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: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{3169})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1585x^{2} + 627264 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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
\( x^{4} + 1585x^{2} + 627264 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 1583\nu ) / 396 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{3} + 3965\nu ) / 396 \)
|
| \(\beta_{3}\) | \(=\) |
\( 60\nu^{2} + 47550 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 5\beta_1 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 47550 ) / 60 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1583\beta_{2} - 3965\beta_1 ) / 30 \)
|
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 |
|
− | 323.882i | − | 532.237i | −72131.3 | 0 | −172382. | − | 502271.i | 1.27490e7i | 1.40656e7 | 0 | |||||||||||||||||||||||||||
| 24.2 | − | 13.8816i | 1207.76i | 32575.3 | 0 | 16765.7 | 2.91863e6i | − | 907071.i | 1.28902e7 | 0 | |||||||||||||||||||||||||||||
| 24.3 | 13.8816i | − | 1207.76i | 32575.3 | 0 | 16765.7 | − | 2.91863e6i | 907071.i | 1.28902e7 | 0 | |||||||||||||||||||||||||||||
| 24.4 | 323.882i | 532.237i | −72131.3 | 0 | −172382. | 502271.i | − | 1.27490e7i | 1.40656e7 | 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.b | 4 | |
| 5.b | even | 2 | 1 | inner | 25.16.b.b | 4 | |
| 5.c | odd | 4 | 1 | 5.16.a.a | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 25.16.a.b | 2 | ||
| 15.e | even | 4 | 1 | 45.16.a.c | 2 | ||
| 20.e | even | 4 | 1 | 80.16.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.16.a.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 25.16.a.b | 2 | 5.c | odd | 4 | 1 | ||
| 25.16.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 25.16.b.b | 4 | 5.b | even | 2 | 1 | inner | |
| 45.16.a.c | 2 | 15.e | even | 4 | 1 | ||
| 80.16.a.d | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 105092T_{2}^{2} + 20214016 \)
acting on \(S_{16}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 105092 T^{2} + 20214016 \)
|
| $3$ |
\( T^{4} + \cdots + 413212409856 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 21\!\cdots\!96 \)
|
| $11$ |
\( (T^{2} + \cdots - 97\!\cdots\!96)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 12\!\cdots\!16 \)
|
| $17$ |
\( T^{4} + \cdots + 61\!\cdots\!56 \)
|
| $19$ |
\( (T^{2} + \cdots - 54\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 28\!\cdots\!76 \)
|
| $29$ |
\( (T^{2} + \cdots - 84\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots - 17\!\cdots\!96)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 29\!\cdots\!76 \)
|
| $41$ |
\( (T^{2} + \cdots + 22\!\cdots\!04)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 20\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 24\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 10\!\cdots\!56 \)
|
| $59$ |
\( (T^{2} + \cdots + 20\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots + 89\!\cdots\!04)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 55\!\cdots\!56 \)
|
| $71$ |
\( (T^{2} + \cdots + 31\!\cdots\!04)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 24\!\cdots\!76 \)
|
| $79$ |
\( (T^{2} + \cdots + 59\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 14\!\cdots\!36 \)
|
| $89$ |
\( (T^{2} + \cdots + 81\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 45\!\cdots\!36 \)
|
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