Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [273,4,Mod(1,273)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(273, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("273.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 273.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: | \(16.1075214316\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1038472.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 22x^{2} + 6x + 104 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 22x^{2} + 6x + 104 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 11\nu + 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 14\beta _1 + 9 \)
|
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 |
|
−4.33161 | 3.00000 | 10.7629 | 6.96257 | −12.9948 | −7.00000 | −11.9677 | 9.00000 | −30.1592 | ||||||||||||||||||||||||||||||
| 1.2 | −3.71034 | 3.00000 | 5.76665 | −15.1555 | −11.1310 | −7.00000 | 8.28649 | 9.00000 | 56.2319 | |||||||||||||||||||||||||||||||
| 1.3 | 1.58295 | 3.00000 | −5.49427 | −1.38810 | 4.74885 | −7.00000 | −21.3608 | 9.00000 | −2.19729 | |||||||||||||||||||||||||||||||
| 1.4 | 3.45901 | 3.00000 | 3.96474 | −14.4190 | 10.3770 | −7.00000 | −13.9580 | 9.00000 | −49.8755 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(13\) | \(1\) |
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 | 273.4.a.f | ✓ | 4 |
| 3.b | odd | 2 | 1 | 819.4.a.g | 4 | ||
| 7.b | odd | 2 | 1 | 1911.4.a.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.4.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 819.4.a.g | 4 | 3.b | odd | 2 | 1 | ||
| 1911.4.a.l | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(273))\):
|
\( T_{2}^{4} + 3T_{2}^{3} - 19T_{2}^{2} - 37T_{2} + 88 \)
|
|
\( T_{5}^{4} + 24T_{5}^{3} + 44T_{5}^{2} - 1504T_{5} - 2112 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3 T^{3} + \cdots + 88 \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} + 24 T^{3} + \cdots - 2112 \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( T^{4} - 8 T^{3} + \cdots + 2244864 \)
|
| $13$ |
\( (T + 13)^{4} \)
|
| $17$ |
\( T^{4} + 6 T^{3} + \cdots + 15592864 \)
|
| $19$ |
\( T^{4} + 332 T^{3} + \cdots + 38566912 \)
|
| $23$ |
\( T^{4} - 6 T^{3} + \cdots + 28332096 \)
|
| $29$ |
\( T^{4} - 8 T^{3} + \cdots + 110171792 \)
|
| $31$ |
\( T^{4} + 184 T^{3} + \cdots + 884861952 \)
|
| $37$ |
\( T^{4} + 244 T^{3} + \cdots - 476435344 \)
|
| $41$ |
\( T^{4} + 296 T^{3} + \cdots - 585736128 \)
|
| $43$ |
\( T^{4} + \cdots + 15629953792 \)
|
| $47$ |
\( T^{4} + \cdots - 9913668544 \)
|
| $53$ |
\( T^{4} + \cdots - 2289233712 \)
|
| $59$ |
\( T^{4} + \cdots + 8780572864 \)
|
| $61$ |
\( T^{4} + \cdots - 133456059152 \)
|
| $67$ |
\( T^{4} + \cdots - 16010591232 \)
|
| $71$ |
\( T^{4} + \cdots - 12060510592 \)
|
| $73$ |
\( T^{4} + \cdots + 9634081264 \)
|
| $79$ |
\( T^{4} + \cdots + 3700506624 \)
|
| $83$ |
\( T^{4} + \cdots + 237457983936 \)
|
| $89$ |
\( T^{4} + \cdots + 52442520576 \)
|
| $97$ |
\( T^{4} + \cdots + 201413916944 \)
|
| show more | |
| show less |