Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [140,2,Mod(111,140)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(140, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("140.111");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 140 = 2^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 140.g (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: | \(1.11790562830\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.342102016.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + \nu^{5} + 4\nu^{3} + 4\nu ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + 3\nu^{5} - 3\nu^{4} + 2\nu^{3} - 10\nu^{2} + 16\nu - 8 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} - 3\nu^{5} + 3\nu^{4} - 2\nu^{3} - 6\nu^{2} + 8 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} + 10\nu^{2} + 16\nu + 8 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} - 6\nu^{2} + 8 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{6} + \nu^{4} + 2\nu^{2} ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 4\beta_{2} + 4\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4\beta_{7} + 3\beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{3} - 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5\beta_{6} + \beta_{5} - 5\beta_{4} + \beta_{3} - 4\beta_{2} + 4\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -12\beta_{7} + \beta_{6} + 3\beta_{5} + \beta_{4} - 3\beta_{3} - 4 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 3\beta_{6} - \beta_{5} - 3\beta_{4} - \beta_{3} + 4\beta_{2} - 20\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).
| \(n\) | \(57\) | \(71\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 111.1 |
|
−0.780776 | − | 1.17915i | −3.02045 | −0.780776 | + | 1.84130i | 1.00000i | 2.35829 | + | 3.56155i | 2.17238 | − | 1.51022i | 2.78078 | − | 0.516994i | 6.12311 | 1.17915 | − | 0.780776i | ||||||||||||||||||||||||||||||
| 111.2 | −0.780776 | − | 1.17915i | 3.02045 | −0.780776 | + | 1.84130i | − | 1.00000i | −2.35829 | − | 3.56155i | −2.17238 | − | 1.51022i | 2.78078 | − | 0.516994i | 6.12311 | −1.17915 | + | 0.780776i | ||||||||||||||||||||||||||||||
| 111.3 | −0.780776 | + | 1.17915i | −3.02045 | −0.780776 | − | 1.84130i | − | 1.00000i | 2.35829 | − | 3.56155i | 2.17238 | + | 1.51022i | 2.78078 | + | 0.516994i | 6.12311 | 1.17915 | + | 0.780776i | ||||||||||||||||||||||||||||||
| 111.4 | −0.780776 | + | 1.17915i | 3.02045 | −0.780776 | − | 1.84130i | 1.00000i | −2.35829 | + | 3.56155i | −2.17238 | + | 1.51022i | 2.78078 | + | 0.516994i | 6.12311 | −1.17915 | − | 0.780776i | |||||||||||||||||||||||||||||||
| 111.5 | 1.28078 | − | 0.599676i | −0.936426 | 1.28078 | − | 1.53610i | − | 1.00000i | −1.19935 | + | 0.561553i | 2.60399 | + | 0.468213i | 0.719224 | − | 2.73546i | −2.12311 | −0.599676 | − | 1.28078i | ||||||||||||||||||||||||||||||
| 111.6 | 1.28078 | − | 0.599676i | 0.936426 | 1.28078 | − | 1.53610i | 1.00000i | 1.19935 | − | 0.561553i | −2.60399 | + | 0.468213i | 0.719224 | − | 2.73546i | −2.12311 | 0.599676 | + | 1.28078i | |||||||||||||||||||||||||||||||
| 111.7 | 1.28078 | + | 0.599676i | −0.936426 | 1.28078 | + | 1.53610i | 1.00000i | −1.19935 | − | 0.561553i | 2.60399 | − | 0.468213i | 0.719224 | + | 2.73546i | −2.12311 | −0.599676 | + | 1.28078i | |||||||||||||||||||||||||||||||
| 111.8 | 1.28078 | + | 0.599676i | 0.936426 | 1.28078 | + | 1.53610i | − | 1.00000i | 1.19935 | + | 0.561553i | −2.60399 | − | 0.468213i | 0.719224 | + | 2.73546i | −2.12311 | 0.599676 | − | 1.28078i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 28.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 140.2.g.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1260.2.c.c | 8 | ||
| 4.b | odd | 2 | 1 | inner | 140.2.g.c | ✓ | 8 |
| 5.b | even | 2 | 1 | 700.2.g.j | 8 | ||
| 5.c | odd | 4 | 1 | 700.2.c.i | 8 | ||
| 5.c | odd | 4 | 1 | 700.2.c.j | 8 | ||
| 7.b | odd | 2 | 1 | inner | 140.2.g.c | ✓ | 8 |
| 7.c | even | 3 | 2 | 980.2.o.e | 16 | ||
| 7.d | odd | 6 | 2 | 980.2.o.e | 16 | ||
| 8.b | even | 2 | 1 | 2240.2.k.e | 8 | ||
| 8.d | odd | 2 | 1 | 2240.2.k.e | 8 | ||
| 12.b | even | 2 | 1 | 1260.2.c.c | 8 | ||
| 20.d | odd | 2 | 1 | 700.2.g.j | 8 | ||
| 20.e | even | 4 | 1 | 700.2.c.i | 8 | ||
| 20.e | even | 4 | 1 | 700.2.c.j | 8 | ||
| 21.c | even | 2 | 1 | 1260.2.c.c | 8 | ||
| 28.d | even | 2 | 1 | inner | 140.2.g.c | ✓ | 8 |
| 28.f | even | 6 | 2 | 980.2.o.e | 16 | ||
| 28.g | odd | 6 | 2 | 980.2.o.e | 16 | ||
| 35.c | odd | 2 | 1 | 700.2.g.j | 8 | ||
| 35.f | even | 4 | 1 | 700.2.c.i | 8 | ||
| 35.f | even | 4 | 1 | 700.2.c.j | 8 | ||
| 56.e | even | 2 | 1 | 2240.2.k.e | 8 | ||
| 56.h | odd | 2 | 1 | 2240.2.k.e | 8 | ||
| 84.h | odd | 2 | 1 | 1260.2.c.c | 8 | ||
| 140.c | even | 2 | 1 | 700.2.g.j | 8 | ||
| 140.j | odd | 4 | 1 | 700.2.c.i | 8 | ||
| 140.j | odd | 4 | 1 | 700.2.c.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.2.g.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 140.2.g.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 140.2.g.c | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 140.2.g.c | ✓ | 8 | 28.d | even | 2 | 1 | inner |
| 700.2.c.i | 8 | 5.c | odd | 4 | 1 | ||
| 700.2.c.i | 8 | 20.e | even | 4 | 1 | ||
| 700.2.c.i | 8 | 35.f | even | 4 | 1 | ||
| 700.2.c.i | 8 | 140.j | odd | 4 | 1 | ||
| 700.2.c.j | 8 | 5.c | odd | 4 | 1 | ||
| 700.2.c.j | 8 | 20.e | even | 4 | 1 | ||
| 700.2.c.j | 8 | 35.f | even | 4 | 1 | ||
| 700.2.c.j | 8 | 140.j | odd | 4 | 1 | ||
| 700.2.g.j | 8 | 5.b | even | 2 | 1 | ||
| 700.2.g.j | 8 | 20.d | odd | 2 | 1 | ||
| 700.2.g.j | 8 | 35.c | odd | 2 | 1 | ||
| 700.2.g.j | 8 | 140.c | even | 2 | 1 | ||
| 980.2.o.e | 16 | 7.c | even | 3 | 2 | ||
| 980.2.o.e | 16 | 7.d | odd | 6 | 2 | ||
| 980.2.o.e | 16 | 28.f | even | 6 | 2 | ||
| 980.2.o.e | 16 | 28.g | odd | 6 | 2 | ||
| 1260.2.c.c | 8 | 3.b | odd | 2 | 1 | ||
| 1260.2.c.c | 8 | 12.b | even | 2 | 1 | ||
| 1260.2.c.c | 8 | 21.c | even | 2 | 1 | ||
| 1260.2.c.c | 8 | 84.h | odd | 2 | 1 | ||
| 2240.2.k.e | 8 | 8.b | even | 2 | 1 | ||
| 2240.2.k.e | 8 | 8.d | odd | 2 | 1 | ||
| 2240.2.k.e | 8 | 56.e | even | 2 | 1 | ||
| 2240.2.k.e | 8 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(140, [\chi])\):
|
\( T_{3}^{4} - 10T_{3}^{2} + 8 \)
|
|
\( T_{19}^{4} - 28T_{19}^{2} + 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} - 2 T + 4)^{2} \)
|
| $3$ |
\( (T^{4} - 10 T^{2} + 8)^{2} \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( T^{8} - 18 T^{6} + \cdots + 2401 \)
|
| $11$ |
\( (T^{4} + 28 T^{2} + 128)^{2} \)
|
| $13$ |
\( (T^{2} + 4)^{4} \)
|
| $17$ |
\( (T^{4} + 52 T^{2} + 64)^{2} \)
|
| $19$ |
\( (T^{4} - 28 T^{2} + 128)^{2} \)
|
| $23$ |
\( (T^{4} + 74 T^{2} + 1352)^{2} \)
|
| $29$ |
\( (T + 2)^{8} \)
|
| $31$ |
\( (T^{4} - 56 T^{2} + 512)^{2} \)
|
| $37$ |
\( (T - 2)^{8} \)
|
| $41$ |
\( (T^{4} + 52 T^{2} + 64)^{2} \)
|
| $43$ |
\( (T^{4} + 58 T^{2} + 8)^{2} \)
|
| $47$ |
\( (T^{4} - 126 T^{2} + 2592)^{2} \)
|
| $53$ |
\( (T - 2)^{8} \)
|
| $59$ |
\( (T^{4} - 124 T^{2} + 512)^{2} \)
|
| $61$ |
\( (T^{2} + 4)^{4} \)
|
| $67$ |
\( (T^{4} + 218 T^{2} + 2888)^{2} \)
|
| $71$ |
\( (T^{4} + 92 T^{2} + 2048)^{2} \)
|
| $73$ |
\( (T^{4} + 324 T^{2} + 20736)^{2} \)
|
| $79$ |
\( (T^{4} + 20 T^{2} + 32)^{2} \)
|
| $83$ |
\( (T^{4} - 10 T^{2} + 8)^{2} \)
|
| $89$ |
\( (T^{2} + 144)^{4} \)
|
| $97$ |
\( (T^{4} + 52 T^{2} + 64)^{2} \)
|
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