Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [28,3,Mod(15,28)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("28.15");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 28 = 2^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 28.c (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: | \(0.762944740209\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.1539727.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{3} - 8x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 3x^{3} - 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + \nu^{2} - 2\nu + 2 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{3} + \nu^{2} + 4\nu - 4 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} + 3\nu^{2} + 4\nu - 8 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 2\nu^{3} - \nu^{2} - 2\nu + 6 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + \nu^{3} - 2\nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{3} + \beta_{2} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 2\beta _1 + 3 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{5} + 4\beta_{4} - \beta_{3} + 3\beta_{2} - 5 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3\beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - 6\beta _1 + 9 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 15.1 |
|
−1.92411 | − | 0.545716i | 4.54500i | 3.40439 | + | 2.10003i | 1.36794 | 2.48028 | − | 8.74507i | 2.64575i | −5.40439 | − | 5.89853i | −11.6570 | −2.63206 | − | 0.746506i | ||||||||||||||||||||||||||
| 15.2 | −1.92411 | + | 0.545716i | − | 4.54500i | 3.40439 | − | 2.10003i | 1.36794 | 2.48028 | + | 8.74507i | − | 2.64575i | −5.40439 | + | 5.89853i | −11.6570 | −2.63206 | + | 0.746506i | |||||||||||||||||||||||||
| 15.3 | −0.163664 | − | 1.99329i | − | 1.56028i | −3.94643 | + | 0.652459i | 3.43742 | −3.11009 | + | 0.255361i | 2.64575i | 1.94643 | + | 7.75960i | 6.56553 | −0.562581 | − | 6.85178i | ||||||||||||||||||||||||||
| 15.4 | −0.163664 | + | 1.99329i | 1.56028i | −3.94643 | − | 0.652459i | 3.43742 | −3.11009 | − | 0.255361i | − | 2.64575i | 1.94643 | − | 7.75960i | 6.56553 | −0.562581 | + | 6.85178i | ||||||||||||||||||||||||||
| 15.5 | 1.58777 | − | 1.21613i | 2.98472i | 1.04204 | − | 3.86188i | −6.80536 | 3.62981 | + | 4.73905i | − | 2.64575i | −3.04204 | − | 7.39905i | 0.0914622 | −10.8054 | + | 8.27622i | ||||||||||||||||||||||||||
| 15.6 | 1.58777 | + | 1.21613i | − | 2.98472i | 1.04204 | + | 3.86188i | −6.80536 | 3.62981 | − | 4.73905i | 2.64575i | −3.04204 | + | 7.39905i | 0.0914622 | −10.8054 | − | 8.27622i | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 28.3.c.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 252.3.g.a | 6 | ||
| 4.b | odd | 2 | 1 | inner | 28.3.c.a | ✓ | 6 |
| 7.b | odd | 2 | 1 | 196.3.c.g | 6 | ||
| 7.c | even | 3 | 2 | 196.3.g.k | 12 | ||
| 7.d | odd | 6 | 2 | 196.3.g.j | 12 | ||
| 8.b | even | 2 | 1 | 448.3.d.d | 6 | ||
| 8.d | odd | 2 | 1 | 448.3.d.d | 6 | ||
| 12.b | even | 2 | 1 | 252.3.g.a | 6 | ||
| 16.e | even | 4 | 2 | 1792.3.g.g | 12 | ||
| 16.f | odd | 4 | 2 | 1792.3.g.g | 12 | ||
| 28.d | even | 2 | 1 | 196.3.c.g | 6 | ||
| 28.f | even | 6 | 2 | 196.3.g.j | 12 | ||
| 28.g | odd | 6 | 2 | 196.3.g.k | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.3.c.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 28.3.c.a | ✓ | 6 | 4.b | odd | 2 | 1 | inner |
| 196.3.c.g | 6 | 7.b | odd | 2 | 1 | ||
| 196.3.c.g | 6 | 28.d | even | 2 | 1 | ||
| 196.3.g.j | 12 | 7.d | odd | 6 | 2 | ||
| 196.3.g.j | 12 | 28.f | even | 6 | 2 | ||
| 196.3.g.k | 12 | 7.c | even | 3 | 2 | ||
| 196.3.g.k | 12 | 28.g | odd | 6 | 2 | ||
| 252.3.g.a | 6 | 3.b | odd | 2 | 1 | ||
| 252.3.g.a | 6 | 12.b | even | 2 | 1 | ||
| 448.3.d.d | 6 | 8.b | even | 2 | 1 | ||
| 448.3.d.d | 6 | 8.d | odd | 2 | 1 | ||
| 1792.3.g.g | 12 | 16.e | even | 4 | 2 | ||
| 1792.3.g.g | 12 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(28, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots + 64 \)
|
| $3$ |
\( T^{6} + 32 T^{4} + \cdots + 448 \)
|
| $5$ |
\( (T^{3} + 2 T^{2} - 28 T + 32)^{2} \)
|
| $7$ |
\( (T^{2} + 7)^{3} \)
|
| $11$ |
\( T^{6} + 512 T^{4} + \cdots + 3469312 \)
|
| $13$ |
\( (T^{3} - 6 T^{2} + \cdots + 1712)^{2} \)
|
| $17$ |
\( (T^{3} + 2 T^{2} + \cdots - 488)^{2} \)
|
| $19$ |
\( T^{6} + 1600 T^{4} + \cdots + 81683392 \)
|
| $23$ |
\( T^{6} + 928 T^{4} + \cdots + 114688 \)
|
| $29$ |
\( (T^{3} + 18 T^{2} + \cdots + 4952)^{2} \)
|
| $31$ |
\( T^{6} + 2048 T^{4} + \cdots + 1404928 \)
|
| $37$ |
\( (T^{3} - 14 T^{2} + \cdots - 4328)^{2} \)
|
| $41$ |
\( (T^{3} + 10 T^{2} + \cdots - 15368)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 1477439488 \)
|
| $47$ |
\( T^{6} + \cdots + 4610486272 \)
|
| $53$ |
\( (T^{3} - 46 T^{2} + \cdots + 8536)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 3560148928 \)
|
| $61$ |
\( (T^{3} + 82 T^{2} + \cdots - 7552)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 92073361408 \)
|
| $71$ |
\( T^{6} + \cdots + 1438646272 \)
|
| $73$ |
\( (T^{3} + 66 T^{2} + \cdots - 3688)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 5754585088 \)
|
| $83$ |
\( T^{6} + 25600 T^{4} + \cdots + 95209408 \)
|
| $89$ |
\( (T^{3} - 174 T^{2} + \cdots + 620888)^{2} \)
|
| $97$ |
\( (T^{3} - 126 T^{2} + \cdots + 196184)^{2} \)
|
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