Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 11 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(5.66539419780\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2659452.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 306x - 836 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 306x - 836 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 6\nu - 204 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 3\beta _1 + 204 \)
|
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 |
|
−37.4510 | 70.6582 | 890.580 | 613.897 | −2646.22 | −6611.82 | −14178.2 | −14690.4 | −22991.1 | |||||||||||||||||||||||||||
| 1.2 | 5.60816 | 5.22371 | −480.549 | −529.708 | 29.2954 | −3708.24 | −5566.37 | −19655.7 | −2970.69 | ||||||||||||||||||||||||||||
| 1.3 | 31.8429 | −261.882 | 501.969 | −1908.19 | −8339.07 | 3060.06 | −319.425 | 48899.1 | −60762.2 | ||||||||||||||||||||||||||||
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 | 11.10.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 99.10.a.b | 3 | ||
| 4.b | odd | 2 | 1 | 176.10.a.g | 3 | ||
| 5.b | even | 2 | 1 | 275.10.a.a | 3 | ||
| 11.b | odd | 2 | 1 | 121.10.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.10.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 99.10.a.b | 3 | 3.b | odd | 2 | 1 | ||
| 121.10.a.b | 3 | 11.b | odd | 2 | 1 | ||
| 176.10.a.g | 3 | 4.b | odd | 2 | 1 | ||
| 275.10.a.a | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 1224T_{2} + 6688 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(11))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 1224T + 6688 \)
|
| $3$ |
\( T^{3} + 186 T^{2} + \cdots + 96660 \)
|
| $5$ |
\( T^{3} + 1824 T^{2} + \cdots - 620517350 \)
|
| $7$ |
\( T^{3} + \cdots - 75027235360 \)
|
| $11$ |
\( (T + 14641)^{3} \)
|
| $13$ |
\( T^{3} + \cdots - 73087940648800 \)
|
| $17$ |
\( T^{3} + \cdots + 13\!\cdots\!12 \)
|
| $19$ |
\( T^{3} + \cdots - 20\!\cdots\!76 \)
|
| $23$ |
\( T^{3} + \cdots + 44\!\cdots\!32 \)
|
| $29$ |
\( T^{3} + \cdots - 61\!\cdots\!96 \)
|
| $31$ |
\( T^{3} + \cdots + 14\!\cdots\!36 \)
|
| $37$ |
\( T^{3} + \cdots + 95\!\cdots\!26 \)
|
| $41$ |
\( T^{3} + \cdots - 53\!\cdots\!12 \)
|
| $43$ |
\( T^{3} + \cdots + 12\!\cdots\!88 \)
|
| $47$ |
\( T^{3} + \cdots + 27\!\cdots\!48 \)
|
| $53$ |
\( T^{3} + \cdots - 34\!\cdots\!76 \)
|
| $59$ |
\( T^{3} + \cdots - 11\!\cdots\!88 \)
|
| $61$ |
\( T^{3} + \cdots - 99\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots + 92\!\cdots\!16 \)
|
| $71$ |
\( T^{3} + \cdots - 24\!\cdots\!84 \)
|
| $73$ |
\( T^{3} + \cdots - 66\!\cdots\!08 \)
|
| $79$ |
\( T^{3} + \cdots - 10\!\cdots\!48 \)
|
| $83$ |
\( T^{3} + \cdots - 48\!\cdots\!84 \)
|
| $89$ |
\( T^{3} + \cdots - 39\!\cdots\!30 \)
|
| $97$ |
\( T^{3} + \cdots + 24\!\cdots\!50 \)
|
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