Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [39,6,Mod(1,39)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 39.a (trivial) |
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: | yes |
| Analytic conductor: | \(6.25496897271\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.125308.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 55x - 78 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 55x - 78 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 37 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 37 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−6.56673 | −9.00000 | 11.1219 | −34.5338 | 59.1006 | −207.734 | 137.101 | 81.0000 | 226.774 | |||||||||||||||||||||||||||
| 1.2 | −1.47673 | −9.00000 | −29.8193 | 6.18613 | 13.2906 | 190.614 | 91.2906 | 81.0000 | −9.13526 | ||||||||||||||||||||||||||||
| 1.3 | 8.04346 | −9.00000 | 32.6973 | 82.3477 | −72.3912 | 101.120 | 5.60882 | 81.0000 | 662.361 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(13\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 39.6.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 117.6.a.e | 3 | ||
| 4.b | odd | 2 | 1 | 624.6.a.t | 3 | ||
| 5.b | even | 2 | 1 | 975.6.a.d | 3 | ||
| 13.b | even | 2 | 1 | 507.6.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.6.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 117.6.a.e | 3 | 3.b | odd | 2 | 1 | ||
| 507.6.a.d | 3 | 13.b | even | 2 | 1 | ||
| 624.6.a.t | 3 | 4.b | odd | 2 | 1 | ||
| 975.6.a.d | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 55T_{2} - 78 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(39))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 55T - 78 \)
|
| $3$ |
\( (T + 9)^{3} \)
|
| $5$ |
\( T^{3} - 54 T^{2} + \cdots + 17592 \)
|
| $7$ |
\( T^{3} - 84 T^{2} + \cdots + 4004032 \)
|
| $11$ |
\( T^{3} - 876 T^{2} + \cdots - 12661440 \)
|
| $13$ |
\( (T - 169)^{3} \)
|
| $17$ |
\( T^{3} + \cdots + 1885945464 \)
|
| $19$ |
\( T^{3} + 16 T^{2} + \cdots + 681737472 \)
|
| $23$ |
\( T^{3} + \cdots - 25559359488 \)
|
| $29$ |
\( T^{3} + \cdots + 90911765352 \)
|
| $31$ |
\( T^{3} + \cdots - 254089992896 \)
|
| $37$ |
\( T^{3} + \cdots + 6699990856 \)
|
| $41$ |
\( T^{3} + \cdots - 1582267635864 \)
|
| $43$ |
\( T^{3} + \cdots - 41306695488 \)
|
| $47$ |
\( T^{3} + \cdots - 610928697600 \)
|
| $53$ |
\( T^{3} + \cdots + 33449916663720 \)
|
| $59$ |
\( T^{3} + \cdots - 16648262130240 \)
|
| $61$ |
\( T^{3} + \cdots + 1605495639880 \)
|
| $67$ |
\( T^{3} + \cdots - 10950814258688 \)
|
| $71$ |
\( T^{3} + \cdots - 24576764449536 \)
|
| $73$ |
\( T^{3} + \cdots - 15764054009096 \)
|
| $79$ |
\( T^{3} + \cdots - 120600155477504 \)
|
| $83$ |
\( T^{3} + \cdots + 288133886745408 \)
|
| $89$ |
\( T^{3} + \cdots - 15094444854456 \)
|
| $97$ |
\( T^{3} + \cdots + 93230572586392 \)
|
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