Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,14,Mod(1,11)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 11 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 11.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: | \(11.7954021847\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 1179x^{3} + 1520x^{2} + 251749x + 900864 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 1179x^{3} + 1520x^{2} + 251749x + 900864 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{4} - 75\nu^{3} - 2689\nu^{2} + 64812\nu + 177344 ) / 119 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 25\nu^{3} + 1531\nu^{2} - 21128\nu - 358598 ) / 119 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{4} + 6\nu^{3} - 1830\nu^{2} + 1264\nu + 155670 ) / 21 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + \beta_{2} - 3\beta _1 + 7547 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{4} + \beta_{3} - 11\beta_{2} + 1497\beta _1 - 1335 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 150\beta_{4} + 2739\beta_{3} + 981\beta_{2} - 14255\beta _1 + 5665627 ) / 16 \)
|
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 |
|
−124.996 | −1750.38 | 7432.04 | 27373.3 | 218790. | 75214.1 | 94992.6 | 1.46950e6 | −3.42156e6 | |||||||||||||||||||||||||||||||||
| 1.2 | −93.8416 | 867.783 | 614.253 | −16509.6 | −81434.2 | 435884. | 711108. | −841275. | 1.54929e6 | ||||||||||||||||||||||||||||||||||
| 1.3 | 3.83503 | 2253.71 | −8177.29 | −38770.4 | 8643.04 | −547112. | −62776.7 | 3.48488e6 | −148686. | ||||||||||||||||||||||||||||||||||
| 1.4 | 38.6633 | −456.812 | −6697.15 | 62419.0 | −17661.8 | −258773. | −575664. | −1.38565e6 | 2.41332e6 | ||||||||||||||||||||||||||||||||||
| 1.5 | 112.339 | −434.303 | 4428.15 | −34966.3 | −48789.3 | −19134.4 | −422828. | −1.40570e6 | −3.92809e6 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
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 | 11.14.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 99.14.a.e | 5 | ||
| 4.b | odd | 2 | 1 | 176.14.a.e | 5 | ||
| 11.b | odd | 2 | 1 | 121.14.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.14.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 99.14.a.e | 5 | 3.b | odd | 2 | 1 | ||
| 121.14.a.b | 5 | 11.b | odd | 2 | 1 | ||
| 176.14.a.e | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 64T_{2}^{4} - 17232T_{2}^{3} - 755648T_{2}^{2} + 54095104T_{2} - 195385344 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(11))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 64 T^{4} + \cdots - 195385344 \)
|
| $3$ |
\( T^{5} + \cdots + 679157507188980 \)
|
| $5$ |
\( T^{5} + \cdots + 38\!\cdots\!50 \)
|
| $7$ |
\( T^{5} + \cdots + 88\!\cdots\!80 \)
|
| $11$ |
\( (T + 1771561)^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 87\!\cdots\!00 \)
|
| $17$ |
\( T^{5} + \cdots - 22\!\cdots\!84 \)
|
| $19$ |
\( T^{5} + \cdots + 16\!\cdots\!44 \)
|
| $23$ |
\( T^{5} + \cdots + 31\!\cdots\!24 \)
|
| $29$ |
\( T^{5} + \cdots + 71\!\cdots\!24 \)
|
| $31$ |
\( T^{5} + \cdots + 16\!\cdots\!92 \)
|
| $37$ |
\( T^{5} + \cdots - 10\!\cdots\!22 \)
|
| $41$ |
\( T^{5} + \cdots + 86\!\cdots\!88 \)
|
| $43$ |
\( T^{5} + \cdots + 15\!\cdots\!64 \)
|
| $47$ |
\( T^{5} + \cdots + 91\!\cdots\!12 \)
|
| $53$ |
\( T^{5} + \cdots - 63\!\cdots\!92 \)
|
| $59$ |
\( T^{5} + \cdots + 95\!\cdots\!44 \)
|
| $61$ |
\( T^{5} + \cdots + 34\!\cdots\!80 \)
|
| $67$ |
\( T^{5} + \cdots + 22\!\cdots\!36 \)
|
| $71$ |
\( T^{5} + \cdots + 10\!\cdots\!88 \)
|
| $73$ |
\( T^{5} + \cdots + 17\!\cdots\!44 \)
|
| $79$ |
\( T^{5} + \cdots + 60\!\cdots\!32 \)
|
| $83$ |
\( T^{5} + \cdots - 21\!\cdots\!96 \)
|
| $89$ |
\( T^{5} + \cdots + 64\!\cdots\!50 \)
|
| $97$ |
\( T^{5} + \cdots - 11\!\cdots\!70 \)
|
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