Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7569,2,Mod(1,7569)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7569, 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("7569.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7569 = 3^{2} \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7569.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: | \(60.4387692899\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.14884000000.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 8\nu^{4} + 16\nu^{2} - 7 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 8\nu^{5} + 16\nu^{3} - 7\nu ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 12\nu^{5} - 40\nu^{3} + 27\nu ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 12\nu^{5} + 44\nu^{3} - 43\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 5\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{7} + 7\beta_{6} + \beta_{5} + 19\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4\beta_{4} + 8\beta_{3} + 24\beta_{2} + 71 \)
|
| \(\nu^{7}\) | \(=\) |
\( 32\beta_{7} + 40\beta_{6} + 12\beta_{5} + 95\beta_1 \)
|
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 |
|
−2.30927 | 0 | 3.33275 | −3.03582 | 0 | −4.39250 | −3.07768 | 0 | 7.01054 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.08529 | 0 | 2.34841 | 0.992398 | 0 | 2.45140 | −0.726543 | 0 | −2.06943 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.13370 | 0 | −0.714715 | −0.407162 | 0 | 2.15643 | 3.07768 | 0 | 0.461601 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.183172 | 0 | −1.96645 | −3.26086 | 0 | −0.215332 | 0.726543 | 0 | 0.597298 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.183172 | 0 | −1.96645 | 3.26086 | 0 | −0.215332 | −0.726543 | 0 | 0.597298 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.13370 | 0 | −0.714715 | 0.407162 | 0 | 2.15643 | −3.07768 | 0 | 0.461601 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.08529 | 0 | 2.34841 | −0.992398 | 0 | 2.45140 | 0.726543 | 0 | −2.06943 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.30927 | 0 | 3.33275 | 3.03582 | 0 | −4.39250 | 3.07768 | 0 | 7.01054 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 7569.2.a.bg | yes | 8 |
| 3.b | odd | 2 | 1 | inner | 7569.2.a.bg | yes | 8 |
| 29.b | even | 2 | 1 | 7569.2.a.bf | ✓ | 8 | |
| 87.d | odd | 2 | 1 | 7569.2.a.bf | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7569.2.a.bf | ✓ | 8 | 29.b | even | 2 | 1 | |
| 7569.2.a.bf | ✓ | 8 | 87.d | odd | 2 | 1 | |
| 7569.2.a.bg | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 7569.2.a.bg | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):
|
\( T_{2}^{8} - 11T_{2}^{6} + 36T_{2}^{4} - 31T_{2}^{2} + 1 \)
|
|
\( T_{5}^{8} - 21T_{5}^{6} + 121T_{5}^{4} - 116T_{5}^{2} + 16 \)
|
|
\( T_{7}^{4} - 15T_{7}^{2} + 20T_{7} + 5 \)
|
|
\( T_{19}^{2} + 5T_{19} + 5 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 11 T^{6} + \cdots + 1 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 21 T^{6} + \cdots + 16 \)
|
| $7$ |
\( (T^{4} - 15 T^{2} + 20 T + 5)^{2} \)
|
| $11$ |
\( T^{8} - 54 T^{6} + \cdots + 1936 \)
|
| $13$ |
\( (T^{4} - T^{3} - 14 T^{2} + \cdots + 1)^{2} \)
|
| $17$ |
\( T^{8} - 84 T^{6} + \cdots + 4096 \)
|
| $19$ |
\( (T^{2} + 5 T + 5)^{4} \)
|
| $23$ |
\( T^{8} - 149 T^{6} + \cdots + 633616 \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( (T^{4} + 5 T^{3} - 10 T^{2} + \cdots + 5)^{2} \)
|
| $37$ |
\( (T^{4} + 9 T^{3} - 39 T^{2} + \cdots - 4)^{2} \)
|
| $41$ |
\( T^{8} - 166 T^{6} + \cdots + 38416 \)
|
| $43$ |
\( (T^{4} + 14 T^{3} + \cdots + 3041)^{2} \)
|
| $47$ |
\( T^{8} - 221 T^{6} + \cdots + 1936 \)
|
| $53$ |
\( (T^{4} - 68 T^{2} + 976)^{2} \)
|
| $59$ |
\( T^{8} - 261 T^{6} + \cdots + 246016 \)
|
| $61$ |
\( (T^{4} + 27 T^{3} + \cdots + 941)^{2} \)
|
| $67$ |
\( (T^{4} - 19 T^{3} + \cdots - 4519)^{2} \)
|
| $71$ |
\( T^{8} - 465 T^{6} + \cdots + 93702400 \)
|
| $73$ |
\( (T^{4} - 8 T^{3} + \cdots + 6151)^{2} \)
|
| $79$ |
\( (T^{4} + 7 T^{3} + \cdots + 1991)^{2} \)
|
| $83$ |
\( T^{8} - 526 T^{6} + \cdots + 91317136 \)
|
| $89$ |
\( T^{8} - 274 T^{6} + \cdots + 190096 \)
|
| $97$ |
\( (T^{4} - 3 T^{3} + \cdots + 4516)^{2} \)
|
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