Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3240,2,Mod(649,3240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3240.649");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3240.f (of order \(2\), degree \(1\), 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: | \(25.8715302549\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.29160000.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 20x^{3} + 125 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - 20x^{3} + 125 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 10\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 15\nu^{2} ) / 25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 40\nu^{2} ) / 25 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{3} - 20 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{5} + 20 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{2} + 5\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 15\beta_{4} + 40\beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).
| \(n\) | \(1297\) | \(1621\) | \(2431\) | \(3161\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 649.1 |
|
0 | 0 | 0 | −1.74131 | − | 1.40280i | 0 | − | 2.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 649.2 | 0 | 0 | 0 | −1.74131 | + | 1.40280i | 0 | 2.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 649.3 | 0 | 0 | 0 | −0.344208 | − | 2.20942i | 0 | 2.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 649.4 | 0 | 0 | 0 | −0.344208 | + | 2.20942i | 0 | − | 2.00000i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 649.5 | 0 | 0 | 0 | 2.08551 | − | 0.806615i | 0 | − | 2.00000i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 649.6 | 0 | 0 | 0 | 2.08551 | + | 0.806615i | 0 | 2.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3240.2.f.h | yes | 6 |
| 3.b | odd | 2 | 1 | 3240.2.f.g | ✓ | 6 | |
| 5.b | even | 2 | 1 | inner | 3240.2.f.h | yes | 6 |
| 15.d | odd | 2 | 1 | 3240.2.f.g | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3240.2.f.g | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 3240.2.f.g | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 3240.2.f.h | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 3240.2.f.h | yes | 6 | 5.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3240, [\chi])\):
|
\( T_{7}^{2} + 4 \)
|
|
\( T_{11}^{3} - 3T_{11}^{2} - 12T_{11} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 10T^{3} + 125 \)
|
| $7$ |
\( (T^{2} + 4)^{3} \)
|
| $11$ |
\( (T^{3} - 3 T^{2} - 12 T + 4)^{2} \)
|
| $13$ |
\( T^{6} + 30 T^{4} + \cdots + 400 \)
|
| $17$ |
\( T^{6} + 42 T^{4} + \cdots + 1764 \)
|
| $19$ |
\( (T^{3} + 9 T^{2} + 12 T - 28)^{2} \)
|
| $23$ |
\( T^{6} + 120 T^{4} + \cdots + 6400 \)
|
| $29$ |
\( (T^{3} + 9 T^{2} + \cdots - 313)^{2} \)
|
| $31$ |
\( (T^{3} + 3 T^{2} + \cdots - 144)^{2} \)
|
| $37$ |
\( T^{6} + 162 T^{4} + \cdots + 145924 \)
|
| $41$ |
\( (T^{3} - 3 T^{2} - 72 T + 4)^{2} \)
|
| $43$ |
\( (T^{2} + 36)^{3} \)
|
| $47$ |
\( T^{6} + 132 T^{4} + \cdots + 36864 \)
|
| $53$ |
\( T^{6} + 228 T^{4} + \cdots + 4096 \)
|
| $59$ |
\( (T^{3} + 9 T^{2} - 48 T - 48)^{2} \)
|
| $61$ |
\( (T^{3} - 15 T + 10)^{2} \)
|
| $67$ |
\( T^{6} + 120 T^{4} + \cdots + 6400 \)
|
| $71$ |
\( (T^{3} - 3 T^{2} - 12 T + 4)^{2} \)
|
| $73$ |
\( T^{6} + 138 T^{4} + \cdots + 11236 \)
|
| $79$ |
\( (T^{3} - 60 T + 160)^{2} \)
|
| $83$ |
\( T^{6} + 228 T^{4} + \cdots + 4096 \)
|
| $89$ |
\( (T + 3)^{6} \)
|
| $97$ |
\( T^{6} + 132 T^{4} + \cdots + 1024 \)
|
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