Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2475,4,Mod(1,2475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2475.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2475.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: | \(146.029727264\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.91035289.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 285x^{2} + 286x + 19616 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 825) |
| 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
\( x^{4} - 2x^{3} - 285x^{2} + 286x + 19616 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 150\nu ) / 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - \nu - 150 ) / 14 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 56\beta_{3} + \beta _1 + 602 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 56\beta_{3} + 56\beta_{2} + 151\beta _1 + 902 ) / 4 \)
|
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 |
|
−2.56155 | 0 | −1.43845 | 0 | 0 | −25.6772 | 24.1771 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −2.56155 | 0 | −1.43845 | 0 | 0 | 29.1157 | 24.1771 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1.56155 | 0 | −5.56155 | 0 | 0 | −16.7054 | −21.1771 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 1.56155 | 0 | −5.56155 | 0 | 0 | 24.2670 | −21.1771 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
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 | 2475.4.a.bb | 4 | |
| 3.b | odd | 2 | 1 | 825.4.a.v | yes | 4 | |
| 5.b | even | 2 | 1 | 2475.4.a.bd | 4 | ||
| 15.d | odd | 2 | 1 | 825.4.a.u | ✓ | 4 | |
| 15.e | even | 4 | 2 | 825.4.c.q | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 825.4.a.u | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 825.4.a.v | yes | 4 | 3.b | odd | 2 | 1 | |
| 825.4.c.q | 8 | 15.e | even | 4 | 2 | ||
| 2475.4.a.bb | 4 | 1.a | even | 1 | 1 | trivial | |
| 2475.4.a.bd | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):
|
\( T_{2}^{2} + T_{2} - 4 \)
|
|
\( T_{7}^{4} - 11T_{7}^{3} - 1127T_{7}^{2} + 7047T_{7} + 303074 \)
|
|
\( T_{29}^{4} + 251T_{29}^{3} - 32818T_{29}^{2} - 8773676T_{29} - 42323912 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T - 4)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 11 T^{3} + \cdots + 303074 \)
|
| $11$ |
\( (T - 11)^{4} \)
|
| $13$ |
\( T^{4} + 25 T^{3} + \cdots - 1706944 \)
|
| $17$ |
\( T^{4} - 85 T^{3} + \cdots - 1313888 \)
|
| $19$ |
\( T^{4} - 118 T^{3} + \cdots - 6797601 \)
|
| $23$ |
\( T^{4} + 10 T^{3} + \cdots + 20292331 \)
|
| $29$ |
\( T^{4} + 251 T^{3} + \cdots - 42323912 \)
|
| $31$ |
\( T^{4} + 135 T^{3} + \cdots + 92275200 \)
|
| $37$ |
\( T^{4} + 419 T^{3} + \cdots - 881580586 \)
|
| $41$ |
\( T^{4} + \cdots + 1343813418 \)
|
| $43$ |
\( T^{4} + 15 T^{3} + \cdots + 795036672 \)
|
| $47$ |
\( T^{4} + \cdots + 1019998564 \)
|
| $53$ |
\( T^{4} + \cdots + 3274784192 \)
|
| $59$ |
\( T^{4} + \cdots - 37958294854 \)
|
| $61$ |
\( T^{4} + \cdots + 10189762592 \)
|
| $67$ |
\( T^{4} + \cdots + 122607629152 \)
|
| $71$ |
\( T^{4} + \cdots + 6876641459 \)
|
| $73$ |
\( T^{4} + \cdots + 625767485216 \)
|
| $79$ |
\( T^{4} + \cdots - 51952998816 \)
|
| $83$ |
\( T^{4} + \cdots - 55004349600 \)
|
| $89$ |
\( T^{4} + \cdots - 134213353376 \)
|
| $97$ |
\( T^{4} + \cdots + 27014010329 \)
|
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