Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [140,2,Mod(13,140)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(140, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("140.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 140 = 2^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 140.m (of order \(4\), degree \(2\), 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: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.11574317056.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 45x^{4} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 45x^{4} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 47\nu^{3} ) / 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 47\nu^{2} ) / 14 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{6} + 2\nu^{5} + 2\nu^{4} + 127\nu^{2} + 94\nu + 38 ) / 28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} - 2\nu^{5} + 2\nu^{4} - 127\nu^{2} - 94\nu + 38 ) / 28 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{7} + 2\nu^{5} + 315\nu^{3} + 94\nu ) / 28 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -7\nu^{7} + 2\nu^{5} - 315\nu^{3} + 94\nu ) / 28 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 7\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 7\beta_{4} - 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{7} + 7\beta_{6} - 47\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -47\beta_{7} - 47\beta_{6} - 47\beta_{5} + 47\beta_{4} + 127\beta_{3} \)
|
| \(\nu^{7}\) | \(=\) |
\( -47\beta_{7} + 47\beta_{6} - 315\beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).
| \(n\) | \(57\) | \(71\) | \(101\) |
| \(\chi(n)\) | \(-\beta_{3}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
0 | −1.83051 | + | 1.83051i | 0 | −1.83051 | − | 1.28422i | 0 | −1.57763 | + | 2.12393i | 0 | − | 3.70156i | 0 | |||||||||||||||||||||||||||||||||||
| 13.2 | 0 | −0.386289 | + | 0.386289i | 0 | −0.386289 | + | 2.20245i | 0 | 2.64515 | − | 0.0564123i | 0 | 2.70156i | 0 | |||||||||||||||||||||||||||||||||||||
| 13.3 | 0 | 0.386289 | − | 0.386289i | 0 | 0.386289 | − | 2.20245i | 0 | 0.0564123 | − | 2.64515i | 0 | 2.70156i | 0 | |||||||||||||||||||||||||||||||||||||
| 13.4 | 0 | 1.83051 | − | 1.83051i | 0 | 1.83051 | + | 1.28422i | 0 | −2.12393 | + | 1.57763i | 0 | − | 3.70156i | 0 | ||||||||||||||||||||||||||||||||||||
| 97.1 | 0 | −1.83051 | − | 1.83051i | 0 | −1.83051 | + | 1.28422i | 0 | −1.57763 | − | 2.12393i | 0 | 3.70156i | 0 | |||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | −0.386289 | − | 0.386289i | 0 | −0.386289 | − | 2.20245i | 0 | 2.64515 | + | 0.0564123i | 0 | − | 2.70156i | 0 | ||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | 0.386289 | + | 0.386289i | 0 | 0.386289 | + | 2.20245i | 0 | 0.0564123 | + | 2.64515i | 0 | − | 2.70156i | 0 | ||||||||||||||||||||||||||||||||||||
| 97.4 | 0 | 1.83051 | + | 1.83051i | 0 | 1.83051 | − | 1.28422i | 0 | −2.12393 | − | 1.57763i | 0 | 3.70156i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 140.2.m.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1260.2.ba.a | 8 | ||
| 4.b | odd | 2 | 1 | 560.2.bj.b | 8 | ||
| 5.b | even | 2 | 1 | 700.2.m.c | 8 | ||
| 5.c | odd | 4 | 1 | inner | 140.2.m.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | 700.2.m.c | 8 | ||
| 7.b | odd | 2 | 1 | inner | 140.2.m.a | ✓ | 8 |
| 7.c | even | 3 | 2 | 980.2.v.b | 16 | ||
| 7.d | odd | 6 | 2 | 980.2.v.b | 16 | ||
| 15.e | even | 4 | 1 | 1260.2.ba.a | 8 | ||
| 20.e | even | 4 | 1 | 560.2.bj.b | 8 | ||
| 21.c | even | 2 | 1 | 1260.2.ba.a | 8 | ||
| 28.d | even | 2 | 1 | 560.2.bj.b | 8 | ||
| 35.c | odd | 2 | 1 | 700.2.m.c | 8 | ||
| 35.f | even | 4 | 1 | inner | 140.2.m.a | ✓ | 8 |
| 35.f | even | 4 | 1 | 700.2.m.c | 8 | ||
| 35.k | even | 12 | 2 | 980.2.v.b | 16 | ||
| 35.l | odd | 12 | 2 | 980.2.v.b | 16 | ||
| 105.k | odd | 4 | 1 | 1260.2.ba.a | 8 | ||
| 140.j | odd | 4 | 1 | 560.2.bj.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.2.m.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 140.2.m.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 140.2.m.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 140.2.m.a | ✓ | 8 | 35.f | even | 4 | 1 | inner |
| 560.2.bj.b | 8 | 4.b | odd | 2 | 1 | ||
| 560.2.bj.b | 8 | 20.e | even | 4 | 1 | ||
| 560.2.bj.b | 8 | 28.d | even | 2 | 1 | ||
| 560.2.bj.b | 8 | 140.j | odd | 4 | 1 | ||
| 700.2.m.c | 8 | 5.b | even | 2 | 1 | ||
| 700.2.m.c | 8 | 5.c | odd | 4 | 1 | ||
| 700.2.m.c | 8 | 35.c | odd | 2 | 1 | ||
| 700.2.m.c | 8 | 35.f | even | 4 | 1 | ||
| 980.2.v.b | 16 | 7.c | even | 3 | 2 | ||
| 980.2.v.b | 16 | 7.d | odd | 6 | 2 | ||
| 980.2.v.b | 16 | 35.k | even | 12 | 2 | ||
| 980.2.v.b | 16 | 35.l | odd | 12 | 2 | ||
| 1260.2.ba.a | 8 | 3.b | odd | 2 | 1 | ||
| 1260.2.ba.a | 8 | 15.e | even | 4 | 1 | ||
| 1260.2.ba.a | 8 | 21.c | even | 2 | 1 | ||
| 1260.2.ba.a | 8 | 105.k | odd | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(140, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 45T^{4} + 4 \)
|
| $5$ |
\( T^{8} + 6 T^{6} + \cdots + 625 \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( (T^{2} + 3 T - 8)^{4} \)
|
| $13$ |
\( T^{8} + 45T^{4} + 4 \)
|
| $17$ |
\( T^{8} + 2109 T^{4} + 2500 \)
|
| $19$ |
\( (T^{4} - 58 T^{2} + 800)^{2} \)
|
| $23$ |
\( (T^{4} - 2 T^{3} + \cdots + 400)^{2} \)
|
| $29$ |
\( (T^{4} + 33 T^{2} + 16)^{2} \)
|
| $31$ |
\( (T^{4} + 142 T^{2} + 5000)^{2} \)
|
| $37$ |
\( (T^{4} + 2 T^{3} + \cdots + 400)^{2} \)
|
| $41$ |
\( (T^{4} + 58 T^{2} + 800)^{2} \)
|
| $43$ |
\( (T^{2} + 10 T + 50)^{4} \)
|
| $47$ |
\( T^{8} + 17325 T^{4} + 7496644 \)
|
| $53$ |
\( (T^{2} - 10 T + 50)^{4} \)
|
| $59$ |
\( (T^{4} - 58 T^{2} + 800)^{2} \)
|
| $61$ |
\( (T^{4} + 232 T^{2} + 12800)^{2} \)
|
| $67$ |
\( (T^{2} - 10 T + 50)^{4} \)
|
| $71$ |
\( (T^{2} - 2 T - 40)^{4} \)
|
| $73$ |
\( T^{8} + 420 T^{4} + 16384 \)
|
| $79$ |
\( (T^{4} + 45 T^{2} + 4)^{2} \)
|
| $83$ |
\( T^{8} + 16500 T^{4} + 17909824 \)
|
| $89$ |
\( (T^{4} - 188 T^{2} + 800)^{2} \)
|
| $97$ |
\( T^{8} + 3645 T^{4} + 26244 \)
|
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