Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8712,2,Mod(1,8712)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("8712.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8712.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: | \(69.5656702409\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.13625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 11x^{2} + 12x + 31 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 264) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 11x^{2} + 12x + 31 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 7\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - \nu - 7 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} + \beta _1 + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} + 8\beta _1 + 7 \)
|
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 |
|
0 | 0 | 0 | −3.41309 | 0 | −1.12667 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −2.50348 | 0 | −2.81465 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 1.50348 | 0 | 3.66875 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 2.41309 | 0 | −4.72744 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8712.2.a.bz | 4 | |
| 3.b | odd | 2 | 1 | 2904.2.a.bb | 4 | ||
| 11.b | odd | 2 | 1 | 8712.2.a.cc | 4 | ||
| 11.c | even | 5 | 2 | 792.2.r.f | 8 | ||
| 12.b | even | 2 | 1 | 5808.2.a.co | 4 | ||
| 33.d | even | 2 | 1 | 2904.2.a.be | 4 | ||
| 33.h | odd | 10 | 2 | 264.2.q.e | ✓ | 8 | |
| 132.d | odd | 2 | 1 | 5808.2.a.cl | 4 | ||
| 132.o | even | 10 | 2 | 528.2.y.k | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 264.2.q.e | ✓ | 8 | 33.h | odd | 10 | 2 | |
| 528.2.y.k | 8 | 132.o | even | 10 | 2 | ||
| 792.2.r.f | 8 | 11.c | even | 5 | 2 | ||
| 2904.2.a.bb | 4 | 3.b | odd | 2 | 1 | ||
| 2904.2.a.be | 4 | 33.d | even | 2 | 1 | ||
| 5808.2.a.cl | 4 | 132.d | odd | 2 | 1 | ||
| 5808.2.a.co | 4 | 12.b | even | 2 | 1 | ||
| 8712.2.a.bz | 4 | 1.a | even | 1 | 1 | trivial | |
| 8712.2.a.cc | 4 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8712))\):
|
\( T_{5}^{4} + 2T_{5}^{3} - 11T_{5}^{2} - 12T_{5} + 31 \)
|
|
\( T_{7}^{4} + 5T_{7}^{3} - 10T_{7}^{2} - 65T_{7} - 55 \)
|
|
\( T_{13}^{4} - 2T_{13}^{3} - 61T_{13}^{2} + 62T_{13} + 836 \)
|
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\( T_{17}^{4} + 13T_{17}^{3} + 49T_{17}^{2} + 52T_{17} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 31 \)
|
| $7$ |
\( T^{4} + 5 T^{3} + \cdots - 55 \)
|
| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} - 2 T^{3} + \cdots + 836 \)
|
| $17$ |
\( T^{4} + 13 T^{3} + \cdots + 16 \)
|
| $19$ |
\( T^{4} + 11 T^{3} + \cdots - 484 \)
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| $23$ |
\( T^{4} + 8 T^{3} + \cdots - 4 \)
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| $29$ |
\( T^{4} - 11 T^{3} + \cdots - 764 \)
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| $31$ |
\( T^{4} - 2 T^{3} + \cdots + 1441 \)
|
| $37$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
|
| $41$ |
\( T^{4} + 6 T^{3} + \cdots - 524 \)
|
| $43$ |
\( T^{4} + 24 T^{3} + \cdots + 716 \)
|
| $47$ |
\( T^{4} + 5 T^{3} + \cdots - 880 \)
|
| $53$ |
\( T^{4} + 2 T^{3} + \cdots - 19 \)
|
| $59$ |
\( T^{4} - 4 T^{3} + \cdots + 341 \)
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| $61$ |
\( T^{4} - 27 T^{3} + \cdots - 2644 \)
|
| $67$ |
\( T^{4} - 19 T^{3} + \cdots - 9424 \)
|
| $71$ |
\( T^{4} + 17 T^{3} + \cdots + 836 \)
|
| $73$ |
\( T^{4} - 15 T^{3} + \cdots - 100 \)
|
| $79$ |
\( T^{4} + 7 T^{3} + \cdots - 289 \)
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| $83$ |
\( T^{4} - T^{3} + \cdots + 121 \)
|
| $89$ |
\( T^{4} + 6 T^{3} + \cdots + 1156 \)
|
| $97$ |
\( T^{4} - 8 T^{3} + \cdots - 1109 \)
|
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