Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(115.797117905\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 133x - 30 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{3} - 133x - 30 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 89 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 89 \)
|
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 |
|
4.00000 | −27.8362 | 16.0000 | −69.2088 | −111.345 | −195.345 | 64.0000 | 531.852 | −276.835 | |||||||||||||||||||||||||||
| 1.2 | 4.00000 | −5.45130 | 16.0000 | 83.4978 | −21.8052 | −105.805 | 64.0000 | −213.283 | 333.991 | ||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 18.2875 | 16.0000 | −28.2890 | 73.1499 | −10.8501 | 64.0000 | 91.4313 | −113.156 | ||||||||||||||||||||||||||||
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 | 722.6.a.f | 3 | |
| 19.b | odd | 2 | 1 | 722.6.a.e | 3 | ||
| 19.d | odd | 6 | 2 | 38.6.c.a | ✓ | 6 | |
| 57.f | even | 6 | 2 | 342.6.g.a | 6 | ||
| 76.f | even | 6 | 2 | 304.6.i.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.6.c.a | ✓ | 6 | 19.d | odd | 6 | 2 | |
| 304.6.i.a | 6 | 76.f | even | 6 | 2 | ||
| 342.6.g.a | 6 | 57.f | even | 6 | 2 | ||
| 722.6.a.e | 3 | 19.b | odd | 2 | 1 | ||
| 722.6.a.f | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 15T_{3}^{2} - 457T_{3} - 2775 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(722))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{3} \)
|
| $3$ |
\( T^{3} + 15 T^{2} + \cdots - 2775 \)
|
| $5$ |
\( T^{3} + 14 T^{2} + \cdots - 163476 \)
|
| $7$ |
\( T^{3} + 312 T^{2} + \cdots + 224256 \)
|
| $11$ |
\( T^{3} + 469 T^{2} + \cdots + 2905680 \)
|
| $13$ |
\( T^{3} + 736 T^{2} + \cdots - 19859022 \)
|
| $17$ |
\( T^{3} - 1000 T^{2} + \cdots - 67385250 \)
|
| $19$ |
\( T^{3} \)
|
| $23$ |
\( T^{3} + \cdots - 1990597134 \)
|
| $29$ |
\( T^{3} + \cdots + 2147316084 \)
|
| $31$ |
\( T^{3} + \cdots - 281768267232 \)
|
| $37$ |
\( T^{3} + \cdots - 34115602072 \)
|
| $41$ |
\( T^{3} + \cdots - 406888455423 \)
|
| $43$ |
\( T^{3} + \cdots - 16394204748 \)
|
| $47$ |
\( T^{3} + \cdots + 440199642180 \)
|
| $53$ |
\( T^{3} + \cdots - 13538278025010 \)
|
| $59$ |
\( T^{3} + \cdots + 200402275455 \)
|
| $61$ |
\( T^{3} + \cdots - 320566542320 \)
|
| $67$ |
\( T^{3} + \cdots + 5436230212089 \)
|
| $71$ |
\( T^{3} + \cdots - 21020361026454 \)
|
| $73$ |
\( T^{3} + \cdots - 23704910119239 \)
|
| $79$ |
\( T^{3} + \cdots + 780775782070780 \)
|
| $83$ |
\( T^{3} + \cdots - 518524918263648 \)
|
| $89$ |
\( T^{3} + \cdots + 113429508943950 \)
|
| $97$ |
\( T^{3} + \cdots + 57457989701689 \)
|
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