Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [272,10,Mod(1,272)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(272, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("272.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 272 = 2^{4} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 272.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: | \(140.089747437\) |
| 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: |
\( x^{5} - 2x^{4} - 1596x^{3} + 5754x^{2} + 488987x - 2711704 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 17) |
| 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
\( x^{5} - 2x^{4} - 1596x^{3} + 5754x^{2} + 488987x - 2711704 \)
:
| \(\beta_{1}\) | \(=\) |
\( 8\nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{4} + 207\nu^{3} + 6301\nu^{2} - 192831\nu - 517832 ) / 8096 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{4} + 115\nu^{3} - 13679\nu^{2} - 91523\nu + 4588376 ) / 16192 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 59\nu^{4} + 391\nu^{3} - 75971\nu^{2} - 352151\nu + 12509496 ) / 16192 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 3 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{4} - 28\beta_{3} + 4\beta_{2} + 3\beta _1 + 5109 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 48\beta_{4} - 16\beta_{3} + 272\beta_{2} + 929\beta _1 - 12365 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7028\beta_{4} - 35948\beta_{3} + 3348\beta_{2} + 3675\beta _1 + 4982221 ) / 8 \)
|
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 | −177.437 | 0 | −1620.18 | 0 | 1834.42 | 0 | 11801.0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 3.02373 | 0 | 762.851 | 0 | −5573.11 | 0 | −19673.9 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 67.6654 | 0 | 2390.67 | 0 | 11355.8 | 0 | −15104.4 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 85.9747 | 0 | −1460.58 | 0 | 446.232 | 0 | −12291.3 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 256.773 | 0 | 1407.25 | 0 | 5138.64 | 0 | 46249.6 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(1\) |
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 | 272.10.a.f | 5 | |
| 4.b | odd | 2 | 1 | 17.10.a.a | ✓ | 5 | |
| 12.b | even | 2 | 1 | 153.10.a.c | 5 | ||
| 68.d | odd | 2 | 1 | 289.10.a.a | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.10.a.a | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 153.10.a.c | 5 | 12.b | even | 2 | 1 | ||
| 272.10.a.f | 5 | 1.a | even | 1 | 1 | trivial | |
| 289.10.a.a | 5 | 68.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 236T_{3}^{4} - 26850T_{3}^{3} + 6621804T_{3}^{2} - 284823432T_{3} + 801447696 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(272))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 236 T^{4} + \cdots + 801447696 \)
|
| $5$ |
\( T^{5} + \cdots - 60\!\cdots\!20 \)
|
| $7$ |
\( T^{5} + \cdots + 26\!\cdots\!44 \)
|
| $11$ |
\( T^{5} + \cdots - 18\!\cdots\!36 \)
|
| $13$ |
\( T^{5} + \cdots + 46\!\cdots\!08 \)
|
| $17$ |
\( (T + 83521)^{5} \)
|
| $19$ |
\( T^{5} + \cdots + 23\!\cdots\!16 \)
|
| $23$ |
\( T^{5} + \cdots - 28\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 63\!\cdots\!72 \)
|
| $37$ |
\( T^{5} + \cdots + 12\!\cdots\!48 \)
|
| $41$ |
\( T^{5} + \cdots + 84\!\cdots\!52 \)
|
| $43$ |
\( T^{5} + \cdots + 35\!\cdots\!88 \)
|
| $47$ |
\( T^{5} + \cdots + 53\!\cdots\!68 \)
|
| $53$ |
\( T^{5} + \cdots + 46\!\cdots\!44 \)
|
| $59$ |
\( T^{5} + \cdots - 43\!\cdots\!28 \)
|
| $61$ |
\( T^{5} + \cdots - 81\!\cdots\!20 \)
|
| $67$ |
\( T^{5} + \cdots + 17\!\cdots\!00 \)
|
| $71$ |
\( T^{5} + \cdots - 27\!\cdots\!64 \)
|
| $73$ |
\( T^{5} + \cdots + 39\!\cdots\!92 \)
|
| $79$ |
\( T^{5} + \cdots - 88\!\cdots\!72 \)
|
| $83$ |
\( T^{5} + \cdots + 48\!\cdots\!16 \)
|
| $89$ |
\( T^{5} + \cdots + 26\!\cdots\!96 \)
|
| $97$ |
\( T^{5} + \cdots + 10\!\cdots\!28 \)
|
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