133\beta_{5} + 133\beta_{4} - 11837\beta_{3} + 15\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 60\beta_{5} + 13320\beta_{3} - 17689\beta_{2} - 17689\beta _1 - 13320 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) |
| \(\chi(n)\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
2.00000 | + | 3.46410i | −13.9181 | − | 24.1068i | −8.00000 | + | 13.8564i | 34.6044 | + | 59.9365i | 55.6723 | − | 96.4273i | −195.345 | −64.0000 | −265.926 | + | 460.597i | −138.418 | + | 239.746i | ||||||||||||||||||||||
| 7.2 | 2.00000 | + | 3.46410i | −2.72565 | − | 4.72096i | −8.00000 | + | 13.8564i | −41.7489 | − | 72.3112i | 10.9026 | − | 18.8839i | −105.805 | −64.0000 | 106.642 | − | 184.709i | 166.996 | − | 289.245i | |||||||||||||||||||||||
| 7.3 | 2.00000 | + | 3.46410i | 9.14373 | + | 15.8374i | −8.00000 | + | 13.8564i | 14.1445 | + | 24.4990i | −36.5749 | + | 63.3496i | −10.8501 | −64.0000 | −45.7157 | + | 79.1819i | −56.5781 | + | 97.9961i | |||||||||||||||||||||||
| 11.1 | 2.00000 | − | 3.46410i | −13.9181 | + | 24.1068i | −8.00000 | − | 13.8564i | 34.6044 | − | 59.9365i | 55.6723 | + | 96.4273i | −195.345 | −64.0000 | −265.926 | − | 460.597i | −138.418 | − | 239.746i | |||||||||||||||||||||||
| 11.2 | 2.00000 | − | 3.46410i | −2.72565 | + | 4.72096i | −8.00000 | − | 13.8564i | −41.7489 | + | 72.3112i | 10.9026 | + | 18.8839i | −105.805 | −64.0000 | 106.642 | + | 184.709i | 166.996 | + | 289.245i | |||||||||||||||||||||||
| 11.3 | 2.00000 | − | 3.46410i | 9.14373 | − | 15.8374i | −8.00000 | − | 13.8564i | 14.1445 | − | 24.4990i | −36.5749 | − | 63.3496i | −10.8501 | −64.0000 | −45.7157 | − | 79.1819i | −56.5781 | − | 97.9961i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 38.6.c.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 342.6.g.a | 6 | ||
| 4.b | odd | 2 | 1 | 304.6.i.a | 6 | ||
| 19.c | even | 3 | 1 | inner | 38.6.c.a | ✓ | 6 |
| 19.c | even | 3 | 1 | 722.6.a.e | 3 | ||
| 19.d | odd | 6 | 1 | 722.6.a.f | 3 | ||
| 57.h | odd | 6 | 1 | 342.6.g.a | 6 | ||
| 76.g | odd | 6 | 1 | 304.6.i.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.6.c.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 38.6.c.a | ✓ | 6 | 19.c | even | 3 | 1 | inner |
| 304.6.i.a | 6 | 4.b | odd | 2 | 1 | ||
| 304.6.i.a | 6 | 76.g | odd | 6 | 1 | ||
| 342.6.g.a | 6 | 3.b | odd | 2 | 1 | ||
| 342.6.g.a | 6 | 57.h | odd | 6 | 1 | ||
| 722.6.a.e | 3 | 19.c | even | 3 | 1 | ||
| 722.6.a.f | 3 | 19.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 15T_{3}^{5} + 682T_{3}^{4} - 1305T_{3}^{3} + 250474T_{3}^{2} + 1268175T_{3} + 7700625 \)
acting on \(S_{6}^{\mathrm{new}}(38, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 4 T + 16)^{3} \)
|
| $3$ |
\( T^{6} + 15 T^{5} + \cdots + 7700625 \)
|
| $5$ |
\( T^{6} + \cdots + 26724402576 \)
|
| $7$ |
\( (T^{3} + 312 T^{2} + \cdots + 224256)^{2} \)
|
| $11$ |
\( (T^{3} + 469 T^{2} + \cdots + 2905680)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 394380754796484 \)
|
| $17$ |
\( T^{6} + \cdots + 45\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots + 15\!\cdots\!99 \)
|
| $23$ |
\( T^{6} + \cdots + 39\!\cdots\!56 \)
|
| $29$ |
\( T^{6} + \cdots + 46\!\cdots\!56 \)
|
| $31$ |
\( (T^{3} - 198 T^{2} + \cdots + 281768267232)^{2} \)
|
| $37$ |
\( (T^{3} - 10030 T^{2} + \cdots + 34115602072)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 16\!\cdots\!29 \)
|
| $43$ |
\( T^{6} + \cdots + 26\!\cdots\!04 \)
|
| $47$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 40\!\cdots\!25 \)
|
| $61$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 29\!\cdots\!21 \)
|
| $71$ |
\( T^{6} + \cdots + 44\!\cdots\!16 \)
|
| $73$ |
\( T^{6} + \cdots + 56\!\cdots\!21 \)
|
| $79$ |
\( T^{6} + \cdots + 60\!\cdots\!00 \)
|
| $83$ |
\( (T^{3} + \cdots - 518524918263648)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 33\!\cdots\!21 \)
|
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