Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,5,Mod(26,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.26");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.c (of order \(2\), degree \(1\), 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: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 4x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 7\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 7\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
− | 6.47871i | 0 | −25.9737 | − | 11.1803i | 0 | −78.4078 | 64.6165i | 0 | −72.4342 | ||||||||||||||||||||||||||||
| 26.2 | − | 2.00657i | 0 | 11.9737 | 11.1803i | 0 | 54.4078 | − | 56.1312i | 0 | 22.4342 | |||||||||||||||||||||||||||||
| 26.3 | 2.00657i | 0 | 11.9737 | − | 11.1803i | 0 | 54.4078 | 56.1312i | 0 | 22.4342 | ||||||||||||||||||||||||||||||
| 26.4 | 6.47871i | 0 | −25.9737 | 11.1803i | 0 | −78.4078 | − | 64.6165i | 0 | −72.4342 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 45.5.c.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 45.5.c.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 720.5.l.c | 4 | ||
| 5.b | even | 2 | 1 | 225.5.c.b | 4 | ||
| 5.c | odd | 4 | 2 | 225.5.d.b | 8 | ||
| 12.b | even | 2 | 1 | 720.5.l.c | 4 | ||
| 15.d | odd | 2 | 1 | 225.5.c.b | 4 | ||
| 15.e | even | 4 | 2 | 225.5.d.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.5.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 45.5.c.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 225.5.c.b | 4 | 5.b | even | 2 | 1 | ||
| 225.5.c.b | 4 | 15.d | odd | 2 | 1 | ||
| 225.5.d.b | 8 | 5.c | odd | 4 | 2 | ||
| 225.5.d.b | 8 | 15.e | even | 4 | 2 | ||
| 720.5.l.c | 4 | 4.b | odd | 2 | 1 | ||
| 720.5.l.c | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(45, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 46T^{2} + 169 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 125)^{2} \)
|
| $7$ |
\( (T^{2} + 24 T - 4266)^{2} \)
|
| $11$ |
\( T^{4} + 11116 T^{2} + 1444804 \)
|
| $13$ |
\( (T^{2} - 404 T + 38554)^{2} \)
|
| $17$ |
\( T^{4} + 195136 T^{2} + 776848384 \)
|
| $19$ |
\( (T^{2} + 880 T + 193240)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 186717323664 \)
|
| $29$ |
\( T^{4} + \cdots + 269556179344 \)
|
| $31$ |
\( (T^{2} + 828 T - 998244)^{2} \)
|
| $37$ |
\( (T^{2} - 172 T - 1681814)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 377091790084 \)
|
| $43$ |
\( (T^{2} - 2888 T + 1356136)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 9788338191424 \)
|
| $53$ |
\( T^{4} + \cdots + 226348173496384 \)
|
| $59$ |
\( T^{4} + \cdots + 50218227136324 \)
|
| $61$ |
\( (T^{2} + 10084 T + 25392604)^{2} \)
|
| $67$ |
\( (T^{2} + 792 T - 2631024)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 184124057069824 \)
|
| $73$ |
\( (T^{2} - 44 T - 13970756)^{2} \)
|
| $79$ |
\( (T^{2} - 8868 T + 18044316)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 1785558717504 \)
|
| $89$ |
\( T^{4} + \cdots + 54\!\cdots\!84 \)
|
| $97$ |
\( (T^{2} - 13708 T + 40220476)^{2} \)
|
| show more | |
| show less |