Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,6,Mod(1,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("950.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 950.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: | \(152.364628822\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{163}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 163 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 190) |
| 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 \(\beta = \sqrt{163}\). 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 |
|
4.00000 | −11.7671 | 16.0000 | 0 | −47.0686 | −129.069 | 64.0000 | −104.534 | 0 | ||||||||||||||||||||||||
| 1.2 | 4.00000 | 13.7671 | 16.0000 | 0 | 55.0686 | −26.9314 | 64.0000 | −53.4657 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(19\) | \( -1 \) |
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 | 950.6.a.e | 2 | |
| 5.b | even | 2 | 1 | 190.6.a.b | ✓ | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 190.6.a.b | ✓ | 2 | 5.b | even | 2 | 1 | |
| 950.6.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2T_{3} - 162 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(950))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{2} \)
|
| $3$ |
\( T^{2} - 2T - 162 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 156T + 3476 \)
|
| $11$ |
\( T^{2} + 512T - 13356 \)
|
| $13$ |
\( T^{2} - 1286 T + 411982 \)
|
| $17$ |
\( T^{2} + 1080 T - 2812572 \)
|
| $19$ |
\( (T - 361)^{2} \)
|
| $23$ |
\( T^{2} - 2224 T - 4162668 \)
|
| $29$ |
\( T^{2} + 2568 T + 1595844 \)
|
| $31$ |
\( T^{2} + 1908 T - 19980616 \)
|
| $37$ |
\( T^{2} + 7806 T + 15205862 \)
|
| $41$ |
\( T^{2} - 3060 T - 56339100 \)
|
| $43$ |
\( T^{2} - 2596 T + 422532 \)
|
| $47$ |
\( T^{2} - 14412 T + 18032868 \)
|
| $53$ |
\( T^{2} - 9638 T - 58252626 \)
|
| $59$ |
\( T^{2} + 43008 T + 433668816 \)
|
| $61$ |
\( T^{2} + 25852 T - 608981736 \)
|
| $67$ |
\( T^{2} - 27466 T + 43647702 \)
|
| $71$ |
\( T^{2} + \cdots + 1178461368 \)
|
| $73$ |
\( T^{2} - 25904 T - 823474124 \)
|
| $79$ |
\( T^{2} + 3720 T - 33472288 \)
|
| $83$ |
\( T^{2} + \cdots - 2207100588 \)
|
| $89$ |
\( T^{2} + \cdots + 4420105236 \)
|
| $97$ |
\( T^{2} - 4038 T - 414769906 \)
|
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