Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8800,2,Mod(1,8800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8800, 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("8800.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8800 = 2^{5} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8800.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: | \(70.2683537787\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 1760) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 6\nu^{4} - 2\nu^{3} - 8\nu^{2} - 3\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 7\nu^{4} + 5\nu^{3} + 11\nu^{2} - 6\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{6} + 8\nu^{4} + 2\nu^{3} - 16\nu^{2} - 5\nu + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + 2\nu^{5} + 6\nu^{4} - 10\nu^{3} - 10\nu^{2} + 11\nu + 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( -2\nu^{6} + 16\nu^{4} + 2\nu^{3} - 30\nu^{2} + 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - 2\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{6} - 2\beta_{5} - 8\beta_{4} + 3\beta_{3} + 7\beta_{2} + 7\beta _1 + 2 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -6\beta_{5} - 6\beta_{4} - 5\beta_{3} + 7\beta_{2} + 9\beta _1 + 15 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5\beta_{6} - 8\beta_{5} - 20\beta_{4} + 3\beta_{3} + 21\beta_{2} + 23\beta _1 + 10 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( \beta_{6} - 68\beta_{5} - 74\beta_{4} - 47\beta_{3} + 89\beta_{2} + 121\beta _1 + 136 ) / 4 \)
|
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 | −3.19254 | 0 | 0 | 0 | −4.81353 | 0 | 7.19231 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.13376 | 0 | 0 | 0 | −1.20437 | 0 | 1.55295 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.874674 | 0 | 0 | 0 | 4.39214 | 0 | −2.23495 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.441641 | 0 | 0 | 0 | −4.16263 | 0 | −2.80495 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.45210 | 0 | 0 | 0 | 1.03231 | 0 | −0.891413 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.64481 | 0 | 0 | 0 | −1.42015 | 0 | −0.294593 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.54571 | 0 | 0 | 0 | −0.823764 | 0 | 3.48064 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 8800.2.a.cd | 7 | |
| 4.b | odd | 2 | 1 | 8800.2.a.ci | 7 | ||
| 5.b | even | 2 | 1 | 8800.2.a.cj | 7 | ||
| 5.c | odd | 4 | 2 | 1760.2.b.f | yes | 14 | |
| 20.d | odd | 2 | 1 | 8800.2.a.cc | 7 | ||
| 20.e | even | 4 | 2 | 1760.2.b.e | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1760.2.b.e | ✓ | 14 | 20.e | even | 4 | 2 | |
| 1760.2.b.f | yes | 14 | 5.c | odd | 4 | 2 | |
| 8800.2.a.cc | 7 | 20.d | odd | 2 | 1 | ||
| 8800.2.a.cd | 7 | 1.a | even | 1 | 1 | trivial | |
| 8800.2.a.ci | 7 | 4.b | odd | 2 | 1 | ||
| 8800.2.a.cj | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8800))\):
|
\( T_{3}^{7} + T_{3}^{6} - 13T_{3}^{5} - 7T_{3}^{4} + 46T_{3}^{3} + 12T_{3}^{2} - 40T_{3} - 16 \)
|
|
\( T_{7}^{7} + 7T_{7}^{6} - 8T_{7}^{5} - 136T_{7}^{4} - 232T_{7}^{3} + 16T_{7}^{2} + 256T_{7} + 128 \)
|
|
\( T_{13}^{7} + 10T_{13}^{6} - 240T_{13}^{4} - 448T_{13}^{3} + 1120T_{13}^{2} + 2624T_{13} + 1024 \)
|
|
\( T_{17}^{7} - 3T_{17}^{6} - 34T_{17}^{5} + 172T_{17}^{4} - 168T_{17}^{3} - 240T_{17}^{2} + 384T_{17} - 128 \)
|
|
\( T_{19}^{7} - 7T_{19}^{6} - 52T_{19}^{5} + 340T_{19}^{4} + 608T_{19}^{3} - 3584T_{19}^{2} + 3584T_{19} - 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} + T^{6} + \cdots - 16 \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} + 7 T^{6} + \cdots + 128 \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} + 10 T^{6} + \cdots + 1024 \)
|
| $17$ |
\( T^{7} - 3 T^{6} + \cdots - 128 \)
|
| $19$ |
\( T^{7} - 7 T^{6} + \cdots - 1024 \)
|
| $23$ |
\( T^{7} + 14 T^{6} + \cdots + 10816 \)
|
| $29$ |
\( T^{7} + 11 T^{6} + \cdots + 256 \)
|
| $31$ |
\( T^{7} + 11 T^{6} + \cdots + 7424 \)
|
| $37$ |
\( T^{7} + 13 T^{6} + \cdots - 256 \)
|
| $41$ |
\( T^{7} - 4 T^{6} + \cdots - 652544 \)
|
| $43$ |
\( T^{7} + 2 T^{6} + \cdots - 20864 \)
|
| $47$ |
\( T^{7} + 22 T^{6} + \cdots + 1262336 \)
|
| $53$ |
\( T^{7} - 3 T^{6} + \cdots + 8192 \)
|
| $59$ |
\( T^{7} - 26 T^{6} + \cdots + 708608 \)
|
| $61$ |
\( T^{7} + 5 T^{6} + \cdots + 20224 \)
|
| $67$ |
\( T^{7} + 14 T^{6} + \cdots - 5312 \)
|
| $71$ |
\( T^{7} - 3 T^{6} + \cdots + 256 \)
|
| $73$ |
\( T^{7} + 16 T^{6} + \cdots - 5921408 \)
|
| $79$ |
\( T^{7} + 32 T^{6} + \cdots + 483328 \)
|
| $83$ |
\( T^{7} + 16 T^{6} + \cdots - 14848 \)
|
| $89$ |
\( T^{7} - 11 T^{6} + \cdots - 859696 \)
|
| $97$ |
\( T^{7} - 8 T^{6} + \cdots - 842752 \)
|
| show more | |
| show less |