Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6030,2,Mod(1,6030)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, 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("6030.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6030.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: | \(48.1497924188\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.6517908.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 12x^{3} + 33x - 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 2010) |
| 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
\( x^{5} - 12x^{3} + 33x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 7\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 7\nu^{2} + 5\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} + 2\beta_{2} + 7\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 33 \)
|
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 |
|
−1.00000 | 0 | 1.00000 | −1.00000 | 0 | −4.96611 | −1.00000 | 0 | 1.00000 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | −1.00000 | 0 | −1.49054 | −1.00000 | 0 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | −1.00000 | 0 | 0.360792 | −1.00000 | 0 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | −1.00000 | 0 | 1.84016 | −1.00000 | 0 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0 | 1.00000 | −1.00000 | 0 | 3.25571 | −1.00000 | 0 | 1.00000 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(67\) | \(1\) |
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 | 6030.2.a.bv | 5 | |
| 3.b | odd | 2 | 1 | 2010.2.a.u | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2010.2.a.u | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 6030.2.a.bv | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6030))\):
|
\( T_{7}^{5} + T_{7}^{4} - 20T_{7}^{3} + 8T_{7}^{2} + 44T_{7} - 16 \)
|
|
\( T_{11}^{5} + T_{11}^{4} - 26T_{11}^{3} - 40T_{11}^{2} + 80T_{11} + 32 \)
|
|
\( T_{13}^{5} - 36T_{13}^{3} - 60T_{13}^{2} + 96T_{13} + 48 \)
|
|
\( T_{17}^{5} + 10T_{17}^{4} - 8T_{17}^{3} - 208T_{17}^{2} + 32T_{17} + 512 \)
|
|
\( T_{23}^{5} + 8T_{23}^{4} - 44T_{23}^{3} - 476T_{23}^{2} - 928T_{23} - 512 \)
|
|
\( T_{29}^{5} + 4T_{29}^{4} - 116T_{29}^{3} - 460T_{29}^{2} + 2840T_{29} + 11168 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T + 1)^{5} \)
|
| $7$ |
\( T^{5} + T^{4} + \cdots - 16 \)
|
| $11$ |
\( T^{5} + T^{4} + \cdots + 32 \)
|
| $13$ |
\( T^{5} - 36 T^{3} + \cdots + 48 \)
|
| $17$ |
\( T^{5} + 10 T^{4} + \cdots + 512 \)
|
| $19$ |
\( T^{5} - 10 T^{4} + \cdots - 896 \)
|
| $23$ |
\( T^{5} + 8 T^{4} + \cdots - 512 \)
|
| $29$ |
\( T^{5} + 4 T^{4} + \cdots + 11168 \)
|
| $31$ |
\( T^{5} - 12 T^{4} + \cdots - 5376 \)
|
| $37$ |
\( T^{5} + T^{4} + \cdots - 256 \)
|
| $41$ |
\( T^{5} - 120 T^{3} + \cdots + 2688 \)
|
| $43$ |
\( T^{5} + 2 T^{4} + \cdots + 1024 \)
|
| $47$ |
\( T^{5} + 10 T^{4} + \cdots + 992 \)
|
| $53$ |
\( T^{5} + 12 T^{4} + \cdots + 576 \)
|
| $59$ |
\( T^{5} - 120 T^{3} + \cdots - 2688 \)
|
| $61$ |
\( T^{5} - 7 T^{4} + \cdots - 1952 \)
|
| $67$ |
\( (T + 1)^{5} \)
|
| $71$ |
\( T^{5} - 21 T^{4} + \cdots - 47712 \)
|
| $73$ |
\( T^{5} + 12 T^{4} + \cdots - 384 \)
|
| $79$ |
\( T^{5} - 10 T^{4} + \cdots - 15056 \)
|
| $83$ |
\( T^{5} - 27 T^{4} + \cdots - 218496 \)
|
| $89$ |
\( T^{5} + 5 T^{4} + \cdots + 50704 \)
|
| $97$ |
\( T^{5} + 3 T^{4} + \cdots + 17184 \)
|
| show more | |
| show less |