Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2025,2,Mod(649,2025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2025, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2025.649");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2025 = 3^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2025.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: | \(16.1697064093\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 405) |
| 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
| \(\beta_{1}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{12}^{3} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( 2\beta_{3} + \beta_{2} + 2 ) / 4 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( \beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(1702\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 649.1 |
|
− | 2.73205i | 0 | −5.46410 | 0 | 0 | − | 1.26795i | 9.46410i | 0 | 0 | ||||||||||||||||||||||||||||
| 649.2 | − | 0.732051i | 0 | 1.46410 | 0 | 0 | 4.73205i | − | 2.53590i | 0 | 0 | |||||||||||||||||||||||||||||
| 649.3 | 0.732051i | 0 | 1.46410 | 0 | 0 | − | 4.73205i | 2.53590i | 0 | 0 | ||||||||||||||||||||||||||||||
| 649.4 | 2.73205i | 0 | −5.46410 | 0 | 0 | 1.26795i | − | 9.46410i | 0 | 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 | 2025.2.b.g | 4 | |
| 3.b | odd | 2 | 1 | 2025.2.b.h | 4 | ||
| 5.b | even | 2 | 1 | inner | 2025.2.b.g | 4 | |
| 5.c | odd | 4 | 1 | 405.2.a.g | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 2025.2.a.m | 2 | ||
| 15.d | odd | 2 | 1 | 2025.2.b.h | 4 | ||
| 15.e | even | 4 | 1 | 405.2.a.h | yes | 2 | |
| 15.e | even | 4 | 1 | 2025.2.a.g | 2 | ||
| 20.e | even | 4 | 1 | 6480.2.a.br | 2 | ||
| 45.k | odd | 12 | 2 | 405.2.e.l | 4 | ||
| 45.l | even | 12 | 2 | 405.2.e.i | 4 | ||
| 60.l | odd | 4 | 1 | 6480.2.a.bi | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 405.2.a.g | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 405.2.a.h | yes | 2 | 15.e | even | 4 | 1 | |
| 405.2.e.i | 4 | 45.l | even | 12 | 2 | ||
| 405.2.e.l | 4 | 45.k | odd | 12 | 2 | ||
| 2025.2.a.g | 2 | 15.e | even | 4 | 1 | ||
| 2025.2.a.m | 2 | 5.c | odd | 4 | 1 | ||
| 2025.2.b.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 2025.2.b.g | 4 | 5.b | even | 2 | 1 | inner | |
| 2025.2.b.h | 4 | 3.b | odd | 2 | 1 | ||
| 2025.2.b.h | 4 | 15.d | odd | 2 | 1 | ||
| 6480.2.a.bi | 2 | 60.l | odd | 4 | 1 | ||
| 6480.2.a.br | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2025, [\chi])\):
|
\( T_{2}^{4} + 8T_{2}^{2} + 4 \)
|
|
\( T_{11}^{2} + 8T_{11} + 13 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 8T^{2} + 4 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 24T^{2} + 36 \)
|
| $11$ |
\( (T^{2} + 8 T + 13)^{2} \)
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| $13$ |
\( T^{4} + 32T^{2} + 64 \)
|
| $17$ |
\( T^{4} + 8T^{2} + 4 \)
|
| $19$ |
\( (T^{2} + 2 T - 11)^{2} \)
|
| $23$ |
\( (T^{2} + 12)^{2} \)
|
| $29$ |
\( (T^{2} - 4 T - 23)^{2} \)
|
| $31$ |
\( (T + 3)^{4} \)
|
| $37$ |
\( T^{4} + 8T^{2} + 4 \)
|
| $41$ |
\( (T^{2} + 4 T - 23)^{2} \)
|
| $43$ |
\( T^{4} + 104T^{2} + 4 \)
|
| $47$ |
\( T^{4} + 104T^{2} + 2116 \)
|
| $53$ |
\( T^{4} + 56T^{2} + 484 \)
|
| $59$ |
\( (T^{2} - 20 T + 97)^{2} \)
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| $61$ |
\( (T - 4)^{4} \)
|
| $67$ |
\( (T^{2} + 12)^{2} \)
|
| $71$ |
\( (T^{2} + 4 T + 1)^{2} \)
|
| $73$ |
\( T^{4} + 152T^{2} + 5476 \)
|
| $79$ |
\( (T^{2} + 24 T + 132)^{2} \)
|
| $83$ |
\( T^{4} + 72T^{2} + 324 \)
|
| $89$ |
\( (T^{2} - 27)^{2} \)
|
| $97$ |
\( T^{4} + 152T^{2} + 5476 \)
|
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