Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,16,Mod(24,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.24");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.b (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: | \(35.6733762750\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 24.1 |
|
− | 216.000i | − | 3348.00i | −13888.0 | 0 | −723168. | − | 2.82246e6i | − | 4.07808e6i | 3.13980e6 | 0 | ||||||||||||||||||||
| 24.2 | 216.000i | 3348.00i | −13888.0 | 0 | −723168. | 2.82246e6i | 4.07808e6i | 3.13980e6 | 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 | 25.16.b.a | 2 | |
| 5.b | even | 2 | 1 | inner | 25.16.b.a | 2 | |
| 5.c | odd | 4 | 1 | 1.16.a.a | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 25.16.a.a | 1 | ||
| 15.e | even | 4 | 1 | 9.16.a.a | 1 | ||
| 20.e | even | 4 | 1 | 16.16.a.d | 1 | ||
| 35.f | even | 4 | 1 | 49.16.a.a | 1 | ||
| 35.k | even | 12 | 2 | 49.16.c.b | 2 | ||
| 35.l | odd | 12 | 2 | 49.16.c.c | 2 | ||
| 40.i | odd | 4 | 1 | 64.16.a.i | 1 | ||
| 40.k | even | 4 | 1 | 64.16.a.c | 1 | ||
| 55.e | even | 4 | 1 | 121.16.a.a | 1 | ||
| 60.l | odd | 4 | 1 | 144.16.a.f | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.16.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 9.16.a.a | 1 | 15.e | even | 4 | 1 | ||
| 16.16.a.d | 1 | 20.e | even | 4 | 1 | ||
| 25.16.a.a | 1 | 5.c | odd | 4 | 1 | ||
| 25.16.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 25.16.b.a | 2 | 5.b | even | 2 | 1 | inner | |
| 49.16.a.a | 1 | 35.f | even | 4 | 1 | ||
| 49.16.c.b | 2 | 35.k | even | 12 | 2 | ||
| 49.16.c.c | 2 | 35.l | odd | 12 | 2 | ||
| 64.16.a.c | 1 | 40.k | even | 4 | 1 | ||
| 64.16.a.i | 1 | 40.i | odd | 4 | 1 | ||
| 121.16.a.a | 1 | 55.e | even | 4 | 1 | ||
| 144.16.a.f | 1 | 60.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 46656 \)
acting on \(S_{16}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 46656 \)
|
| $3$ |
\( T^{2} + 11209104 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 7966257871936 \)
|
| $11$ |
\( (T - 20586852)^{2} \)
|
| $13$ |
\( T^{2} + 36\!\cdots\!44 \)
|
| $17$ |
\( T^{2} + 27\!\cdots\!96 \)
|
| $19$ |
\( (T + 1563257180)^{2} \)
|
| $23$ |
\( T^{2} + 89\!\cdots\!84 \)
|
| $29$ |
\( (T - 36902568330)^{2} \)
|
| $31$ |
\( (T - 71588483552)^{2} \)
|
| $37$ |
\( T^{2} + 10\!\cdots\!16 \)
|
| $41$ |
\( (T - 1641974018202)^{2} \)
|
| $43$ |
\( T^{2} + 24\!\cdots\!64 \)
|
| $47$ |
\( T^{2} + 11\!\cdots\!76 \)
|
| $53$ |
\( T^{2} + 46\!\cdots\!04 \)
|
| $59$ |
\( (T + 9858856815540)^{2} \)
|
| $61$ |
\( (T - 4931842626902)^{2} \)
|
| $67$ |
\( T^{2} + 83\!\cdots\!96 \)
|
| $71$ |
\( (T - 125050114914552)^{2} \)
|
| $73$ |
\( T^{2} + 67\!\cdots\!84 \)
|
| $79$ |
\( (T - 25413078694480)^{2} \)
|
| $83$ |
\( T^{2} + 79\!\cdots\!24 \)
|
| $89$ |
\( (T + 715618564776810)^{2} \)
|
| $97$ |
\( T^{2} + 37\!\cdots\!76 \)
|
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