Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [145,2,Mod(144,145)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(145, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("145.144");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 145 = 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 145.d (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: | \(1.15783082931\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 4 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 5\nu^{2} - 5\nu + 6 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{2} - 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(117\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 144.1 |
|
−1.00000 | −1.56155 | −1.00000 | 1.78078 | − | 1.35234i | 1.56155 | 3.46410i | 3.00000 | −0.561553 | −1.78078 | + | 1.35234i | ||||||||||||||||||||||||||
| 144.2 | −1.00000 | −1.56155 | −1.00000 | 1.78078 | + | 1.35234i | 1.56155 | − | 3.46410i | 3.00000 | −0.561553 | −1.78078 | − | 1.35234i | ||||||||||||||||||||||||||
| 144.3 | −1.00000 | 2.56155 | −1.00000 | −0.280776 | − | 2.21837i | −2.56155 | − | 3.46410i | 3.00000 | 3.56155 | 0.280776 | + | 2.21837i | ||||||||||||||||||||||||||
| 144.4 | −1.00000 | 2.56155 | −1.00000 | −0.280776 | + | 2.21837i | −2.56155 | 3.46410i | 3.00000 | 3.56155 | 0.280776 | − | 2.21837i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 145.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 145.2.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1305.2.f.i | 4 | ||
| 4.b | odd | 2 | 1 | 2320.2.j.a | 4 | ||
| 5.b | even | 2 | 1 | 145.2.d.c | yes | 4 | |
| 5.c | odd | 4 | 2 | 725.2.c.f | 8 | ||
| 15.d | odd | 2 | 1 | 1305.2.f.e | 4 | ||
| 20.d | odd | 2 | 1 | 2320.2.j.c | 4 | ||
| 29.b | even | 2 | 1 | 145.2.d.c | yes | 4 | |
| 87.d | odd | 2 | 1 | 1305.2.f.e | 4 | ||
| 116.d | odd | 2 | 1 | 2320.2.j.c | 4 | ||
| 145.d | even | 2 | 1 | inner | 145.2.d.a | ✓ | 4 |
| 145.h | odd | 4 | 2 | 725.2.c.f | 8 | ||
| 435.b | odd | 2 | 1 | 1305.2.f.i | 4 | ||
| 580.e | odd | 2 | 1 | 2320.2.j.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 145.2.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 145.2.d.a | ✓ | 4 | 145.d | even | 2 | 1 | inner |
| 145.2.d.c | yes | 4 | 5.b | even | 2 | 1 | |
| 145.2.d.c | yes | 4 | 29.b | even | 2 | 1 | |
| 725.2.c.f | 8 | 5.c | odd | 4 | 2 | ||
| 725.2.c.f | 8 | 145.h | odd | 4 | 2 | ||
| 1305.2.f.e | 4 | 15.d | odd | 2 | 1 | ||
| 1305.2.f.e | 4 | 87.d | odd | 2 | 1 | ||
| 1305.2.f.i | 4 | 3.b | odd | 2 | 1 | ||
| 1305.2.f.i | 4 | 435.b | odd | 2 | 1 | ||
| 2320.2.j.a | 4 | 4.b | odd | 2 | 1 | ||
| 2320.2.j.a | 4 | 580.e | odd | 2 | 1 | ||
| 2320.2.j.c | 4 | 20.d | odd | 2 | 1 | ||
| 2320.2.j.c | 4 | 116.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(145, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
|
| $3$ |
\( (T^{2} - T - 4)^{2} \)
|
| $5$ |
\( T^{4} - 3 T^{3} + \cdots + 25 \)
|
| $7$ |
\( (T^{2} + 12)^{2} \)
|
| $11$ |
\( T^{4} + 39T^{2} + 36 \)
|
| $13$ |
\( T^{4} + 27T^{2} + 144 \)
|
| $17$ |
\( (T - 2)^{4} \)
|
| $19$ |
\( (T^{2} + 12)^{2} \)
|
| $23$ |
\( (T^{2} + 12)^{2} \)
|
| $29$ |
\( T^{4} - 10T^{2} + 841 \)
|
| $31$ |
\( T^{4} + 63T^{2} + 36 \)
|
| $37$ |
\( (T^{2} + 6 T - 8)^{2} \)
|
| $41$ |
\( T^{4} + 108T^{2} + 2304 \)
|
| $43$ |
\( (T^{2} + 15 T + 52)^{2} \)
|
| $47$ |
\( (T^{2} - 11 T - 8)^{2} \)
|
| $53$ |
\( T^{4} + 27T^{2} + 144 \)
|
| $59$ |
\( (T^{2} - 6 T - 8)^{2} \)
|
| $61$ |
\( T^{4} + 156T^{2} + 576 \)
|
| $67$ |
\( T^{4} + 156T^{2} + 576 \)
|
| $71$ |
\( (T^{2} - 12 T - 32)^{2} \)
|
| $73$ |
\( (T^{2} - 6 T - 8)^{2} \)
|
| $79$ |
\( T^{4} + 279 T^{2} + 12996 \)
|
| $83$ |
\( (T^{2} + 108)^{2} \)
|
| $89$ |
\( T^{4} + 108T^{2} + 2304 \)
|
| $97$ |
\( (T^{2} - 68)^{2} \)
|
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