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: | \(0\) |
| 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: |
\( x^{3} - x^{2} - 454x + 3760 \)
|
| 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
\( x^{3} - x^{2} - 454x + 3760 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 11\nu - 306 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} - 11\beta _1 + 306 \)
|
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 | −18.3926 | 16.0000 | 42.3408 | 73.5705 | −127.237 | −64.0000 | 95.2889 | −169.363 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | −14.7990 | 16.0000 | −19.0956 | 59.1959 | 212.287 | −64.0000 | −23.9901 | 76.3823 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 20.1916 | 16.0000 | 57.7548 | −80.7665 | 142.950 | −64.0000 | 164.701 | −231.019 | ||||||||||||||||||||||||||||
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.d | 3 | |
| 19.b | odd | 2 | 1 | 38.6.a.d | ✓ | 3 | |
| 57.d | even | 2 | 1 | 342.6.a.l | 3 | ||
| 76.d | even | 2 | 1 | 304.6.a.h | 3 | ||
| 95.d | odd | 2 | 1 | 950.6.a.f | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.6.a.d | ✓ | 3 | 19.b | odd | 2 | 1 | |
| 304.6.a.h | 3 | 76.d | even | 2 | 1 | ||
| 342.6.a.l | 3 | 57.d | even | 2 | 1 | ||
| 722.6.a.d | 3 | 1.a | even | 1 | 1 | trivial | |
| 950.6.a.f | 3 | 95.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 13T_{3}^{2} - 398T_{3} - 5496 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(722))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( T^{3} + 13 T^{2} + \cdots - 5496 \)
|
| $5$ |
\( T^{3} - 81 T^{2} + \cdots + 46696 \)
|
| $7$ |
\( T^{3} - 228 T^{2} + \cdots + 3861216 \)
|
| $11$ |
\( T^{3} - 363 T^{2} + \cdots + 162880120 \)
|
| $13$ |
\( T^{3} + 501 T^{2} + \cdots - 93082696 \)
|
| $17$ |
\( T^{3} + 1206 T^{2} + \cdots - 963841518 \)
|
| $19$ |
\( T^{3} \)
|
| $23$ |
\( T^{3} + \cdots - 18644491520 \)
|
| $29$ |
\( T^{3} + \cdots + 14808989500 \)
|
| $31$ |
\( T^{3} + \cdots + 164301107200 \)
|
| $37$ |
\( T^{3} + \cdots + 19509523912 \)
|
| $41$ |
\( T^{3} + \cdots - 164480699264 \)
|
| $43$ |
\( T^{3} + \cdots - 2228453451472 \)
|
| $47$ |
\( T^{3} + \cdots - 1331645125760 \)
|
| $53$ |
\( T^{3} + \cdots - 5330002936312 \)
|
| $59$ |
\( T^{3} + \cdots - 56358292470552 \)
|
| $61$ |
\( T^{3} + \cdots + 3097994159068 \)
|
| $67$ |
\( T^{3} + \cdots + 6192188856432 \)
|
| $71$ |
\( T^{3} + \cdots - 33641692455520 \)
|
| $73$ |
\( T^{3} + \cdots - 23123416049802 \)
|
| $79$ |
\( T^{3} + \cdots - 2286937169920 \)
|
| $83$ |
\( T^{3} + \cdots + 183940117843104 \)
|
| $89$ |
\( T^{3} + \cdots + 63263160328320 \)
|
| $97$ |
\( T^{3} + \cdots - 11475767642528 \)
|
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