Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1682,4,Mod(1,1682)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1682, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1682.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1682 = 2 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1682.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: | \(99.2412126297\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{6}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 58) |
| 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 \(\beta = \sqrt{6}\). 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 | −1.44949 | 4.00000 | −19.6969 | −2.89898 | 11.5959 | 8.00000 | −24.8990 | −39.3939 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 3.44949 | 4.00000 | 9.69694 | 6.89898 | −27.5959 | 8.00000 | −15.1010 | 19.3939 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(29\) | \(1\) |
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 | 1682.4.a.c | 2 | |
| 29.b | even | 2 | 1 | 58.4.a.c | ✓ | 2 | |
| 87.d | odd | 2 | 1 | 522.4.a.j | 2 | ||
| 116.d | odd | 2 | 1 | 464.4.a.e | 2 | ||
| 145.d | even | 2 | 1 | 1450.4.a.g | 2 | ||
| 232.b | odd | 2 | 1 | 1856.4.a.i | 2 | ||
| 232.g | even | 2 | 1 | 1856.4.a.l | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.4.a.c | ✓ | 2 | 29.b | even | 2 | 1 | |
| 464.4.a.e | 2 | 116.d | odd | 2 | 1 | ||
| 522.4.a.j | 2 | 87.d | odd | 2 | 1 | ||
| 1450.4.a.g | 2 | 145.d | even | 2 | 1 | ||
| 1682.4.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 1856.4.a.i | 2 | 232.b | odd | 2 | 1 | ||
| 1856.4.a.l | 2 | 232.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2T_{3} - 5 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1682))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( T^{2} - 2T - 5 \)
|
| $5$ |
\( T^{2} + 10T - 191 \)
|
| $7$ |
\( T^{2} + 16T - 320 \)
|
| $11$ |
\( T^{2} - 90T + 1971 \)
|
| $13$ |
\( T^{2} + 50T + 241 \)
|
| $17$ |
\( T^{2} - 44T - 1052 \)
|
| $19$ |
\( T^{2} + 108T + 2820 \)
|
| $23$ |
\( T^{2} + 28T - 36308 \)
|
| $29$ |
\( T^{2} \)
|
| $31$ |
\( T^{2} + 66T - 70197 \)
|
| $37$ |
\( T^{2} + 40T + 304 \)
|
| $41$ |
\( T^{2} + 304T - 15296 \)
|
| $43$ |
\( T^{2} - 130T - 12629 \)
|
| $47$ |
\( T^{2} - 514T + 7243 \)
|
| $53$ |
\( T^{2} + 958T + 213217 \)
|
| $59$ |
\( T^{2} + 180T - 366900 \)
|
| $61$ |
\( T^{2} + 1028 T + 263332 \)
|
| $67$ |
\( T^{2} + 912T + 205536 \)
|
| $71$ |
\( T^{2} - 796T + 151468 \)
|
| $73$ |
\( T^{2} - 856T - 2672 \)
|
| $79$ |
\( T^{2} - 318T - 756645 \)
|
| $83$ |
\( T^{2} + 1828 T + 826732 \)
|
| $89$ |
\( T^{2} - 944T + 191680 \)
|
| $97$ |
\( T^{2} + 368T - 26144 \)
|
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