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: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{13}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 34) |
| 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 \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.
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 | −6.60555 | 4.00000 | 16.4222 | −13.2111 | 0 | 8.00000 | 16.6333 | 32.8444 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 0.605551 | 4.00000 | −12.4222 | 1.21110 | 0 | 8.00000 | −26.6333 | −24.8444 | |||||||||||||||||||||||||
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.d | 2 | |
| 7.b | odd | 2 | 1 | 34.4.a.c | ✓ | 2 | |
| 21.c | even | 2 | 1 | 306.4.a.j | 2 | ||
| 28.d | even | 2 | 1 | 272.4.a.e | 2 | ||
| 35.c | odd | 2 | 1 | 850.4.a.e | 2 | ||
| 35.f | even | 4 | 2 | 850.4.c.i | 4 | ||
| 56.e | even | 2 | 1 | 1088.4.a.q | 2 | ||
| 56.h | odd | 2 | 1 | 1088.4.a.m | 2 | ||
| 84.h | odd | 2 | 1 | 2448.4.a.y | 2 | ||
| 119.d | odd | 2 | 1 | 578.4.a.h | 2 | ||
| 119.f | odd | 4 | 2 | 578.4.b.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 34.4.a.c | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 272.4.a.e | 2 | 28.d | even | 2 | 1 | ||
| 306.4.a.j | 2 | 21.c | even | 2 | 1 | ||
| 578.4.a.h | 2 | 119.d | odd | 2 | 1 | ||
| 578.4.b.e | 4 | 119.f | odd | 4 | 2 | ||
| 850.4.a.e | 2 | 35.c | odd | 2 | 1 | ||
| 850.4.c.i | 4 | 35.f | even | 4 | 2 | ||
| 1088.4.a.m | 2 | 56.h | odd | 2 | 1 | ||
| 1088.4.a.q | 2 | 56.e | even | 2 | 1 | ||
| 1666.4.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 2448.4.a.y | 2 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1666))\):
|
\( T_{3}^{2} + 6T_{3} - 4 \)
|
|
\( T_{5}^{2} - 4T_{5} - 204 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( T^{2} + 6T - 4 \)
|
| $5$ |
\( T^{2} - 4T - 204 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 6T - 2916 \)
|
| $13$ |
\( T^{2} - 64T - 1524 \)
|
| $17$ |
\( (T + 17)^{2} \)
|
| $19$ |
\( T^{2} - 36T - 2224 \)
|
| $23$ |
\( T^{2} - 42T - 612 \)
|
| $29$ |
\( T^{2} - 428T + 40596 \)
|
| $31$ |
\( T^{2} + 86T - 5028 \)
|
| $37$ |
\( T^{2} - 340T + 21412 \)
|
| $41$ |
\( T^{2} + 404T - 59868 \)
|
| $43$ |
\( T^{2} + 620T + 91888 \)
|
| $47$ |
\( T^{2} - 56T - 283968 \)
|
| $53$ |
\( T^{2} - 28T - 300156 \)
|
| $59$ |
\( T^{2} + 276T + 7344 \)
|
| $61$ |
\( T^{2} - 236T - 16028 \)
|
| $67$ |
\( T^{2} - 536T - 141168 \)
|
| $71$ |
\( T^{2} - 1542 T + 574668 \)
|
| $73$ |
\( T^{2} - 164T - 309644 \)
|
| $79$ |
\( T^{2} + 1854 T + 704876 \)
|
| $83$ |
\( T^{2} - 372T - 22032 \)
|
| $89$ |
\( T^{2} - 1976 T + 974844 \)
|
| $97$ |
\( T^{2} - 220T - 71100 \)
|
| show more | |
| show less |