Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1521,4,Mod(1,1521)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1521.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1521 = 3^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1521.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: | \(89.7419051187\) |
| 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: |
\( x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 117) |
| 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
\( x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 35\nu^{3} + 210\nu ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 51\nu^{3} - 546\nu ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 43\nu^{4} + 426\nu^{2} - 688 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{4} - 33\nu^{2} + 156 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 46\nu^{5} + 547\nu^{3} - 1590\nu ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 21\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} + 33\beta_{2} + 273 \)
|
| \(\nu^{5}\) | \(=\) |
\( 35\beta_{4} + 51\beta_{3} + 525\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 86\beta_{6} + 16\beta_{5} + 993\beta_{2} + 6889 \)
|
| \(\nu^{7}\) | \(=\) |
\( 16\beta_{7} + 1063\beta_{4} + 1799\beta_{3} + 14253\beta_1 \)
|
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 |
|
−5.39589 | 0 | 21.1156 | −13.0421 | 0 | −6.42494 | −70.7705 | 0 | 70.3740 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.69212 | 0 | 5.63172 | 18.7574 | 0 | −24.1383 | 8.74396 | 0 | −69.2546 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.75628 | 0 | −0.402937 | −0.313209 | 0 | 28.5660 | 23.1608 | 0 | 0.863291 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.28670 | 0 | −6.34441 | −12.4484 | 0 | −9.00273 | 18.4569 | 0 | 16.0173 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.28670 | 0 | −6.34441 | 12.4484 | 0 | −9.00273 | −18.4569 | 0 | 16.0173 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.75628 | 0 | −0.402937 | 0.313209 | 0 | 28.5660 | −23.1608 | 0 | 0.863291 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.69212 | 0 | 5.63172 | −18.7574 | 0 | −24.1383 | −8.74396 | 0 | −69.2546 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.39589 | 0 | 21.1156 | 13.0421 | 0 | −6.42494 | 70.7705 | 0 | 70.3740 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(13\) | \(1\) |
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 | 1521.4.a.bc | 8 | |
| 3.b | odd | 2 | 1 | inner | 1521.4.a.bc | 8 | |
| 13.b | even | 2 | 1 | 1521.4.a.bd | 8 | ||
| 13.c | even | 3 | 2 | 117.4.g.f | ✓ | 16 | |
| 39.d | odd | 2 | 1 | 1521.4.a.bd | 8 | ||
| 39.i | odd | 6 | 2 | 117.4.g.f | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.4.g.f | ✓ | 16 | 13.c | even | 3 | 2 | |
| 117.4.g.f | ✓ | 16 | 39.i | odd | 6 | 2 | |
| 1521.4.a.bc | 8 | 1.a | even | 1 | 1 | trivial | |
| 1521.4.a.bc | 8 | 3.b | odd | 2 | 1 | inner | |
| 1521.4.a.bd | 8 | 13.b | even | 2 | 1 | ||
| 1521.4.a.bd | 8 | 39.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):
|
\( T_{2}^{8} - 52T_{2}^{6} + 805T_{2}^{4} - 4210T_{2}^{2} + 4992 \)
|
|
\( T_{5}^{8} - 677T_{5}^{6} + 140795T_{5}^{4} - 9287935T_{5}^{2} + 909792 \)
|
|
\( T_{7}^{4} + 11T_{7}^{3} - 700T_{7}^{2} - 10894T_{7} - 39884 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 52 T^{6} + \cdots + 4992 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 677 T^{6} + \cdots + 909792 \)
|
| $7$ |
\( (T^{4} + 11 T^{3} + \cdots - 39884)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 17384221390848 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} + \cdots + 30103640387808 \)
|
| $19$ |
\( (T^{4} - 122 T^{3} + \cdots + 14520480)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 203262208261632 \)
|
| $29$ |
\( T^{8} + \cdots + 24\!\cdots\!48 \)
|
| $31$ |
\( (T^{4} - 121 T^{3} + \cdots - 349762400)^{2} \)
|
| $37$ |
\( (T^{4} - 509 T^{3} + \cdots - 1178984222)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 14\!\cdots\!92 \)
|
| $43$ |
\( (T^{4} - 37 T^{3} + \cdots + 1323590892)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 36\!\cdots\!52 \)
|
| $53$ |
\( T^{8} + \cdots + 44\!\cdots\!28 \)
|
| $59$ |
\( T^{8} + \cdots + 18\!\cdots\!88 \)
|
| $61$ |
\( (T^{4} - 574 T^{3} + \cdots + 684529261)^{2} \)
|
| $67$ |
\( (T^{4} + 1099 T^{3} + \cdots + 2700497644)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 12\!\cdots\!00 \)
|
| $73$ |
\( (T^{4} + 1088 T^{3} + \cdots + 20997802657)^{2} \)
|
| $79$ |
\( (T^{4} - 931 T^{3} + \cdots - 9861650240)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 23\!\cdots\!32 \)
|
| $89$ |
\( T^{8} + \cdots + 14\!\cdots\!68 \)
|
| $97$ |
\( (T^{4} + 2185 T^{3} + \cdots + 291511171744)^{2} \)
|
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