Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1575,2,Mod(1574,1575)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1575, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1575.1574");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1575.g (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(12.5764383184\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.157351936.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + x^{4} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 63) |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + x^{4} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{4} + 1 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 5\nu^{2} ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{3} - \nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 2\nu^{5} - 5\nu^{3} - 10\nu ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{2} ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} - \nu^{3} + 8\nu ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} - 8\nu^{5} - 11\nu^{3} + 32\nu ) / 24 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 3\beta_{6} - 2\beta_{4} ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 3\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{7} - 5\beta_{4} + 3\beta_{3} ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta _1 - 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -5\beta_{7} - 3\beta_{6} - 8\beta_{4} - 6\beta_{3} ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -5\beta_{5} + 9\beta_{2} ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 10\beta_{7} - 24\beta_{6} - 11\beta_{4} - 3\beta_{3} ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(451\) | \(1226\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1574.1 |
|
−2.57794 | 0 | 4.64575 | 0 | 0 | − | 2.64575i | −6.82058 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1574.2 | −2.57794 | 0 | 4.64575 | 0 | 0 | 2.64575i | −6.82058 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1574.3 | −1.16372 | 0 | −0.645751 | 0 | 0 | − | 2.64575i | 3.07892 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1574.4 | −1.16372 | 0 | −0.645751 | 0 | 0 | 2.64575i | 3.07892 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1574.5 | 1.16372 | 0 | −0.645751 | 0 | 0 | − | 2.64575i | −3.07892 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1574.6 | 1.16372 | 0 | −0.645751 | 0 | 0 | 2.64575i | −3.07892 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1574.7 | 2.57794 | 0 | 4.64575 | 0 | 0 | − | 2.64575i | 6.82058 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1574.8 | 2.57794 | 0 | 4.64575 | 0 | 0 | 2.64575i | 6.82058 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
| 35.c | odd | 2 | 1 | inner |
| 105.g | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1575, [\chi])\):
|
\( T_{2}^{4} - 8T_{2}^{2} + 9 \)
|
|
\( T_{59} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 8 T^{2} + 9)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{2} + 7)^{4} \)
|
| $11$ |
\( (T^{4} + 44 T^{2} + 36)^{2} \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( (T^{4} - 92 T^{2} + 324)^{2} \)
|
| $29$ |
\( (T^{4} + 116 T^{2} + 2916)^{2} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( (T^{2} + 112)^{4} \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( (T^{2} + 28)^{4} \)
|
| $47$ |
\( T^{8} \)
|
| $53$ |
\( (T^{4} - 212 T^{2} + 36)^{2} \)
|
| $59$ |
\( T^{8} \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( (T^{2} + 16)^{4} \)
|
| $71$ |
\( (T^{4} + 284 T^{2} + 12996)^{2} \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( (T + 8)^{8} \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( T^{8} \)
|
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