Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,16,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, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 11 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| 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: | \(15.6962855610\) |
| 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: |
\( x^{5} - x^{4} - 33269x^{3} - 1263797x^{2} + 175826600x + 6964408066 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{3}\cdot 5 \) |
| 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
\( x^{5} - x^{4} - 33269x^{3} - 1263797x^{2} + 175826600x + 6964408066 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -129\nu^{4} + 11992\nu^{3} + 2876941\nu^{2} - 92824630\nu - 8498906126 ) / 718624 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 143\nu^{4} - 2152\nu^{3} - 5450883\nu^{2} - 17607382\nu + 30874127474 ) / 2155872 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -223\nu^{4} + 48584\nu^{3} + 3630771\nu^{2} - 716122186\nu - 18391940914 ) / 1077936 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - 5\beta_{3} - 3\beta_{2} + 118\beta _1 + 53187 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 203\beta_{4} - 241\beta_{3} - 323\beta_{2} + 45586\beta _1 + 3094967 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 41173\beta_{4} - 133913\beta_{3} - 119215\beta_{2} + 5430210\beta _1 + 1210349463 ) / 4 \)
|
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 |
|
−267.433 | −6454.90 | 38752.4 | −194092. | 1.72625e6 | 1.17300e6 | −1.60043e6 | 2.73168e7 | 5.19067e7 | |||||||||||||||||||||||||||||||||
| 1.2 | −228.747 | 4981.89 | 19557.0 | −128217. | −1.13959e6 | −162296. | 3.02197e6 | 1.04704e7 | 2.93293e7 | ||||||||||||||||||||||||||||||||||
| 1.3 | −104.988 | −3030.53 | −21745.6 | 215347. | 318169. | 1.27489e6 | 5.72326e6 | −5.16477e6 | −2.26088e7 | ||||||||||||||||||||||||||||||||||
| 1.4 | 130.234 | 2698.11 | −15807.0 | −40440.6 | 351387. | −1.83046e6 | −6.32613e6 | −7.06910e6 | −5.26675e6 | ||||||||||||||||||||||||||||||||||
| 1.5 | 342.933 | −3618.57 | 84835.1 | −311587. | −1.24093e6 | −2.65873e6 | 1.78556e7 | −1.25483e6 | −1.06854e8 | ||||||||||||||||||||||||||||||||||
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 | 11.16.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 99.16.a.a | 5 | ||
| 11.b | odd | 2 | 1 | 121.16.a.c | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.16.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 99.16.a.a | 5 | 3.b | odd | 2 | 1 | ||
| 121.16.a.c | 5 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 128T_{2}^{4} - 126524T_{2}^{3} - 20322656T_{2}^{2} + 2019752192T_{2} + 286842333184 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(11))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 286842333184 \)
|
| $3$ |
\( T^{5} + \cdots + 95\!\cdots\!76 \)
|
| $5$ |
\( T^{5} + \cdots - 67\!\cdots\!50 \)
|
| $7$ |
\( T^{5} + \cdots + 11\!\cdots\!04 \)
|
| $11$ |
\( (T - 19487171)^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 27\!\cdots\!44 \)
|
| $17$ |
\( T^{5} + \cdots + 89\!\cdots\!68 \)
|
| $19$ |
\( T^{5} + \cdots + 47\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 33\!\cdots\!56 \)
|
| $29$ |
\( T^{5} + \cdots - 58\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots + 49\!\cdots\!82 \)
|
| $41$ |
\( T^{5} + \cdots + 14\!\cdots\!32 \)
|
| $43$ |
\( T^{5} + \cdots - 12\!\cdots\!32 \)
|
| $47$ |
\( T^{5} + \cdots + 19\!\cdots\!32 \)
|
| $53$ |
\( T^{5} + \cdots + 18\!\cdots\!32 \)
|
| $59$ |
\( T^{5} + \cdots + 23\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 22\!\cdots\!28 \)
|
| $67$ |
\( T^{5} + \cdots + 12\!\cdots\!44 \)
|
| $71$ |
\( T^{5} + \cdots + 61\!\cdots\!44 \)
|
| $73$ |
\( T^{5} + \cdots - 12\!\cdots\!96 \)
|
| $79$ |
\( T^{5} + \cdots + 16\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 59\!\cdots\!48 \)
|
| $89$ |
\( T^{5} + \cdots + 14\!\cdots\!50 \)
|
| $97$ |
\( T^{5} + \cdots - 29\!\cdots\!58 \)
|
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