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: | 6.6.46840000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 13\nu - 6 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 3 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 14\nu^{3} - 11\nu^{2} - 43\nu + 24 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{5} + 2\beta_{4} + 15\beta_{3} + 10\beta_{2} + 29\beta _1 + 16 \)
|
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 |
|
−2.38719 | −1.00000 | 3.69868 | 0 | 2.38719 | −3.31671 | −4.05506 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.02791 | −1.00000 | 2.11242 | 0 | 2.02791 | 0.505614 | −0.227977 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.364088 | −1.00000 | −1.86744 | 0 | 0.364088 | 2.24941 | 1.40809 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.858825 | −1.00000 | −1.26242 | 0 | −0.858825 | 3.88045 | −2.80185 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.13324 | −1.00000 | 2.55073 | 0 | −2.13324 | −2.16876 | 1.17484 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.78712 | −1.00000 | 5.76803 | 0 | −2.78712 | −3.15000 | 10.5020 | 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.l | yes | 6 |
| 3.b | odd | 2 | 1 | 5625.2.a.o | 6 | ||
| 5.b | even | 2 | 1 | 1875.2.a.i | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 1875.2.b.e | 12 | ||
| 15.d | odd | 2 | 1 | 5625.2.a.r | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1875.2.a.i | ✓ | 6 | 5.b | even | 2 | 1 | |
| 1875.2.a.l | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 1875.2.b.e | 12 | 5.c | odd | 4 | 2 | ||
| 5625.2.a.o | 6 | 3.b | odd | 2 | 1 | ||
| 5625.2.a.r | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - T_{2}^{5} - 11T_{2}^{4} + 8T_{2}^{3} + 31T_{2}^{2} - 15T_{2} - 9 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} - 11 T^{4} + \cdots - 9 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots - 100 \)
|
| $11$ |
\( T^{6} - 29 T^{4} + \cdots - 144 \)
|
| $13$ |
\( T^{6} - 56 T^{4} + \cdots + 349 \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots - 576 \)
|
| $19$ |
\( T^{6} + 2 T^{5} + \cdots - 5725 \)
|
| $23$ |
\( T^{6} + T^{5} + \cdots - 720 \)
|
| $29$ |
\( T^{6} - 31 T^{5} + \cdots + 6480 \)
|
| $31$ |
\( T^{6} + 2 T^{5} + \cdots + 3155 \)
|
| $37$ |
\( T^{6} + 22 T^{5} + \cdots - 46100 \)
|
| $41$ |
\( T^{6} - 33 T^{5} + \cdots + 720 \)
|
| $43$ |
\( T^{6} + 3 T^{5} + \cdots - 1289 \)
|
| $47$ |
\( T^{6} + 6 T^{5} + \cdots + 80064 \)
|
| $53$ |
\( T^{6} - 14 T^{5} + \cdots + 14400 \)
|
| $59$ |
\( T^{6} + 8 T^{5} + \cdots - 2880 \)
|
| $61$ |
\( T^{6} - 34 T^{5} + \cdots - 72001 \)
|
| $67$ |
\( T^{6} - 2 T^{5} + \cdots + 59 \)
|
| $71$ |
\( T^{6} + 3 T^{5} + \cdots - 12816 \)
|
| $73$ |
\( T^{6} + 36 T^{5} + \cdots + 20380 \)
|
| $79$ |
\( T^{6} - 25 T^{5} + \cdots + 2725 \)
|
| $83$ |
\( T^{6} + 12 T^{5} + \cdots - 23616 \)
|
| $89$ |
\( T^{6} - 18 T^{5} + \cdots - 42480 \)
|
| $97$ |
\( T^{6} - 7 T^{5} + \cdots - 32291 \)
|
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