Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [840,2,Mod(421,840)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(840, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("840.421");
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.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: | \(6.70743376979\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{16})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( 2\zeta_{16}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{16}^{4} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{16}^{5} + \zeta_{16} \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\zeta_{16}^{6} \)
|
| \(\beta_{5}\) | \(=\) |
\( \zeta_{16}^{7} + \zeta_{16}^{3} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{16}^{5} + \zeta_{16} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{16}^{7} + \zeta_{16}^{3} \)
|
| \(\zeta_{16}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} ) / 2 \)
|
| \(\zeta_{16}^{2}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\zeta_{16}^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} ) / 2 \)
|
| \(\zeta_{16}^{4}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{16}^{5}\) | \(=\) |
\( ( -\beta_{6} + \beta_{3} ) / 2 \)
|
| \(\zeta_{16}^{6}\) | \(=\) |
\( ( \beta_{4} ) / 2 \)
|
| \(\zeta_{16}^{7}\) | \(=\) |
\( ( -\beta_{7} + \beta_{5} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(281\) | \(337\) | \(421\) | \(631\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 421.1 |
|
−1.30656 | − | 0.541196i | 1.00000i | 1.41421 | + | 1.41421i | 1.00000i | 0.541196 | − | 1.30656i | 1.00000 | −1.08239 | − | 2.61313i | −1.00000 | 0.541196 | − | 1.30656i | ||||||||||||||||||||||||||||||||
| 421.2 | −1.30656 | + | 0.541196i | − | 1.00000i | 1.41421 | − | 1.41421i | − | 1.00000i | 0.541196 | + | 1.30656i | 1.00000 | −1.08239 | + | 2.61313i | −1.00000 | 0.541196 | + | 1.30656i | |||||||||||||||||||||||||||||||
| 421.3 | −0.541196 | − | 1.30656i | − | 1.00000i | −1.41421 | + | 1.41421i | − | 1.00000i | −1.30656 | + | 0.541196i | 1.00000 | 2.61313 | + | 1.08239i | −1.00000 | −1.30656 | + | 0.541196i | |||||||||||||||||||||||||||||||
| 421.4 | −0.541196 | + | 1.30656i | 1.00000i | −1.41421 | − | 1.41421i | 1.00000i | −1.30656 | − | 0.541196i | 1.00000 | 2.61313 | − | 1.08239i | −1.00000 | −1.30656 | − | 0.541196i | |||||||||||||||||||||||||||||||||
| 421.5 | 0.541196 | − | 1.30656i | 1.00000i | −1.41421 | − | 1.41421i | 1.00000i | 1.30656 | + | 0.541196i | 1.00000 | −2.61313 | + | 1.08239i | −1.00000 | 1.30656 | + | 0.541196i | |||||||||||||||||||||||||||||||||
| 421.6 | 0.541196 | + | 1.30656i | − | 1.00000i | −1.41421 | + | 1.41421i | − | 1.00000i | 1.30656 | − | 0.541196i | 1.00000 | −2.61313 | − | 1.08239i | −1.00000 | 1.30656 | − | 0.541196i | |||||||||||||||||||||||||||||||
| 421.7 | 1.30656 | − | 0.541196i | − | 1.00000i | 1.41421 | − | 1.41421i | − | 1.00000i | −0.541196 | − | 1.30656i | 1.00000 | 1.08239 | − | 2.61313i | −1.00000 | −0.541196 | − | 1.30656i | |||||||||||||||||||||||||||||||
| 421.8 | 1.30656 | + | 0.541196i | 1.00000i | 1.41421 | + | 1.41421i | 1.00000i | −0.541196 | + | 1.30656i | 1.00000 | 1.08239 | + | 2.61313i | −1.00000 | −0.541196 | + | 1.30656i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 840.2.g.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 3360.2.g.a | 8 | ||
| 8.b | even | 2 | 1 | inner | 840.2.g.a | ✓ | 8 |
| 8.d | odd | 2 | 1 | 3360.2.g.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 840.2.g.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 840.2.g.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 3360.2.g.a | 8 | 4.b | odd | 2 | 1 | ||
| 3360.2.g.a | 8 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{4} + 8T_{11}^{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 16 \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( (T^{4} + 8 T^{2} + 8)^{2} \)
|
| $13$ |
\( T^{8} + 48 T^{6} + \cdots + 64 \)
|
| $17$ |
\( (T^{4} - 16 T^{2} + 32)^{2} \)
|
| $19$ |
\( T^{8} + 144 T^{6} + \cdots + 602176 \)
|
| $23$ |
\( (T^{4} + 16 T^{3} + \cdots + 32)^{2} \)
|
| $29$ |
\( T^{8} + 80 T^{6} + \cdots + 73984 \)
|
| $31$ |
\( (T^{4} - 16 T^{3} + \cdots - 376)^{2} \)
|
| $37$ |
\( (T^{4} + 16 T^{2} + 32)^{2} \)
|
| $41$ |
\( (T^{4} - 48 T^{2} + \cdots - 64)^{2} \)
|
| $43$ |
\( T^{8} + 48 T^{6} + \cdots + 256 \)
|
| $47$ |
\( (T^{4} - 8 T^{3} + 8 T^{2} + \cdots - 16)^{2} \)
|
| $53$ |
\( T^{8} + 144 T^{6} + \cdots + 141376 \)
|
| $59$ |
\( T^{8} + 144 T^{6} + \cdots + 73984 \)
|
| $61$ |
\( T^{8} + 336 T^{6} + \cdots + 73984 \)
|
| $67$ |
\( T^{8} + 336 T^{6} + \cdots + 4129024 \)
|
| $71$ |
\( (T^{4} - 136 T^{2} + \cdots - 632)^{2} \)
|
| $73$ |
\( (T^{4} - 120 T^{2} + \cdots - 376)^{2} \)
|
| $79$ |
\( (T^{4} - 8 T^{3} + \cdots + 4624)^{2} \)
|
| $83$ |
\( T^{8} + 544 T^{6} + \cdots + 66064384 \)
|
| $89$ |
\( (T^{4} - 160 T^{2} + 6272)^{2} \)
|
| $97$ |
\( (T^{4} - 216 T^{2} + \cdots + 776)^{2} \)
|
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