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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1441}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 360 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{1441})\). We also show the integral \(q\)-expansion of the trace form.
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 | −20.4803 | 16.0000 | 34.4408 | −81.9210 | −18.9210 | 64.0000 | 176.441 | 137.763 | ||||||||||||||||||||||||
| 1.2 | 4.00000 | 17.4803 | 16.0000 | −79.4408 | 69.9210 | 132.921 | 64.0000 | 62.5592 | −317.763 | |||||||||||||||||||||||||
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.c | 2 | |
| 19.b | odd | 2 | 1 | 38.6.a.c | ✓ | 2 | |
| 57.d | even | 2 | 1 | 342.6.a.i | 2 | ||
| 76.d | even | 2 | 1 | 304.6.a.f | 2 | ||
| 95.d | odd | 2 | 1 | 950.6.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.6.a.c | ✓ | 2 | 19.b | odd | 2 | 1 | |
| 304.6.a.f | 2 | 76.d | even | 2 | 1 | ||
| 342.6.a.i | 2 | 57.d | even | 2 | 1 | ||
| 722.6.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 950.6.a.d | 2 | 95.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 3T_{3} - 358 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(722))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{2} \)
|
| $3$ |
\( T^{2} + 3T - 358 \)
|
| $5$ |
\( T^{2} + 45T - 2736 \)
|
| $7$ |
\( T^{2} - 114T - 2515 \)
|
| $11$ |
\( T^{2} - 661T + 108870 \)
|
| $13$ |
\( T^{2} + 1613 T + 641436 \)
|
| $17$ |
\( T^{2} - 64T - 35001 \)
|
| $19$ |
\( T^{2} \)
|
| $23$ |
\( T^{2} + 3185 T + 1671096 \)
|
| $29$ |
\( T^{2} - 2481 T - 34206966 \)
|
| $31$ |
\( T^{2} - 1180 T - 35625024 \)
|
| $37$ |
\( T^{2} + 10488 T + 16841900 \)
|
| $41$ |
\( T^{2} + 16630 T - 61416816 \)
|
| $43$ |
\( T^{2} - 11303 T + 3493752 \)
|
| $47$ |
\( T^{2} + 12155 T - 410935800 \)
|
| $53$ |
\( T^{2} + 20585 T - 24187104 \)
|
| $59$ |
\( T^{2} + \cdots + 1541823618 \)
|
| $61$ |
\( T^{2} - 43621 T + 230502754 \)
|
| $67$ |
\( T^{2} + \cdots - 3449006904 \)
|
| $71$ |
\( T^{2} - 62488 T + 396968940 \)
|
| $73$ |
\( T^{2} + \cdots - 3142002343 \)
|
| $79$ |
\( T^{2} + \cdots + 1119872072 \)
|
| $83$ |
\( T^{2} + \cdots - 2126648040 \)
|
| $89$ |
\( T^{2} + \cdots - 2478833568 \)
|
| $97$ |
\( T^{2} + \cdots + 5494062880 \)
|
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