Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [165,4,Mod(1,165)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(165, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("165.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 165.a (trivial) |
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: | yes |
| Analytic conductor: | \(9.73531515095\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1957.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 9x + 10 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 9x + 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + \nu + 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 13 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.38835 | −3.00000 | 11.2577 | 5.00000 | 13.1651 | −11.7304 | −14.2958 | 9.00000 | −21.9418 | |||||||||||||||||||||||||||
| 1.2 | 0.793499 | −3.00000 | −7.37036 | 5.00000 | −2.38050 | −2.90793 | −12.1964 | 9.00000 | 3.96749 | ||||||||||||||||||||||||||||
| 1.3 | 4.59486 | −3.00000 | 13.1127 | 5.00000 | −13.7846 | 20.6383 | 23.4921 | 9.00000 | 22.9743 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 165.4.a.g | ✓ | 3 |
| 3.b | odd | 2 | 1 | 495.4.a.i | 3 | ||
| 5.b | even | 2 | 1 | 825.4.a.p | 3 | ||
| 5.c | odd | 4 | 2 | 825.4.c.m | 6 | ||
| 11.b | odd | 2 | 1 | 1815.4.a.q | 3 | ||
| 15.d | odd | 2 | 1 | 2475.4.a.z | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.4.a.g | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 495.4.a.i | 3 | 3.b | odd | 2 | 1 | ||
| 825.4.a.p | 3 | 5.b | even | 2 | 1 | ||
| 825.4.c.m | 6 | 5.c | odd | 4 | 2 | ||
| 1815.4.a.q | 3 | 11.b | odd | 2 | 1 | ||
| 2475.4.a.z | 3 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 20T_{2} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - T^{2} + \cdots + 16 \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( (T - 5)^{3} \)
|
| $7$ |
\( T^{3} - 6 T^{2} + \cdots - 704 \)
|
| $11$ |
\( (T - 11)^{3} \)
|
| $13$ |
\( T^{3} + 20 T^{2} + \cdots - 78104 \)
|
| $17$ |
\( T^{3} - 32 T^{2} + \cdots - 22424 \)
|
| $19$ |
\( T^{3} - 116 T^{2} + \cdots - 80 \)
|
| $23$ |
\( T^{3} - 240 T^{2} + \cdots + 180224 \)
|
| $29$ |
\( T^{3} - 238 T^{2} + \cdots + 428416 \)
|
| $31$ |
\( T^{3} - 92 T^{2} + \cdots + 6769664 \)
|
| $37$ |
\( T^{3} + 90 T^{2} + \cdots - 6364168 \)
|
| $41$ |
\( T^{3} + 46 T^{2} + \cdots - 245888 \)
|
| $43$ |
\( T^{3} + 134 T^{2} + \cdots - 2381360 \)
|
| $47$ |
\( T^{3} + 220 T^{2} + \cdots + 10980224 \)
|
| $53$ |
\( T^{3} + 798 T^{2} + \cdots - 17262968 \)
|
| $59$ |
\( T^{3} - 1236 T^{2} + \cdots - 32923904 \)
|
| $61$ |
\( T^{3} - 342 T^{2} + \cdots - 2655176 \)
|
| $67$ |
\( T^{3} - 764 T^{2} + \cdots + 153685184 \)
|
| $71$ |
\( T^{3} - 1816 T^{2} + \cdots - 198158720 \)
|
| $73$ |
\( T^{3} - 100 T^{2} + \cdots - 5132984 \)
|
| $79$ |
\( T^{3} + 96 T^{2} + \cdots - 167159872 \)
|
| $83$ |
\( T^{3} - 858 T^{2} + \cdots + 542136176 \)
|
| $89$ |
\( T^{3} - 838 T^{2} + \cdots + 693013592 \)
|
| $97$ |
\( T^{3} + 1322 T^{2} + \cdots - 354601256 \)
|
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