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: | \(8\) |
| Coefficient field: | 8.0.49787136.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 5\nu^{5} + 15\nu^{3} + 42\nu ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + 5\nu^{4} + 15\nu^{2} + 8 ) / 20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 3\nu^{4} - \nu^{2} - 8 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{5} - 5\nu^{3} - 4\nu ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{6} + 5\nu^{4} + 15\nu^{2} + 44 ) / 20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - \nu^{5} - 3\nu^{3} - 6\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{4} + 2\beta_{2} - 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} - \beta_{5} + 5\beta_{3} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{6} - 3\beta_{4} + \beta_{2} - 2 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5\beta_{7} - 6\beta_{5} - 6\beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 5\beta_{6} - 5\beta_{2} - 9 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -10\beta_{7} + 3\beta_{5} - 3\beta_{3} - 7\beta_1 ) / 2 \)
|
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 |
|
− | 2.18890i | 0 | −2.79129 | 0 | 0 | − | 3.79129i | 1.73205i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 649.2 | − | 2.18890i | 0 | −2.79129 | 0 | 0 | 3.79129i | 1.73205i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 649.3 | − | 0.456850i | 0 | 1.79129 | 0 | 0 | − | 0.791288i | − | 1.73205i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 649.4 | − | 0.456850i | 0 | 1.79129 | 0 | 0 | 0.791288i | − | 1.73205i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 649.5 | 0.456850i | 0 | 1.79129 | 0 | 0 | − | 0.791288i | 1.73205i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 649.6 | 0.456850i | 0 | 1.79129 | 0 | 0 | 0.791288i | 1.73205i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 649.7 | 2.18890i | 0 | −2.79129 | 0 | 0 | − | 3.79129i | − | 1.73205i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 649.8 | 2.18890i | 0 | −2.79129 | 0 | 0 | 3.79129i | − | 1.73205i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 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.p | 8 | |
| 3.b | odd | 2 | 1 | inner | 2025.2.b.p | 8 | |
| 5.b | even | 2 | 1 | inner | 2025.2.b.p | 8 | |
| 5.c | odd | 4 | 1 | 2025.2.a.u | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 2025.2.a.v | yes | 4 | |
| 15.d | odd | 2 | 1 | inner | 2025.2.b.p | 8 | |
| 15.e | even | 4 | 1 | 2025.2.a.u | ✓ | 4 | |
| 15.e | even | 4 | 1 | 2025.2.a.v | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2025.2.a.u | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 2025.2.a.u | ✓ | 4 | 15.e | even | 4 | 1 | |
| 2025.2.a.v | yes | 4 | 5.c | odd | 4 | 1 | |
| 2025.2.a.v | yes | 4 | 15.e | even | 4 | 1 | |
| 2025.2.b.p | 8 | 1.a | even | 1 | 1 | trivial | |
| 2025.2.b.p | 8 | 3.b | odd | 2 | 1 | inner | |
| 2025.2.b.p | 8 | 5.b | even | 2 | 1 | inner | |
| 2025.2.b.p | 8 | 15.d | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2025, [\chi])\):
|
\( T_{2}^{4} + 5T_{2}^{2} + 1 \)
|
|
\( T_{11}^{4} - 5T_{11}^{2} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 5 T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} + 15 T^{2} + 9)^{2} \)
|
| $11$ |
\( (T^{4} - 5 T^{2} + 1)^{2} \)
|
| $13$ |
\( (T^{4} + 35 T^{2} + 49)^{2} \)
|
| $17$ |
\( (T^{4} + 89 T^{2} + 1849)^{2} \)
|
| $19$ |
\( (T^{2} + 2 T - 20)^{4} \)
|
| $23$ |
\( (T^{4} + 69 T^{2} + 9)^{2} \)
|
| $29$ |
\( (T^{2} - 28)^{4} \)
|
| $31$ |
\( (T - 3)^{8} \)
|
| $37$ |
\( (T^{4} + 23 T^{2} + 1)^{2} \)
|
| $41$ |
\( (T^{4} - 125 T^{2} + 3481)^{2} \)
|
| $43$ |
\( (T^{4} + 119 T^{2} + 1225)^{2} \)
|
| $47$ |
\( (T^{4} + 38 T^{2} + 25)^{2} \)
|
| $53$ |
\( (T^{4} + 257 T^{2} + 15625)^{2} \)
|
| $59$ |
\( (T^{4} - 185 T^{2} + 7921)^{2} \)
|
| $61$ |
\( (T^{2} + T - 47)^{4} \)
|
| $67$ |
\( (T^{4} + 114 T^{2} + 225)^{2} \)
|
| $71$ |
\( (T^{4} - 230 T^{2} + 11881)^{2} \)
|
| $73$ |
\( (T^{4} + 218 T^{2} + 3481)^{2} \)
|
| $79$ |
\( (T^{2} - 12 T - 48)^{4} \)
|
| $83$ |
\( (T^{2} + 27)^{4} \)
|
| $89$ |
\( (T^{4} - 180 T^{2} + 1296)^{2} \)
|
| $97$ |
\( (T^{4} + 200 T^{2} + 4624)^{2} \)
|
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