Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [95,4,Mod(18,95)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(95, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("95.18");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 95.g (of order \(4\), 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: | \(5.60518145055\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{19})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 9x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 4\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 14\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 7\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(77\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
0 | 0 | − | 8.00000i | −7.00000 | − | 8.71780i | 0 | −13.6411 | + | 13.6411i | 0 | 27.0000i | 0 | |||||||||||||||||||||||||
| 18.2 | 0 | 0 | − | 8.00000i | −7.00000 | + | 8.71780i | 0 | −22.3589 | + | 22.3589i | 0 | 27.0000i | 0 | ||||||||||||||||||||||||||
| 37.1 | 0 | 0 | 8.00000i | −7.00000 | − | 8.71780i | 0 | −22.3589 | − | 22.3589i | 0 | − | 27.0000i | 0 | ||||||||||||||||||||||||||
| 37.2 | 0 | 0 | 8.00000i | −7.00000 | + | 8.71780i | 0 | −13.6411 | − | 13.6411i | 0 | − | 27.0000i | 0 | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-19}) \) |
| 5.c | odd | 4 | 1 | inner |
| 95.g | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 95.4.g.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | inner | 95.4.g.a | ✓ | 4 |
| 19.b | odd | 2 | 1 | CM | 95.4.g.a | ✓ | 4 |
| 95.g | even | 4 | 1 | inner | 95.4.g.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.4.g.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 95.4.g.a | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 95.4.g.a | ✓ | 4 | 19.b | odd | 2 | 1 | CM |
| 95.4.g.a | ✓ | 4 | 95.g | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{4}^{\mathrm{new}}(95, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 14 T + 125)^{2} \)
|
| $7$ |
\( T^{4} + 72 T^{3} + \cdots + 372100 \)
|
| $11$ |
\( (T^{2} - 3724)^{2} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} - 28 T^{3} + \cdots + 92736900 \)
|
| $19$ |
\( (T^{2} + 6859)^{2} \)
|
| $23$ |
\( T^{4} + 424 T^{3} + \cdots + 424772100 \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} + \cdots + 20343316900 \)
|
| $47$ |
\( T^{4} + \cdots + 5647522500 \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( (T^{2} - 511024)^{2} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} + \cdots + 147494402500 \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} + \cdots + 1279228860900 \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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