Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5625,2,Mod(1,5625)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5625, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5625.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5625 = 3^{2} \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5625.a (trivial) |
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: | yes |
| Analytic conductor: | \(44.9158511370\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.46980000000.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} + 14 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{7} + 12\nu^{5} - 44\nu^{3} + 48\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 12\nu^{4} + 45\nu^{2} - 53 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 12\nu^{5} + 45\nu^{3} - 53\nu \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{7} + 13\nu^{5} - 53\nu^{3} + 66\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 8\beta_{2} + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 9\beta_{6} + 8\beta_{4} + 27\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{5} + 12\beta_{3} + 51\beta_{2} + 89 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12\beta_{7} + 64\beta_{6} + 51\beta_{4} + 152\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.44055 | 0 | 3.95630 | 0 | 0 | 0.591023 | −4.77444 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.05160 | 0 | 2.20906 | 0 | 0 | 1.27977 | −0.428901 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.63148 | 0 | 0.661739 | 0 | 0 | −1.44512 | 2.18335 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.47408 | 0 | 0.172909 | 0 | 0 | 4.57433 | 2.69328 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.47408 | 0 | 0.172909 | 0 | 0 | 4.57433 | −2.69328 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.63148 | 0 | 0.661739 | 0 | 0 | −1.44512 | −2.18335 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.05160 | 0 | 2.20906 | 0 | 0 | 1.27977 | 0.428901 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.44055 | 0 | 3.95630 | 0 | 0 | 0.591023 | 4.77444 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5625.2.a.ba | yes | 8 |
| 3.b | odd | 2 | 1 | inner | 5625.2.a.ba | yes | 8 |
| 5.b | even | 2 | 1 | 5625.2.a.y | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 5625.2.a.y | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5625.2.a.y | ✓ | 8 | 5.b | even | 2 | 1 | |
| 5625.2.a.y | ✓ | 8 | 15.d | odd | 2 | 1 | |
| 5625.2.a.ba | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 5625.2.a.ba | yes | 8 | 3.b | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\):
|
\( T_{2}^{8} - 15T_{2}^{6} + 80T_{2}^{4} - 180T_{2}^{2} + 145 \)
|
|
\( T_{7}^{4} - 5T_{7}^{3} + 10T_{7} - 5 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 15 T^{6} + \cdots + 145 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} - 5 T^{3} + 10 T - 5)^{2} \)
|
| $11$ |
\( T^{8} - 60 T^{6} + \cdots + 3625 \)
|
| $13$ |
\( (T^{4} - 5 T^{3} - 10 T^{2} + \cdots - 5)^{2} \)
|
| $17$ |
\( T^{8} - 65 T^{6} + \cdots + 145 \)
|
| $19$ |
\( (T^{4} - T^{3} - 24 T^{2} + \cdots - 59)^{2} \)
|
| $23$ |
\( T^{8} - 60 T^{6} + \cdots + 11745 \)
|
| $29$ |
\( T^{8} - 210 T^{6} + \cdots + 3483625 \)
|
| $31$ |
\( (T^{4} - 2 T^{3} + \cdots + 211)^{2} \)
|
| $37$ |
\( (T^{4} - 90 T^{2} + \cdots - 405)^{2} \)
|
| $41$ |
\( T^{8} - 120 T^{6} + \cdots + 3625 \)
|
| $43$ |
\( (T^{4} - 25 T^{3} + \cdots + 895)^{2} \)
|
| $47$ |
\( T^{8} - 305 T^{6} + \cdots + 1148545 \)
|
| $53$ |
\( T^{8} - 180 T^{6} + \cdots + 145 \)
|
| $59$ |
\( T^{8} - 270 T^{6} + \cdots + 3483625 \)
|
| $61$ |
\( (T^{4} - 23 T^{3} + \cdots - 359)^{2} \)
|
| $67$ |
\( (T^{4} - 20 T^{3} + \cdots - 3455)^{2} \)
|
| $71$ |
\( T^{8} - 575 T^{6} + \cdots + 118758625 \)
|
| $73$ |
\( (T^{4} - 25 T^{3} + \cdots - 1055)^{2} \)
|
| $79$ |
\( (T^{4} + 6 T^{3} + \cdots - 279)^{2} \)
|
| $83$ |
\( T^{8} - 555 T^{6} + \cdots + 10648945 \)
|
| $89$ |
\( T^{8} - 525 T^{6} + \cdots + 76215625 \)
|
| $97$ |
\( (T^{4} - 25 T^{3} + \cdots - 7355)^{2} \)
|
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