Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,4,Mod(1,722)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 722.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: | \(42.5993790241\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.253788.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 63x + 15 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 38) |
| 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
\( x^{3} - x^{2} - 63x + 15 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 43 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 43 \)
|
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 |
|
2.00000 | −9.57650 | 4.00000 | 15.7782 | −19.1530 | 16.5765 | 8.00000 | 64.7093 | 31.5563 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −1.76259 | 4.00000 | −20.7092 | −3.52517 | 8.76259 | 8.00000 | −23.8933 | −41.4185 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 6.33908 | 4.00000 | 5.93108 | 12.6782 | 0.660916 | 8.00000 | 13.1840 | 11.8622 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 722.4.a.k | 3 | |
| 19.b | odd | 2 | 1 | 722.4.a.j | 3 | ||
| 19.d | odd | 6 | 2 | 38.4.c.c | ✓ | 6 | |
| 57.f | even | 6 | 2 | 342.4.g.f | 6 | ||
| 76.f | even | 6 | 2 | 304.4.i.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.4.c.c | ✓ | 6 | 19.d | odd | 6 | 2 | |
| 304.4.i.e | 6 | 76.f | even | 6 | 2 | ||
| 342.4.g.f | 6 | 57.f | even | 6 | 2 | ||
| 722.4.a.j | 3 | 19.b | odd | 2 | 1 | ||
| 722.4.a.k | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(722))\):
|
\( T_{3}^{3} + 5T_{3}^{2} - 55T_{3} - 107 \)
|
|
\( T_{5}^{3} - T_{5}^{2} - 356T_{5} + 1938 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( T^{3} + 5 T^{2} + \cdots - 107 \)
|
| $5$ |
\( T^{3} - T^{2} + \cdots + 1938 \)
|
| $7$ |
\( T^{3} - 26 T^{2} + \cdots - 96 \)
|
| $11$ |
\( T^{3} - 4 T^{2} + \cdots + 49980 \)
|
| $13$ |
\( T^{3} - 129 T^{2} + \cdots + 11372 \)
|
| $17$ |
\( T^{3} - 51 T^{2} + \cdots + 106272 \)
|
| $19$ |
\( T^{3} \)
|
| $23$ |
\( T^{3} + 47 T^{2} + \cdots - 67494 \)
|
| $29$ |
\( T^{3} + 125 T^{2} + \cdots + 2436750 \)
|
| $31$ |
\( T^{3} - 50 T^{2} + \cdots - 3809848 \)
|
| $37$ |
\( T^{3} - 188 T^{2} + \cdots + 88004 \)
|
| $41$ |
\( T^{3} - 475 T^{2} + \cdots + 9010191 \)
|
| $43$ |
\( T^{3} - 73 T^{2} + \cdots + 16102456 \)
|
| $47$ |
\( T^{3} - 241 T^{2} + \cdots + 5066730 \)
|
| $53$ |
\( T^{3} - 29 T^{2} + \cdots + 3295020 \)
|
| $59$ |
\( T^{3} + 1065 T^{2} + \cdots + 35876961 \)
|
| $61$ |
\( T^{3} - 981 T^{2} + \cdots - 18873422 \)
|
| $67$ |
\( T^{3} - 877 T^{2} + \cdots - 981957 \)
|
| $71$ |
\( T^{3} - 2135 T^{2} + \cdots - 260901996 \)
|
| $73$ |
\( T^{3} + 667 T^{2} + \cdots + 844797 \)
|
| $79$ |
\( T^{3} - 1671 T^{2} + \cdots + 393755600 \)
|
| $83$ |
\( T^{3} - 588 T^{2} + \cdots - 162474984 \)
|
| $89$ |
\( T^{3} - 693 T^{2} + \cdots + 52321248 \)
|
| $97$ |
\( T^{3} + \cdots - 2608119145 \)
|
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