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: | \(0\) |
| 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: |
\( x^{5} - x^{4} - 1608x^{3} - 7720x^{2} + 616135x + 6122025 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 1608x^{3} - 7720x^{2} + 616135x + 6122025 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 66\nu^{3} - 438\nu^{2} + 47270\nu + 92625 ) / 1560 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 66\nu^{3} + 1998\nu^{2} - 59750\nu - 1094145 ) / 1560 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 14\nu^{3} - 1322\nu^{2} + 10506\nu + 377949 ) / 312 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 8\beta _1 + 642 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{4} + 17\beta_{3} - 13\beta_{2} + 843\beta _1 + 5427 \)
|
| \(\nu^{4}\) | \(=\) |
\( 396\beta_{4} + 1560\beta_{3} + 1140\beta_{2} + 11872\beta _1 + 546753 \)
|
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 |
|
−25.5978 | −87.4752 | 143.246 | −1673.94 | 2239.17 | 11699.6 | 9439.27 | −12031.1 | 42849.1 | |||||||||||||||||||||||||||||||||
| 1.2 | −15.6530 | −245.723 | −266.983 | 2148.33 | 3846.30 | −7680.33 | 12193.4 | 40696.7 | −33627.8 | ||||||||||||||||||||||||||||||||||
| 1.3 | −9.63895 | 265.389 | −419.091 | 610.442 | −2558.07 | 4856.42 | 8974.73 | 50748.6 | −5884.02 | ||||||||||||||||||||||||||||||||||
| 1.4 | 29.1027 | −6.46828 | 334.967 | 2255.95 | −188.244 | 6425.84 | −5152.14 | −19641.2 | 65654.3 | ||||||||||||||||||||||||||||||||||
| 1.5 | 37.7870 | 186.277 | 915.861 | −1746.78 | 7038.85 | −6901.51 | 15260.7 | 15016.0 | −66005.6 | ||||||||||||||||||||||||||||||||||
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.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 99.10.a.f | 5 | ||
| 4.b | odd | 2 | 1 | 176.10.a.j | 5 | ||
| 5.b | even | 2 | 1 | 275.10.a.b | 5 | ||
| 11.b | odd | 2 | 1 | 121.10.a.c | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.10.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 99.10.a.f | 5 | 3.b | odd | 2 | 1 | ||
| 121.10.a.c | 5 | 11.b | odd | 2 | 1 | ||
| 176.10.a.j | 5 | 4.b | odd | 2 | 1 | ||
| 275.10.a.b | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 16T_{2}^{4} - 1506T_{2}^{3} + 6428T_{2}^{2} + 619552T_{2} + 4247232 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(11))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 16 T^{4} + \cdots + 4247232 \)
|
| $3$ |
\( T^{5} + \cdots + 6873235452 \)
|
| $5$ |
\( T^{5} + \cdots - 86\!\cdots\!50 \)
|
| $7$ |
\( T^{5} + \cdots - 19\!\cdots\!68 \)
|
| $11$ |
\( (T - 14641)^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 12\!\cdots\!88 \)
|
| $17$ |
\( T^{5} + \cdots + 68\!\cdots\!64 \)
|
| $19$ |
\( T^{5} + \cdots - 20\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 28\!\cdots\!52 \)
|
| $29$ |
\( T^{5} + \cdots + 30\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 34\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots + 13\!\cdots\!26 \)
|
| $41$ |
\( T^{5} + \cdots + 78\!\cdots\!72 \)
|
| $43$ |
\( T^{5} + \cdots + 13\!\cdots\!56 \)
|
| $47$ |
\( T^{5} + \cdots - 16\!\cdots\!84 \)
|
| $53$ |
\( T^{5} + \cdots + 50\!\cdots\!64 \)
|
| $59$ |
\( T^{5} + \cdots + 37\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 36\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots + 45\!\cdots\!32 \)
|
| $71$ |
\( T^{5} + \cdots - 23\!\cdots\!76 \)
|
| $73$ |
\( T^{5} + \cdots - 13\!\cdots\!92 \)
|
| $79$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 30\!\cdots\!96 \)
|
| $89$ |
\( T^{5} + \cdots + 11\!\cdots\!50 \)
|
| $97$ |
\( T^{5} + \cdots - 27\!\cdots\!74 \)
|
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