Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [840,2,Mod(13,840)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(840, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 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("840.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 840.bj (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: | \(6.70743376979\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.1871773696.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 31x^{4} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} + 31x^{4} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 19\nu ) / 21 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 40\nu^{2} ) / 63 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{6} + 9\nu^{4} - 97\nu^{2} + 108 ) / 63 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 40\nu^{3} - 63\nu ) / 63 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{6} - 9\nu^{4} - 97\nu^{2} - 108 ) / 63 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 3\nu^{5} + 40\nu^{3} + 120\nu ) / 63 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{7} + 97\nu^{3} ) / 189 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{4} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{3} + 8\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{7} + 2\beta_{6} + 2\beta_{4} - 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -7\beta_{5} + 7\beta_{3} - 24 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -19\beta_{6} + 19\beta_{4} + 61\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -20\beta_{5} - 20\beta_{3} - 97\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 240\beta_{7} - 97\beta_{6} - 97\beta_{4} + 97\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(281\) | \(337\) | \(421\) | \(631\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-\beta_{2}\) | \(-1\) | \(1\) |
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 |
|
−1.00000 | + | 1.00000i | −0.707107 | − | 0.707107i | − | 2.00000i | −2.12132 | + | 0.707107i | 1.41421 | −2.30278 | − | 1.30278i | 2.00000 | + | 2.00000i | 1.00000i | 1.41421 | − | 2.82843i | |||||||||||||||||||||||||||||
| 13.2 | −1.00000 | + | 1.00000i | −0.707107 | − | 0.707107i | − | 2.00000i | −2.12132 | + | 0.707107i | 1.41421 | 1.30278 | + | 2.30278i | 2.00000 | + | 2.00000i | 1.00000i | 1.41421 | − | 2.82843i | ||||||||||||||||||||||||||||||
| 13.3 | −1.00000 | + | 1.00000i | 0.707107 | + | 0.707107i | − | 2.00000i | 2.12132 | − | 0.707107i | −1.41421 | −2.30278 | − | 1.30278i | 2.00000 | + | 2.00000i | 1.00000i | −1.41421 | + | 2.82843i | ||||||||||||||||||||||||||||||
| 13.4 | −1.00000 | + | 1.00000i | 0.707107 | + | 0.707107i | − | 2.00000i | 2.12132 | − | 0.707107i | −1.41421 | 1.30278 | + | 2.30278i | 2.00000 | + | 2.00000i | 1.00000i | −1.41421 | + | 2.82843i | ||||||||||||||||||||||||||||||
| 517.1 | −1.00000 | − | 1.00000i | −0.707107 | + | 0.707107i | 2.00000i | −2.12132 | − | 0.707107i | 1.41421 | −2.30278 | + | 1.30278i | 2.00000 | − | 2.00000i | − | 1.00000i | 1.41421 | + | 2.82843i | ||||||||||||||||||||||||||||||
| 517.2 | −1.00000 | − | 1.00000i | −0.707107 | + | 0.707107i | 2.00000i | −2.12132 | − | 0.707107i | 1.41421 | 1.30278 | − | 2.30278i | 2.00000 | − | 2.00000i | − | 1.00000i | 1.41421 | + | 2.82843i | ||||||||||||||||||||||||||||||
| 517.3 | −1.00000 | − | 1.00000i | 0.707107 | − | 0.707107i | 2.00000i | 2.12132 | + | 0.707107i | −1.41421 | −2.30278 | + | 1.30278i | 2.00000 | − | 2.00000i | − | 1.00000i | −1.41421 | − | 2.82843i | ||||||||||||||||||||||||||||||
| 517.4 | −1.00000 | − | 1.00000i | 0.707107 | − | 0.707107i | 2.00000i | 2.12132 | + | 0.707107i | −1.41421 | 1.30278 | − | 2.30278i | 2.00000 | − | 2.00000i | − | 1.00000i | −1.41421 | − | 2.82843i | ||||||||||||||||||||||||||||||
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 |
| 8.b | even | 2 | 1 | inner |
| 35.f | even | 4 | 1 | inner |
| 40.i | odd | 4 | 1 | inner |
| 56.h | odd | 2 | 1 | inner |
| 280.s | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 840.2.bj.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | inner | 840.2.bj.a | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 840.2.bj.a | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 840.2.bj.a | ✓ | 8 |
| 35.f | even | 4 | 1 | inner | 840.2.bj.a | ✓ | 8 |
| 40.i | odd | 4 | 1 | inner | 840.2.bj.a | ✓ | 8 |
| 56.h | odd | 2 | 1 | inner | 840.2.bj.a | ✓ | 8 |
| 280.s | even | 4 | 1 | inner | 840.2.bj.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 840.2.bj.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 840.2.bj.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 840.2.bj.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 840.2.bj.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 840.2.bj.a | ✓ | 8 | 35.f | even | 4 | 1 | inner |
| 840.2.bj.a | ✓ | 8 | 40.i | odd | 4 | 1 | inner |
| 840.2.bj.a | ✓ | 8 | 56.h | odd | 2 | 1 | inner |
| 840.2.bj.a | ✓ | 8 | 280.s | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{2} + 26 \)
acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{4} \)
|
| $3$ |
\( (T^{4} + 1)^{2} \)
|
| $5$ |
\( (T^{4} - 8 T^{2} + 25)^{2} \)
|
| $7$ |
\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 49)^{2} \)
|
| $11$ |
\( (T^{2} + 26)^{4} \)
|
| $13$ |
\( (T^{4} + 16)^{2} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( (T^{2} + 4 T + 8)^{4} \)
|
| $29$ |
\( (T^{2} - 26)^{4} \)
|
| $31$ |
\( (T^{2} + 52)^{4} \)
|
| $37$ |
\( (T^{4} + 2704)^{2} \)
|
| $41$ |
\( (T^{2} + 52)^{4} \)
|
| $43$ |
\( T^{8} \)
|
| $47$ |
\( (T^{4} + 10816)^{2} \)
|
| $53$ |
\( (T^{4} + 2704)^{2} \)
|
| $59$ |
\( (T^{2} + 98)^{4} \)
|
| $61$ |
\( (T^{2} - 8)^{4} \)
|
| $67$ |
\( (T^{4} + 43264)^{2} \)
|
| $71$ |
\( (T - 12)^{8} \)
|
| $73$ |
\( (T^{4} + 676)^{2} \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( (T^{4} + 1296)^{2} \)
|
| $89$ |
\( (T^{2} - 52)^{4} \)
|
| $97$ |
\( (T^{4} + 54756)^{2} \)
|
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