Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2400,4,Mod(1,2400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2400.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2400.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: | \(141.604584014\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.115636.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 69x - 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 69x - 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{2} - 8\nu - 90 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 2\beta _1 + 92 ) / 2 \)
|
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 |
|
0 | 3.00000 | 0 | 0 | 0 | −26.7458 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | 0 | 0 | 2.13303 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 0 | 0 | 21.6128 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
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 | 2400.4.a.bt | yes | 3 |
| 4.b | odd | 2 | 1 | 2400.4.a.bi | ✓ | 3 | |
| 5.b | even | 2 | 1 | 2400.4.a.bj | yes | 3 | |
| 20.d | odd | 2 | 1 | 2400.4.a.bs | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2400.4.a.bi | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 2400.4.a.bj | yes | 3 | 5.b | even | 2 | 1 | |
| 2400.4.a.bs | yes | 3 | 20.d | odd | 2 | 1 | |
| 2400.4.a.bt | yes | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2400))\):
|
\( T_{7}^{3} + 3T_{7}^{2} - 589T_{7} + 1233 \)
|
|
\( T_{11}^{3} - 44T_{11}^{2} - 656T_{11} + 24864 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} + 3 T^{2} + \cdots + 1233 \)
|
| $11$ |
\( T^{3} - 44 T^{2} + \cdots + 24864 \)
|
| $13$ |
\( T^{3} + 13 T^{2} + \cdots - 119313 \)
|
| $17$ |
\( T^{3} - 36 T^{2} + \cdots - 218016 \)
|
| $19$ |
\( T^{3} - 71 T^{2} + \cdots + 2891 \)
|
| $23$ |
\( T^{3} - 160 T^{2} + \cdots + 2700000 \)
|
| $29$ |
\( T^{3} + 184 T^{2} + \cdots - 1985440 \)
|
| $31$ |
\( T^{3} + 59 T^{2} + \cdots + 5724361 \)
|
| $37$ |
\( T^{3} - 350 T^{2} + \cdots + 223640 \)
|
| $41$ |
\( T^{3} - 166 T^{2} + \cdots - 1460648 \)
|
| $43$ |
\( T^{3} - 341 T^{2} + \cdots + 43261577 \)
|
| $47$ |
\( T^{3} + 394 T^{2} + \cdots - 44505544 \)
|
| $53$ |
\( T^{3} - 314 T^{2} + \cdots + 7884200 \)
|
| $59$ |
\( T^{3} - 538 T^{2} + \cdots + 161430408 \)
|
| $61$ |
\( T^{3} + 523 T^{2} + \cdots - 6303415 \)
|
| $67$ |
\( T^{3} + 543 T^{2} + \cdots - 72402579 \)
|
| $71$ |
\( T^{3} - 910 T^{2} + \cdots + 28003320 \)
|
| $73$ |
\( T^{3} - 870 T^{2} + \cdots + 462224120 \)
|
| $79$ |
\( T^{3} + 96 T^{2} + \cdots - 4064256 \)
|
| $83$ |
\( T^{3} - 210 T^{2} + \cdots + 8776232 \)
|
| $89$ |
\( T^{3} + 64 T^{2} + \cdots - 20736000 \)
|
| $97$ |
\( T^{3} - 943 T^{2} + \cdots - 15001373 \)
|
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