Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(84.2015054018\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 60x^{2} + 8x + 392 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - x^{3} - 60x^{2} + 8x + 392 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 3\nu^{2} - 46\nu + 64 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 30 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 4\beta_{2} + 49\beta _1 + 26 \)
|
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 |
|
−10.7721 | 9.00000 | 84.0388 | 0 | −96.9492 | 49.0000 | −560.569 | 81.0000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −6.72516 | 9.00000 | 13.2278 | 0 | −60.5264 | 49.0000 | 126.246 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | −1.26285 | 9.00000 | −30.4052 | 0 | −11.3656 | 49.0000 | 78.8082 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 3.76014 | 9.00000 | −17.8614 | 0 | 33.8412 | 49.0000 | −187.486 | 81.0000 | 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 | 525.6.a.k | ✓ | 4 |
| 5.b | even | 2 | 1 | 525.6.a.p | yes | 4 | |
| 5.c | odd | 4 | 2 | 525.6.d.m | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 525.6.a.k | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 525.6.a.p | yes | 4 | 5.b | even | 2 | 1 | |
| 525.6.d.m | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 15T_{2}^{3} + 24T_{2}^{2} - 264T_{2} - 344 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(525))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 15 T^{3} + \cdots - 344 \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T - 49)^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 9038084300 \)
|
| $13$ |
\( T^{4} + \cdots + 112995026908 \)
|
| $17$ |
\( T^{4} + \cdots - 30234512012 \)
|
| $19$ |
\( T^{4} + \cdots - 304545071360 \)
|
| $23$ |
\( T^{4} + \cdots + 5588725322923 \)
|
| $29$ |
\( T^{4} + \cdots - 11538421263305 \)
|
| $31$ |
\( T^{4} + \cdots - 11\!\cdots\!56 \)
|
| $37$ |
\( T^{4} + \cdots - 12\!\cdots\!72 \)
|
| $41$ |
\( T^{4} + \cdots - 30\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 19\!\cdots\!53 \)
|
| $47$ |
\( T^{4} + \cdots + 16\!\cdots\!44 \)
|
| $53$ |
\( T^{4} + \cdots + 359909465323660 \)
|
| $59$ |
\( T^{4} + \cdots - 31\!\cdots\!60 \)
|
| $61$ |
\( T^{4} + \cdots + 33\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots + 15\!\cdots\!52 \)
|
| $71$ |
\( T^{4} + \cdots + 36\!\cdots\!64 \)
|
| $73$ |
\( T^{4} + \cdots + 10\!\cdots\!32 \)
|
| $79$ |
\( T^{4} + \cdots - 77\!\cdots\!40 \)
|
| $83$ |
\( T^{4} + \cdots + 19\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots + 16\!\cdots\!20 \)
|
| $97$ |
\( T^{4} + \cdots + 20\!\cdots\!92 \)
|
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