Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1666,4,Mod(1,1666)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1666, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1666.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1666 = 2 \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1666.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: | \(98.2971820696\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.55029.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 59x + 132 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 238) |
| 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
\( x^{3} - x^{2} - 59x + 132 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + \nu - 39 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} - \beta _1 + 39 \)
|
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 |
|
2.00000 | −8.81518 | 4.00000 | 4.56912 | −17.6304 | 0 | 8.00000 | 50.7074 | 9.13823 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −4.36713 | 4.00000 | −14.9761 | −8.73425 | 0 | 8.00000 | −7.92822 | −29.9521 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 6.18230 | 4.00000 | −8.59304 | 12.3646 | 0 | 8.00000 | 11.2209 | −17.1861 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 1666.4.a.h | 3 | |
| 7.b | odd | 2 | 1 | 238.4.a.f | ✓ | 3 | |
| 21.c | even | 2 | 1 | 2142.4.a.n | 3 | ||
| 28.d | even | 2 | 1 | 1904.4.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 238.4.a.f | ✓ | 3 | 7.b | odd | 2 | 1 | |
| 1666.4.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
| 1904.4.a.d | 3 | 28.d | even | 2 | 1 | ||
| 2142.4.a.n | 3 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1666))\):
|
\( T_{3}^{3} + 7T_{3}^{2} - 43T_{3} - 238 \)
|
|
\( T_{5}^{3} + 19T_{5}^{2} + 21T_{5} - 588 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( T^{3} + 7 T^{2} + \cdots - 238 \)
|
| $5$ |
\( T^{3} + 19 T^{2} + \cdots - 588 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + 8 T^{2} + \cdots - 3264 \)
|
| $13$ |
\( T^{3} + 32 T^{2} + \cdots - 4832 \)
|
| $17$ |
\( (T - 17)^{3} \)
|
| $19$ |
\( T^{3} + 122 T^{2} + \cdots - 146048 \)
|
| $23$ |
\( T^{3} - 254 T^{2} + \cdots + 2593248 \)
|
| $29$ |
\( T^{3} - 138 T^{2} + \cdots + 930888 \)
|
| $31$ |
\( T^{3} - 5 T^{2} + \cdots + 1089564 \)
|
| $37$ |
\( T^{3} + 334 T^{2} + \cdots - 10762624 \)
|
| $41$ |
\( T^{3} + 19 T^{2} + \cdots + 5478 \)
|
| $43$ |
\( T^{3} - 123 T^{2} + \cdots + 30533272 \)
|
| $47$ |
\( T^{3} + 162 T^{2} + \cdots - 39351312 \)
|
| $53$ |
\( T^{3} + 213 T^{2} + \cdots - 155918178 \)
|
| $59$ |
\( T^{3} - 538 T^{2} + \cdots + 1156512 \)
|
| $61$ |
\( T^{3} + 19 T^{2} + \cdots - 4142436 \)
|
| $67$ |
\( T^{3} + 845 T^{2} + \cdots - 90604356 \)
|
| $71$ |
\( T^{3} + 656 T^{2} + \cdots - 315850752 \)
|
| $73$ |
\( T^{3} + 529 T^{2} + \cdots - 229749366 \)
|
| $79$ |
\( T^{3} + 1286 T^{2} + \cdots + 6959712 \)
|
| $83$ |
\( T^{3} - 156 T^{2} + \cdots + 89620776 \)
|
| $89$ |
\( T^{3} + 12 T^{2} + \cdots + 15219576 \)
|
| $97$ |
\( T^{3} + 297 T^{2} + \cdots + 143984918 \)
|
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