Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8325,2,Mod(1,8325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8325, 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("8325.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8325.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: | \(66.4754596827\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4400.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 7x^{2} + 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 555) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 7x^{2} + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.61803 | 0 | 0.618034 | 0 | 0 | −3.32813 | 2.23607 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1.61803 | 0 | 0.618034 | 0 | 0 | −0.671869 | 2.23607 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0.618034 | 0 | −1.61803 | 0 | 0 | −4.49721 | −2.23607 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0.618034 | 0 | −1.61803 | 0 | 0 | 0.497212 | −2.23607 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(37\) | \( +1 \) |
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 | 8325.2.a.bt | 4 | |
| 3.b | odd | 2 | 1 | 2775.2.a.y | 4 | ||
| 5.b | even | 2 | 1 | 8325.2.a.bw | 4 | ||
| 5.c | odd | 4 | 2 | 1665.2.c.d | 8 | ||
| 15.d | odd | 2 | 1 | 2775.2.a.w | 4 | ||
| 15.e | even | 4 | 2 | 555.2.c.b | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 555.2.c.b | ✓ | 8 | 15.e | even | 4 | 2 | |
| 1665.2.c.d | 8 | 5.c | odd | 4 | 2 | ||
| 2775.2.a.w | 4 | 15.d | odd | 2 | 1 | ||
| 2775.2.a.y | 4 | 3.b | odd | 2 | 1 | ||
| 8325.2.a.bt | 4 | 1.a | even | 1 | 1 | trivial | |
| 8325.2.a.bw | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):
|
\( T_{2}^{2} + T_{2} - 1 \)
|
|
\( T_{7}^{4} + 8T_{7}^{3} + 16T_{7}^{2} - 5 \)
|
|
\( T_{11}^{4} - 4T_{11}^{3} - T_{11}^{2} + 10T_{11} + 5 \)
|
|
\( T_{13}^{4} + 2T_{13}^{3} - 14T_{13}^{2} + 10T_{13} + 5 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T - 1)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 8 T^{3} + \cdots - 5 \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 5 \)
|
| $13$ |
\( T^{4} + 2 T^{3} + \cdots + 5 \)
|
| $17$ |
\( T^{4} + 2 T^{3} + \cdots + 145 \)
|
| $19$ |
\( T^{4} + 4 T^{3} + \cdots - 5 \)
|
| $23$ |
\( T^{4} - 42 T^{2} + \cdots + 41 \)
|
| $29$ |
\( T^{4} - 12 T^{3} + \cdots - 49 \)
|
| $31$ |
\( T^{4} + 2 T^{3} + \cdots + 59 \)
|
| $37$ |
\( (T + 1)^{4} \)
|
| $41$ |
\( T^{4} - 26 T^{3} + \cdots + 1289 \)
|
| $43$ |
\( T^{4} + 6 T^{3} + \cdots + 5 \)
|
| $47$ |
\( T^{4} + 6 T^{3} + \cdots - 1 \)
|
| $53$ |
\( T^{4} - 68T^{2} + 176 \)
|
| $59$ |
\( T^{4} - 12 T^{3} + \cdots - 1949 \)
|
| $61$ |
\( T^{4} + 6 T^{3} + \cdots - 25 \)
|
| $67$ |
\( T^{4} + 8 T^{3} + \cdots + 655 \)
|
| $71$ |
\( T^{4} - 34 T^{3} + \cdots + 2801 \)
|
| $73$ |
\( T^{4} + 20 T^{3} + \cdots - 3625 \)
|
| $79$ |
\( T^{4} - 6 T^{3} + \cdots - 6245 \)
|
| $83$ |
\( T^{4} - 2 T^{3} + \cdots + 11 \)
|
| $89$ |
\( T^{4} - 24 T^{3} + \cdots - 725 \)
|
| $97$ |
\( T^{4} - 14 T^{3} + \cdots - 631 \)
|
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