Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,4,Mod(1,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, 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("525.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 525.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: | \(30.9760027530\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 24x^{2} - 3x + 46 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 24x^{2} - 3x + 46 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 20\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 20\beta _1 + 2 \)
|
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 |
|
−4.60171 | 3.00000 | 13.1758 | 0 | −13.8051 | 7.00000 | −23.8174 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1.52801 | 3.00000 | −5.66519 | 0 | −4.58402 | 7.00000 | 20.8805 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1.37627 | 3.00000 | −6.10588 | 0 | 4.12881 | 7.00000 | −19.4135 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 4.75345 | 3.00000 | 14.5953 | 0 | 14.2604 | 7.00000 | 31.3504 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
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 | 525.4.a.u | yes | 4 |
| 3.b | odd | 2 | 1 | 1575.4.a.bk | 4 | ||
| 5.b | even | 2 | 1 | 525.4.a.t | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 525.4.d.n | 8 | ||
| 15.d | odd | 2 | 1 | 1575.4.a.bj | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 525.4.a.t | ✓ | 4 | 5.b | even | 2 | 1 | |
| 525.4.a.u | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 525.4.d.n | 8 | 5.c | odd | 4 | 2 | ||
| 1575.4.a.bj | 4 | 15.d | odd | 2 | 1 | ||
| 1575.4.a.bk | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):
|
\( T_{2}^{4} - 24T_{2}^{2} - 3T_{2} + 46 \)
|
|
\( T_{11}^{4} - 21T_{11}^{3} - 2643T_{11}^{2} + 61617T_{11} - 304010 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 24 T^{2} + \cdots + 46 \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T - 7)^{4} \)
|
| $11$ |
\( T^{4} - 21 T^{3} + \cdots - 304010 \)
|
| $13$ |
\( T^{4} - 5 T^{3} + \cdots + 545368 \)
|
| $17$ |
\( T^{4} - 99 T^{3} + \cdots - 25684592 \)
|
| $19$ |
\( T^{4} - 72 T^{3} + \cdots + 1017760 \)
|
| $23$ |
\( T^{4} - 102 T^{3} + \cdots - 390917 \)
|
| $29$ |
\( T^{4} + 240 T^{3} + \cdots + 130344085 \)
|
| $31$ |
\( T^{4} - 351 T^{3} + \cdots + 137611224 \)
|
| $37$ |
\( T^{4} - 399 T^{3} + \cdots - 476873082 \)
|
| $41$ |
\( T^{4} - 381 T^{3} + \cdots - 91750400 \)
|
| $43$ |
\( T^{4} + \cdots + 3696646993 \)
|
| $47$ |
\( T^{4} + \cdots + 11323563904 \)
|
| $53$ |
\( T^{4} + \cdots - 2685647720 \)
|
| $59$ |
\( T^{4} + \cdots - 35121553400 \)
|
| $61$ |
\( T^{4} + \cdots + 13537528704 \)
|
| $67$ |
\( T^{4} + \cdots + 120089209012 \)
|
| $71$ |
\( T^{4} + \cdots - 3046275956 \)
|
| $73$ |
\( T^{4} + \cdots - 498004222688 \)
|
| $79$ |
\( T^{4} + \cdots - 14999641370 \)
|
| $83$ |
\( T^{4} + \cdots - 374122465304 \)
|
| $89$ |
\( T^{4} + \cdots - 25670269520 \)
|
| $97$ |
\( T^{4} + \cdots + 190440971632 \)
|
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