Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,3,Mod(721,722)]
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([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.721");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 722.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(19.6730750868\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 38) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 4\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/722\mathbb{Z}\right)^\times\).
| \(n\) | \(363\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 721.1 |
|
− | 1.41421i | − | 0.317837i | −2.00000 | −1.00000 | −0.449490 | −2.89898 | 2.82843i | 8.89898 | 1.41421i | ||||||||||||||||||||||||||||
| 721.2 | − | 1.41421i | 3.14626i | −2.00000 | −1.00000 | 4.44949 | 6.89898 | 2.82843i | −0.898979 | 1.41421i | ||||||||||||||||||||||||||||||
| 721.3 | 1.41421i | − | 3.14626i | −2.00000 | −1.00000 | 4.44949 | 6.89898 | − | 2.82843i | −0.898979 | − | 1.41421i | ||||||||||||||||||||||||||||
| 721.4 | 1.41421i | 0.317837i | −2.00000 | −1.00000 | −0.449490 | −2.89898 | − | 2.82843i | 8.89898 | − | 1.41421i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 722.3.b.b | 4 | |
| 19.b | odd | 2 | 1 | inner | 722.3.b.b | 4 | |
| 19.c | even | 3 | 1 | 38.3.d.a | ✓ | 4 | |
| 19.d | odd | 6 | 1 | 38.3.d.a | ✓ | 4 | |
| 57.f | even | 6 | 1 | 342.3.m.a | 4 | ||
| 57.h | odd | 6 | 1 | 342.3.m.a | 4 | ||
| 76.f | even | 6 | 1 | 304.3.r.a | 4 | ||
| 76.g | odd | 6 | 1 | 304.3.r.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.3.d.a | ✓ | 4 | 19.c | even | 3 | 1 | |
| 38.3.d.a | ✓ | 4 | 19.d | odd | 6 | 1 | |
| 304.3.r.a | 4 | 76.f | even | 6 | 1 | ||
| 304.3.r.a | 4 | 76.g | odd | 6 | 1 | ||
| 342.3.m.a | 4 | 57.f | even | 6 | 1 | ||
| 342.3.m.a | 4 | 57.h | odd | 6 | 1 | ||
| 722.3.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 722.3.b.b | 4 | 19.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 10T_{3}^{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(722, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{2} \)
|
| $3$ |
\( T^{4} + 10T^{2} + 1 \)
|
| $5$ |
\( (T + 1)^{4} \)
|
| $7$ |
\( (T^{2} - 4 T - 20)^{2} \)
|
| $11$ |
\( (T^{2} + 20 T + 76)^{2} \)
|
| $13$ |
\( T^{4} + 294T^{2} + 9 \)
|
| $17$ |
\( (T^{2} + 14 T + 25)^{2} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( (T^{2} - 10 T - 461)^{2} \)
|
| $29$ |
\( T^{4} + 2022 T^{2} + 81225 \)
|
| $31$ |
\( (T^{2} + 972)^{2} \)
|
| $37$ |
\( T^{4} + 1320 T^{2} + 404496 \)
|
| $41$ |
\( T^{4} + 7110 T^{2} + 12257001 \)
|
| $43$ |
\( (T^{2} + 38 T - 2813)^{2} \)
|
| $47$ |
\( (T^{2} - 70 T - 941)^{2} \)
|
| $53$ |
\( T^{4} + 9510 T^{2} + 19900521 \)
|
| $59$ |
\( T^{4} + 4650 T^{2} + 4124961 \)
|
| $61$ |
\( (T^{2} + 74 T - 167)^{2} \)
|
| $67$ |
\( T^{4} + 17610 T^{2} + 49999041 \)
|
| $71$ |
\( T^{4} + 15930 T^{2} + 58384881 \)
|
| $73$ |
\( (T^{2} + 98 T - 3743)^{2} \)
|
| $79$ |
\( T^{4} + 4410 T^{2} + 194481 \)
|
| $83$ |
\( (T^{2} + 32 T + 40)^{2} \)
|
| $89$ |
\( T^{4} + 150T^{2} + 4761 \)
|
| $97$ |
\( T^{4} + 39510 T^{2} + 384591321 \)
|
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