Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,6,Mod(274,525)]
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, 1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.274");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 525.d (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(84.2015054018\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{73})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 37x^{2} + 324 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 105) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 37x^{2} + 324 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 19\nu ) / 18 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 19 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 19 \)
|
| \(\nu^{3}\) | \(=\) |
\( 18\beta_{2} - 19\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(176\) | \(451\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 274.1 |
|
− | 4.77200i | − | 9.00000i | 9.22800 | 0 | −42.9480 | 49.0000i | − | 196.740i | −81.0000 | 0 | |||||||||||||||||||||||||||
| 274.2 | − | 3.77200i | 9.00000i | 17.7720 | 0 | 33.9480 | − | 49.0000i | − | 187.740i | −81.0000 | 0 | ||||||||||||||||||||||||||||
| 274.3 | 3.77200i | − | 9.00000i | 17.7720 | 0 | 33.9480 | 49.0000i | 187.740i | −81.0000 | 0 | ||||||||||||||||||||||||||||||
| 274.4 | 4.77200i | 9.00000i | 9.22800 | 0 | −42.9480 | − | 49.0000i | 196.740i | −81.0000 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 525.6.d.j | 4 | |
| 5.b | even | 2 | 1 | inner | 525.6.d.j | 4 | |
| 5.c | odd | 4 | 1 | 105.6.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 525.6.a.g | 2 | ||
| 15.e | even | 4 | 1 | 315.6.a.e | 2 | ||
| 35.f | even | 4 | 1 | 735.6.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.6.a.d | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 315.6.a.e | 2 | 15.e | even | 4 | 1 | ||
| 525.6.a.g | 2 | 5.c | odd | 4 | 1 | ||
| 525.6.d.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 525.6.d.j | 4 | 5.b | even | 2 | 1 | inner | |
| 735.6.a.f | 2 | 35.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 37T_{2}^{2} + 324 \)
acting on \(S_{6}^{\mathrm{new}}(525, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 37T^{2} + 324 \)
|
| $3$ |
\( (T^{2} + 81)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 2401)^{2} \)
|
| $11$ |
\( (T^{2} + 396 T - 26496)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 5031348624 \)
|
| $17$ |
\( T^{4} + \cdots + 1600903111824 \)
|
| $19$ |
\( (T^{2} - 1044 T - 5698624)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 13408955006976 \)
|
| $29$ |
\( (T^{2} - 3148 T - 10155612)^{2} \)
|
| $31$ |
\( (T^{2} + 4380 T - 16648672)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 139815205625104 \)
|
| $41$ |
\( (T^{2} + 15780 T - 115400700)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 81261715062784 \)
|
| $47$ |
\( T^{4} + \cdots + 25\!\cdots\!56 \)
|
| $53$ |
\( T^{4} + \cdots + 75\!\cdots\!96 \)
|
| $59$ |
\( (T^{2} - 11584 T - 116985744)^{2} \)
|
| $61$ |
\( (T^{2} - 40932 T - 200164156)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 40\!\cdots\!96 \)
|
| $71$ |
\( (T^{2} + 16476 T - 1677718656)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 29\!\cdots\!36 \)
|
| $79$ |
\( (T^{2} + 60952 T - 2262667136)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 13\!\cdots\!56 \)
|
| $89$ |
\( (T^{2} - 25476 T - 5196718908)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 12\!\cdots\!24 \)
|
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