Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,6,Mod(1,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.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: | \(52.1247414392\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.49857.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 42x - 48 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 65) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 42x - 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3\nu - 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3\beta _1 + 27 \)
|
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 |
|
−6.47640 | 11.4674 | 9.94380 | 0 | −74.2675 | 146.818 | 142.845 | −111.499 | 0 | |||||||||||||||||||||||||||
| 1.2 | 2.22183 | −16.8416 | −27.0635 | 0 | −37.4193 | 177.501 | −131.229 | 40.6407 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 6.25457 | 21.3742 | 7.11967 | 0 | 133.687 | −116.319 | −155.616 | 213.858 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 325.6.a.d | 3 | |
| 5.b | even | 2 | 1 | 65.6.a.b | ✓ | 3 | |
| 5.c | odd | 4 | 2 | 325.6.b.d | 6 | ||
| 15.d | odd | 2 | 1 | 585.6.a.c | 3 | ||
| 20.d | odd | 2 | 1 | 1040.6.a.k | 3 | ||
| 65.d | even | 2 | 1 | 845.6.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.6.a.b | ✓ | 3 | 5.b | even | 2 | 1 | |
| 325.6.a.d | 3 | 1.a | even | 1 | 1 | trivial | |
| 325.6.b.d | 6 | 5.c | odd | 4 | 2 | ||
| 585.6.a.c | 3 | 15.d | odd | 2 | 1 | ||
| 845.6.a.c | 3 | 65.d | even | 2 | 1 | ||
| 1040.6.a.k | 3 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 2T_{2}^{2} - 41T_{2} + 90 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 2 T^{2} + \cdots + 90 \)
|
| $3$ |
\( T^{3} - 16 T^{2} + \cdots + 4128 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} - 208 T^{2} + \cdots + 3031296 \)
|
| $11$ |
\( T^{3} + 616 T^{2} + \cdots + 295968 \)
|
| $13$ |
\( (T + 169)^{3} \)
|
| $17$ |
\( T^{3} - 794 T^{2} + \cdots + 169609992 \)
|
| $19$ |
\( T^{3} + 3360 T^{2} + \cdots + 281302528 \)
|
| $23$ |
\( T^{3} + \cdots - 9011376144 \)
|
| $29$ |
\( T^{3} - 2306 T^{2} + \cdots - 450963192 \)
|
| $31$ |
\( T^{3} + \cdots - 148752304496 \)
|
| $37$ |
\( T^{3} + \cdots - 200939185512 \)
|
| $41$ |
\( T^{3} + \cdots - 340059823848 \)
|
| $43$ |
\( T^{3} + \cdots - 551757458848 \)
|
| $47$ |
\( T^{3} + \cdots - 605944083840 \)
|
| $53$ |
\( T^{3} + \cdots + 754046116056 \)
|
| $59$ |
\( T^{3} + \cdots - 17139260691072 \)
|
| $61$ |
\( T^{3} + \cdots - 58378672604984 \)
|
| $67$ |
\( T^{3} + \cdots + 131580185301056 \)
|
| $71$ |
\( T^{3} + \cdots - 96871211913744 \)
|
| $73$ |
\( T^{3} + \cdots + 63353389001064 \)
|
| $79$ |
\( T^{3} + \cdots - 87960642878336 \)
|
| $83$ |
\( T^{3} + \cdots - 48499203492864 \)
|
| $89$ |
\( T^{3} + \cdots - 433282552140840 \)
|
| $97$ |
\( T^{3} + \cdots - 76035768760440 \)
|
| show more | |
| show less |