Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1875,2,Mod(1,1875)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1875.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1875 = 3 \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1875.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: | \(14.9719503790\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.44400625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 75) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 6\nu^{3} + 5\nu^{2} - 1 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 10\nu^{3} + 5\nu^{2} + 24\nu - 1 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 10\nu^{3} + 9\nu^{2} + 24\nu - 13 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 12\nu^{3} - 7\nu^{2} + 34\nu + 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 6\beta_{4} - 9\beta_{3} + \beta_{2} + \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 10\beta_{3} + 11\beta_{2} + 37\beta _1 + 2 \)
|
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 |
|
−2.44028 | 1.00000 | 3.95498 | 0 | −2.44028 | 3.44028 | −4.77071 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.16056 | 1.00000 | 2.66802 | 0 | −2.16056 | 3.16056 | −1.44329 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.246759 | 1.00000 | −1.93911 | 0 | −0.246759 | 1.24676 | 0.972011 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.141689 | 1.00000 | −1.97992 | 0 | 0.141689 | 0.858311 | −0.563913 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.01887 | 1.00000 | 2.07584 | 0 | 2.01887 | −1.01887 | 0.153106 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.68704 | 1.00000 | 5.22020 | 0 | 2.68704 | −1.68704 | 8.65280 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
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 | 1875.2.a.k | 6 | |
| 3.b | odd | 2 | 1 | 5625.2.a.q | 6 | ||
| 5.b | even | 2 | 1 | 1875.2.a.j | 6 | ||
| 5.c | odd | 4 | 2 | 1875.2.b.f | 12 | ||
| 15.d | odd | 2 | 1 | 5625.2.a.p | 6 | ||
| 25.d | even | 5 | 2 | 375.2.g.c | 12 | ||
| 25.e | even | 10 | 2 | 75.2.g.c | ✓ | 12 | |
| 25.f | odd | 20 | 4 | 375.2.i.d | 24 | ||
| 75.h | odd | 10 | 2 | 225.2.h.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.2.g.c | ✓ | 12 | 25.e | even | 10 | 2 | |
| 225.2.h.d | 12 | 75.h | odd | 10 | 2 | ||
| 375.2.g.c | 12 | 25.d | even | 5 | 2 | ||
| 375.2.i.d | 24 | 25.f | odd | 20 | 4 | ||
| 1875.2.a.j | 6 | 5.b | even | 2 | 1 | ||
| 1875.2.a.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 1875.2.b.f | 12 | 5.c | odd | 4 | 2 | ||
| 5625.2.a.p | 6 | 15.d | odd | 2 | 1 | ||
| 5625.2.a.q | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 11T_{2}^{4} - T_{2}^{3} + 29T_{2}^{2} + 3T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 11 T^{4} + \cdots - 1 \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 6 T^{5} + \cdots + 20 \)
|
| $11$ |
\( T^{6} - 3 T^{5} + \cdots + 244 \)
|
| $13$ |
\( T^{6} - 6 T^{5} + \cdots + 101 \)
|
| $17$ |
\( T^{6} - 13 T^{5} + \cdots + 4639 \)
|
| $19$ |
\( T^{6} - 11 T^{5} + \cdots - 380 \)
|
| $23$ |
\( T^{6} - 13 T^{5} + \cdots - 2020 \)
|
| $29$ |
\( T^{6} + 3 T^{5} + \cdots + 2105 \)
|
| $31$ |
\( T^{6} + 11 T^{5} + \cdots - 2900 \)
|
| $37$ |
\( T^{6} - 21 T^{5} + \cdots - 6025 \)
|
| $41$ |
\( T^{6} + T^{5} + \cdots - 82655 \)
|
| $43$ |
\( T^{6} - 2 T^{5} + \cdots - 6284 \)
|
| $47$ |
\( T^{6} - 14 T^{5} + \cdots + 2284 \)
|
| $53$ |
\( T^{6} - 23 T^{5} + \cdots - 34495 \)
|
| $59$ |
\( T^{6} - 9 T^{5} + \cdots + 3920 \)
|
| $61$ |
\( T^{6} - 11 T^{5} + \cdots + 168269 \)
|
| $67$ |
\( T^{6} - 8 T^{5} + \cdots + 14684 \)
|
| $71$ |
\( T^{6} + 8 T^{5} + \cdots - 196 \)
|
| $73$ |
\( T^{6} - 13 T^{5} + \cdots + 22205 \)
|
| $79$ |
\( T^{6} + 5 T^{5} + \cdots - 8000 \)
|
| $83$ |
\( T^{6} + 20 T^{5} + \cdots + 41036 \)
|
| $89$ |
\( T^{6} + 4 T^{5} + \cdots - 377055 \)
|
| $97$ |
\( T^{6} + 7 T^{5} + \cdots + 2399 \)
|
| show more | |
| show less |