Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,5,Mod(28,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.28");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.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: | \(4.65164833877\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(i\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
5.00000 | − | 5.00000i | 0 | − | 34.0000i | −25.0000 | 0 | 40.0000 | − | 40.0000i | −90.0000 | − | 90.0000i | 0 | −125.000 | + | 125.000i | |||||||||||||||
| 37.1 | 5.00000 | + | 5.00000i | 0 | 34.0000i | −25.0000 | 0 | 40.0000 | + | 40.0000i | −90.0000 | + | 90.0000i | 0 | −125.000 | − | 125.000i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 45.5.g.c | yes | 2 |
| 3.b | odd | 2 | 1 | 45.5.g.a | ✓ | 2 | |
| 5.b | even | 2 | 1 | 225.5.g.a | 2 | ||
| 5.c | odd | 4 | 1 | inner | 45.5.g.c | yes | 2 |
| 5.c | odd | 4 | 1 | 225.5.g.a | 2 | ||
| 15.d | odd | 2 | 1 | 225.5.g.c | 2 | ||
| 15.e | even | 4 | 1 | 45.5.g.a | ✓ | 2 | |
| 15.e | even | 4 | 1 | 225.5.g.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.5.g.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 45.5.g.a | ✓ | 2 | 15.e | even | 4 | 1 | |
| 45.5.g.c | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 45.5.g.c | yes | 2 | 5.c | odd | 4 | 1 | inner |
| 225.5.g.a | 2 | 5.b | even | 2 | 1 | ||
| 225.5.g.a | 2 | 5.c | odd | 4 | 1 | ||
| 225.5.g.c | 2 | 15.d | odd | 2 | 1 | ||
| 225.5.g.c | 2 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 10T_{2} + 50 \)
acting on \(S_{5}^{\mathrm{new}}(45, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 10T + 50 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T + 25)^{2} \)
|
| $7$ |
\( T^{2} - 80T + 3200 \)
|
| $11$ |
\( (T - 100)^{2} \)
|
| $13$ |
\( T^{2} - 410T + 84050 \)
|
| $17$ |
\( T^{2} + 470T + 110450 \)
|
| $19$ |
\( T^{2} + 5184 \)
|
| $23$ |
\( T^{2} + 680T + 231200 \)
|
| $29$ |
\( T^{2} + 202500 \)
|
| $31$ |
\( (T - 428)^{2} \)
|
| $37$ |
\( T^{2} + 1510 T + 1140050 \)
|
| $41$ |
\( (T + 950)^{2} \)
|
| $43$ |
\( T^{2} + 2440 T + 2976800 \)
|
| $47$ |
\( T^{2} - 640T + 204800 \)
|
| $53$ |
\( T^{2} + 1010 T + 510050 \)
|
| $59$ |
\( T^{2} + 39690000 \)
|
| $61$ |
\( (T + 3808)^{2} \)
|
| $67$ |
\( T^{2} - 680T + 231200 \)
|
| $71$ |
\( (T - 3400)^{2} \)
|
| $73$ |
\( T^{2} - 830T + 344450 \)
|
| $79$ |
\( T^{2} + 45319824 \)
|
| $83$ |
\( T^{2} - 1360 T + 924800 \)
|
| $89$ |
\( T^{2} + 5062500 \)
|
| $97$ |
\( T^{2} - 3230 T + 5216450 \)
|
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