Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3900,2,Mod(601,3900)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3900, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3900.601");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3900.q (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: | \(31.1416567883\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 10x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 780) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 10x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 10\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 10\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).
| \(n\) | \(301\) | \(1301\) | \(1951\) | \(3277\) |
| \(\chi(n)\) | \(-1 - \beta_{2}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 601.1 |
|
0 | −0.500000 | − | 0.866025i | 0 | 0 | 0 | −2.08114 | + | 3.60464i | 0 | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||
| 601.2 | 0 | −0.500000 | − | 0.866025i | 0 | 0 | 0 | 1.08114 | − | 1.87259i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||
| 2401.1 | 0 | −0.500000 | + | 0.866025i | 0 | 0 | 0 | −2.08114 | − | 3.60464i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||
| 2401.2 | 0 | −0.500000 | + | 0.866025i | 0 | 0 | 0 | 1.08114 | + | 1.87259i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3900.2.q.j | 4 | |
| 5.b | even | 2 | 1 | 780.2.q.c | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 3900.2.by.g | 8 | ||
| 13.c | even | 3 | 1 | inner | 3900.2.q.j | 4 | |
| 15.d | odd | 2 | 1 | 2340.2.q.g | 4 | ||
| 65.n | even | 6 | 1 | 780.2.q.c | ✓ | 4 | |
| 65.q | odd | 12 | 2 | 3900.2.by.g | 8 | ||
| 195.x | odd | 6 | 1 | 2340.2.q.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 780.2.q.c | ✓ | 4 | 5.b | even | 2 | 1 | |
| 780.2.q.c | ✓ | 4 | 65.n | even | 6 | 1 | |
| 2340.2.q.g | 4 | 15.d | odd | 2 | 1 | ||
| 2340.2.q.g | 4 | 195.x | odd | 6 | 1 | ||
| 3900.2.q.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 3900.2.q.j | 4 | 13.c | even | 3 | 1 | inner | |
| 3900.2.by.g | 8 | 5.c | odd | 4 | 2 | ||
| 3900.2.by.g | 8 | 65.q | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3900, [\chi])\):
|
\( T_{7}^{4} + 2T_{7}^{3} + 13T_{7}^{2} - 18T_{7} + 81 \)
|
|
\( T_{11}^{2} - 2T_{11} + 4 \)
|
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\( T_{23}^{2} - 2T_{23} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + T + 1)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 2 T^{3} + \cdots + 81 \)
|
| $11$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $13$ |
\( T^{4} - 6 T^{3} + \cdots + 169 \)
|
| $17$ |
\( T^{4} - 4 T^{3} + \cdots + 36 \)
|
| $19$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $23$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $29$ |
\( T^{4} - 4 T^{3} + \cdots + 36 \)
|
| $31$ |
\( (T - 1)^{4} \)
|
| $37$ |
\( T^{4} + 4 T^{3} + \cdots + 1296 \)
|
| $41$ |
\( T^{4} - 8 T^{3} + \cdots + 36 \)
|
| $43$ |
\( T^{4} - 2 T^{3} + \cdots + 7921 \)
|
| $47$ |
\( (T^{2} - 90)^{2} \)
|
| $53$ |
\( (T^{2} + 8 T - 24)^{2} \)
|
| $59$ |
\( T^{4} + 4 T^{3} + \cdots + 36 \)
|
| $61$ |
\( (T^{2} + T + 1)^{2} \)
|
| $67$ |
\( T^{4} + 14 T^{3} + \cdots + 1521 \)
|
| $71$ |
\( T^{4} + 4 T^{3} + \cdots + 36 \)
|
| $73$ |
\( (T^{2} + 14 T + 39)^{2} \)
|
| $79$ |
\( (T^{2} - 6 T - 31)^{2} \)
|
| $83$ |
\( (T - 12)^{4} \)
|
| $89$ |
\( T^{4} + 10T^{2} + 100 \)
|
| $97$ |
\( T^{4} + 14 T^{3} + \cdots + 1521 \)
|
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