Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3969,2,Mod(1,3969)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3969, 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("3969.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3969 = 3^{4} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3969.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: | \(31.6926245622\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.59351616.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 12x^{4} + 21x^{2} - 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 441) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 12x^{4} + 21x^{2} - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 12\nu^{3} + 21\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 12\nu^{2} + 15 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 21\nu^{3} + 15\nu ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{4} - 21\nu^{2} + 15 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 11\nu^{3} + 9\nu \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{3} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 2\beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{5} + 4\beta_{3} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{4} - 21\beta_{2} + 45 \)
|
| \(\nu^{5}\) | \(=\) |
\( -29\beta_{5} + 41\beta_{3} + 8\beta_1 \)
|
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 |
|
−2.46050 | 0 | 4.05408 | −3.65808 | 0 | 0 | −5.05408 | 0 | 9.00071 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.46050 | 0 | 4.05408 | 3.65808 | 0 | 0 | −5.05408 | 0 | −9.00071 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.239123 | 0 | −1.94282 | −2.59179 | 0 | 0 | 0.942820 | 0 | 0.619757 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.239123 | 0 | −1.94282 | 2.59179 | 0 | 0 | 0.942820 | 0 | −0.619757 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.69963 | 0 | 0.888736 | −0.949271 | 0 | 0 | −1.88874 | 0 | −1.61341 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.69963 | 0 | 0.888736 | 0.949271 | 0 | 0 | −1.88874 | 0 | 1.61341 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3969.2.a.bd | 6 | |
| 3.b | odd | 2 | 1 | 3969.2.a.be | 6 | ||
| 7.b | odd | 2 | 1 | inner | 3969.2.a.bd | 6 | |
| 9.c | even | 3 | 2 | 1323.2.f.g | 12 | ||
| 9.d | odd | 6 | 2 | 441.2.f.g | ✓ | 12 | |
| 21.c | even | 2 | 1 | 3969.2.a.be | 6 | ||
| 63.g | even | 3 | 2 | 1323.2.g.g | 12 | ||
| 63.h | even | 3 | 2 | 1323.2.h.g | 12 | ||
| 63.i | even | 6 | 2 | 441.2.h.g | 12 | ||
| 63.j | odd | 6 | 2 | 441.2.h.g | 12 | ||
| 63.k | odd | 6 | 2 | 1323.2.g.g | 12 | ||
| 63.l | odd | 6 | 2 | 1323.2.f.g | 12 | ||
| 63.n | odd | 6 | 2 | 441.2.g.g | 12 | ||
| 63.o | even | 6 | 2 | 441.2.f.g | ✓ | 12 | |
| 63.s | even | 6 | 2 | 441.2.g.g | 12 | ||
| 63.t | odd | 6 | 2 | 1323.2.h.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 441.2.f.g | ✓ | 12 | 9.d | odd | 6 | 2 | |
| 441.2.f.g | ✓ | 12 | 63.o | even | 6 | 2 | |
| 441.2.g.g | 12 | 63.n | odd | 6 | 2 | ||
| 441.2.g.g | 12 | 63.s | even | 6 | 2 | ||
| 441.2.h.g | 12 | 63.i | even | 6 | 2 | ||
| 441.2.h.g | 12 | 63.j | odd | 6 | 2 | ||
| 1323.2.f.g | 12 | 9.c | even | 3 | 2 | ||
| 1323.2.f.g | 12 | 63.l | odd | 6 | 2 | ||
| 1323.2.g.g | 12 | 63.g | even | 3 | 2 | ||
| 1323.2.g.g | 12 | 63.k | odd | 6 | 2 | ||
| 1323.2.h.g | 12 | 63.h | even | 3 | 2 | ||
| 1323.2.h.g | 12 | 63.t | odd | 6 | 2 | ||
| 3969.2.a.bd | 6 | 1.a | even | 1 | 1 | trivial | |
| 3969.2.a.bd | 6 | 7.b | odd | 2 | 1 | inner | |
| 3969.2.a.be | 6 | 3.b | odd | 2 | 1 | ||
| 3969.2.a.be | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):
|
\( T_{2}^{3} + T_{2}^{2} - 4T_{2} - 1 \)
|
|
\( T_{5}^{6} - 21T_{5}^{4} + 108T_{5}^{2} - 81 \)
|
|
\( T_{11}^{3} + 4T_{11}^{2} - T_{11} - 1 \)
|
|
\( T_{13}^{6} - 39T_{13}^{4} + 351T_{13}^{2} - 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} + T^{2} - 4 T - 1)^{2} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 21 T^{4} + \cdots - 81 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T^{3} + 4 T^{2} - T - 1)^{2} \)
|
| $13$ |
\( T^{6} - 39 T^{4} + \cdots - 81 \)
|
| $17$ |
\( T^{6} - 84 T^{4} + \cdots - 3969 \)
|
| $19$ |
\( T^{6} - 75 T^{4} + \cdots - 3969 \)
|
| $23$ |
\( (T^{3} + 2 T^{2} - 25 T - 59)^{2} \)
|
| $29$ |
\( (T^{3} + 11 T^{2} + \cdots - 89)^{2} \)
|
| $31$ |
\( T^{6} - 129 T^{4} + \cdots - 77841 \)
|
| $37$ |
\( (T^{3} + 3 T^{2} - 24 T + 27)^{2} \)
|
| $41$ |
\( T^{6} - 162 T^{4} + \cdots - 6561 \)
|
| $43$ |
\( (T^{3} - 3 T^{2} - 24 T - 27)^{2} \)
|
| $47$ |
\( T^{6} - 183 T^{4} + \cdots - 194481 \)
|
| $53$ |
\( (T^{3} + 14 T^{2} + \cdots - 263)^{2} \)
|
| $59$ |
\( T^{6} - 183 T^{4} + \cdots - 149769 \)
|
| $61$ |
\( T^{6} - 264 T^{4} + \cdots - 558009 \)
|
| $67$ |
\( (T^{3} - 111 T + 353)^{2} \)
|
| $71$ |
\( (T^{3} + 19 T^{2} + \cdots + 227)^{2} \)
|
| $73$ |
\( T^{6} - 75 T^{4} + \cdots - 3969 \)
|
| $79$ |
\( (T^{3} + 3 T^{2} + \cdots - 107)^{2} \)
|
| $83$ |
\( T^{6} - 228 T^{4} + \cdots - 227529 \)
|
| $89$ |
\( T^{6} - 246 T^{4} + \cdots - 3969 \)
|
| $97$ |
\( T^{6} - 111 T^{4} + \cdots - 9801 \)
|
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