Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1665,2,Mod(1,1665)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1665, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1665.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1665 = 3^{2} \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1665.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: | \(13.2950919365\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.973904.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 185) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 4\nu^{3} + 2\nu^{2} - 12\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 9\nu + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2\beta_{2} + 7\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 3\beta_{3} + 10\beta_{2} + 20\beta _1 + 18 \)
|
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 |
|
−2.72362 | 0 | 5.41809 | 1.00000 | 0 | 3.82710 | −9.30957 | 0 | −2.72362 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.15510 | 0 | 2.64446 | 1.00000 | 0 | −2.62521 | −1.38887 | 0 | −2.15510 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.728950 | 0 | −1.46863 | 1.00000 | 0 | 2.55244 | 2.52846 | 0 | −0.728950 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.13359 | 0 | −0.714970 | 1.00000 | 0 | 2.46164 | −3.07767 | 0 | 1.13359 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.47408 | 0 | 4.12105 | 1.00000 | 0 | 4.78404 | 5.24765 | 0 | 2.47408 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(37\) | \(-1\) |
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 | 1665.2.a.p | 5 | |
| 3.b | odd | 2 | 1 | 185.2.a.e | ✓ | 5 | |
| 5.b | even | 2 | 1 | 8325.2.a.ch | 5 | ||
| 12.b | even | 2 | 1 | 2960.2.a.w | 5 | ||
| 15.d | odd | 2 | 1 | 925.2.a.f | 5 | ||
| 15.e | even | 4 | 2 | 925.2.b.f | 10 | ||
| 21.c | even | 2 | 1 | 9065.2.a.k | 5 | ||
| 111.d | odd | 2 | 1 | 6845.2.a.f | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 185.2.a.e | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 925.2.a.f | 5 | 15.d | odd | 2 | 1 | ||
| 925.2.b.f | 10 | 15.e | even | 4 | 2 | ||
| 1665.2.a.p | 5 | 1.a | even | 1 | 1 | trivial | |
| 2960.2.a.w | 5 | 12.b | even | 2 | 1 | ||
| 6845.2.a.f | 5 | 111.d | odd | 2 | 1 | ||
| 8325.2.a.ch | 5 | 5.b | even | 2 | 1 | ||
| 9065.2.a.k | 5 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1665))\):
|
\( T_{2}^{5} + 2T_{2}^{4} - 8T_{2}^{3} - 14T_{2}^{2} + 11T_{2} + 12 \)
|
|
\( T_{7}^{5} - 11T_{7}^{4} + 32T_{7}^{3} + 32T_{7}^{2} - 268T_{7} + 302 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 2 T^{4} + \cdots + 12 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} - 11 T^{4} + \cdots + 302 \)
|
| $11$ |
\( T^{5} - 5 T^{4} + \cdots - 96 \)
|
| $13$ |
\( T^{5} - 4 T^{4} + \cdots - 256 \)
|
| $17$ |
\( T^{5} - 52 T^{3} + \cdots - 192 \)
|
| $19$ |
\( T^{5} + 4 T^{4} + \cdots - 8 \)
|
| $23$ |
\( T^{5} + 4 T^{4} + \cdots - 192 \)
|
| $29$ |
\( T^{5} - 4 T^{4} + \cdots + 192 \)
|
| $31$ |
\( T^{5} - 8 T^{4} + \cdots + 324 \)
|
| $37$ |
\( (T - 1)^{5} \)
|
| $41$ |
\( T^{5} - 5 T^{4} + \cdots - 96 \)
|
| $43$ |
\( T^{5} - 10 T^{4} + \cdots - 2528 \)
|
| $47$ |
\( T^{5} + 7 T^{4} + \cdots + 978 \)
|
| $53$ |
\( T^{5} - T^{4} + \cdots - 528 \)
|
| $59$ |
\( T^{5} - 30 T^{4} + \cdots + 576 \)
|
| $61$ |
\( T^{5} + 14 T^{4} + \cdots + 3296 \)
|
| $67$ |
\( T^{5} - 24 T^{4} + \cdots + 10952 \)
|
| $71$ |
\( T^{5} - 7 T^{4} + \cdots + 7104 \)
|
| $73$ |
\( T^{5} - 5 T^{4} + \cdots + 368 \)
|
| $79$ |
\( T^{5} - 28 T^{4} + \cdots - 19508 \)
|
| $83$ |
\( T^{5} + 27 T^{4} + \cdots - 4818 \)
|
| $89$ |
\( T^{5} + 6 T^{4} + \cdots - 22944 \)
|
| $97$ |
\( T^{5} + 26 T^{4} + \cdots - 166976 \)
|
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