Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [572,2,Mod(441,572)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(572, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("572.441");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 572 = 2^{2} \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 572.f (of order \(2\), degree \(1\), 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: | \(4.56744299562\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 21x^{6} + 136x^{4} + 309x^{2} + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{8} + 21x^{6} + 136x^{4} + 309x^{2} + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} - 18\nu^{4} - 88\nu^{2} - 105 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 20\nu^{4} - 112\nu^{2} - 149 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -7\nu^{7} - 132\nu^{5} - 682\nu^{3} - 843\nu ) / 90 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{7} - 15\nu^{6} - 132\nu^{5} - 300\nu^{4} - 652\nu^{3} - 1620\nu^{2} - 663\nu - 1935 ) / 120 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{7} + 15\nu^{6} - 132\nu^{5} + 300\nu^{4} - 652\nu^{3} + 1620\nu^{2} - 663\nu + 1935 ) / 120 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 19\nu^{7} + 384\nu^{5} + 2224\nu^{3} + 3291\nu ) / 180 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{5} - 3\beta_{4} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{6} - 12\beta_{5} + 10\beta_{3} + 3\beta_{2} + 38 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{7} - 29\beta_{6} - 29\beta_{5} + 53\beta_{4} + 48\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -128\beta_{6} + 128\beta_{5} - 92\beta_{3} - 60\beta_{2} - 349 \)
|
| \(\nu^{7}\) | \(=\) |
\( -132\beta_{7} + 352\beta_{6} + 352\beta_{5} - 720\beta_{4} - 441\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).
| \(n\) | \(287\) | \(353\) | \(365\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 441.1 |
|
0 | −3.32727 | 0 | − | 2.74343i | 0 | 4.32727i | 0 | 8.07070 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 441.2 | 0 | −3.32727 | 0 | 2.74343i | 0 | − | 4.32727i | 0 | 8.07070 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 441.3 | 0 | −1.43491 | 0 | − | 4.37595i | 0 | − | 2.43491i | 0 | −0.941037 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 441.4 | 0 | −1.43491 | 0 | 4.37595i | 0 | 2.43491i | 0 | −0.941037 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 441.5 | 0 | 1.25126 | 0 | − | 2.18309i | 0 | 0.251260i | 0 | −1.43435 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 441.6 | 0 | 1.25126 | 0 | 2.18309i | 0 | − | 0.251260i | 0 | −1.43435 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 441.7 | 0 | 2.51091 | 0 | − | 3.81560i | 0 | − | 1.51091i | 0 | 3.30469 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 441.8 | 0 | 2.51091 | 0 | 3.81560i | 0 | 1.51091i | 0 | 3.30469 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 572.2.f.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 5148.2.e.c | 8 | ||
| 4.b | odd | 2 | 1 | 2288.2.j.i | 8 | ||
| 13.b | even | 2 | 1 | inner | 572.2.f.c | ✓ | 8 |
| 13.d | odd | 4 | 1 | 7436.2.a.o | 4 | ||
| 13.d | odd | 4 | 1 | 7436.2.a.p | 4 | ||
| 39.d | odd | 2 | 1 | 5148.2.e.c | 8 | ||
| 52.b | odd | 2 | 1 | 2288.2.j.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 572.2.f.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 572.2.f.c | ✓ | 8 | 13.b | even | 2 | 1 | inner |
| 2288.2.j.i | 8 | 4.b | odd | 2 | 1 | ||
| 2288.2.j.i | 8 | 52.b | odd | 2 | 1 | ||
| 5148.2.e.c | 8 | 3.b | odd | 2 | 1 | ||
| 5148.2.e.c | 8 | 39.d | odd | 2 | 1 | ||
| 7436.2.a.o | 4 | 13.d | odd | 4 | 1 | ||
| 7436.2.a.p | 4 | 13.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + T_{3}^{3} - 10T_{3}^{2} - 3T_{3} + 15 \)
acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + T^{3} - 10 T^{2} + \cdots + 15)^{2} \)
|
| $5$ |
\( T^{8} + 46 T^{6} + \cdots + 10000 \)
|
| $7$ |
\( T^{8} + 27 T^{6} + \cdots + 16 \)
|
| $11$ |
\( (T^{2} + 1)^{4} \)
|
| $13$ |
\( T^{8} + 8 T^{7} + \cdots + 28561 \)
|
| $17$ |
\( (T^{4} - 8 T^{3} + \cdots - 352)^{2} \)
|
| $19$ |
\( T^{8} + 107 T^{6} + \cdots + 106276 \)
|
| $23$ |
\( (T^{4} - 5 T^{3} + \cdots - 111)^{2} \)
|
| $29$ |
\( (T^{4} + 2 T^{3} + \cdots - 344)^{2} \)
|
| $31$ |
\( T^{8} + 190 T^{6} + \cdots + 283024 \)
|
| $37$ |
\( T^{8} + 122 T^{6} + \cdots + 309136 \)
|
| $41$ |
\( T^{8} + 275 T^{6} + \cdots + 9326916 \)
|
| $43$ |
\( (T^{4} + 10 T^{3} + \cdots - 1272)^{2} \)
|
| $47$ |
\( T^{8} + 132 T^{6} + \cdots + 30976 \)
|
| $53$ |
\( (T^{4} - 19 T^{3} + \cdots + 216)^{2} \)
|
| $59$ |
\( T^{8} + 266 T^{6} + \cdots + 156816 \)
|
| $61$ |
\( (T^{4} + 18 T^{3} + \cdots - 2440)^{2} \)
|
| $67$ |
\( T^{8} + 222 T^{6} + \cdots + 5776 \)
|
| $71$ |
\( T^{8} + 318 T^{6} + \cdots + 17424 \)
|
| $73$ |
\( T^{8} + 319 T^{6} + \cdots + 22429696 \)
|
| $79$ |
\( (T^{4} - 20 T^{3} + \cdots + 2456)^{2} \)
|
| $83$ |
\( T^{8} + 207 T^{6} + \cdots + 1119364 \)
|
| $89$ |
\( T^{8} + 446 T^{6} + \cdots + 23931664 \)
|
| $97$ |
\( T^{8} + 302 T^{6} + \cdots + 7683984 \)
|
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