Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [420,2,Mod(239,420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(420, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("420.239");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 420.l (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: | \(3.35371688489\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.386672896.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{5} + 2\nu^{4} + 2\nu^{3} - 4\nu ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - \nu^{5} - 2\nu^{4} + 2\nu^{3} - 4\nu^{2} - 4\nu ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{5} - 2\nu^{4} - 2\nu^{3} + 8\nu^{2} + 4\nu ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - \nu^{4} + 2\nu^{3} + 2\nu^{2} - 4\nu - 8 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} - \nu^{5} - 2\nu^{3} - 4\nu^{2} - 12\nu ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} + 3\nu^{5} + 2\nu^{3} - 4\nu^{2} - 12\nu - 24 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{4} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{4} - 2\beta_{3} + \beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} + 4\beta _1 + 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2\beta_{7} + \beta_{6} - 3\beta_{5} + 5\beta_{4} + 4\beta_{3} + 6\beta_{2} + 4\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(241\) | \(281\) | \(337\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 239.1 |
|
−1.19503 | − | 0.756243i | −0.356193 | − | 1.69503i | 0.856193 | + | 1.80747i | −1.00000 | − | 2.00000i | −0.856193 | + | 2.29498i | 1.00000 | 0.343707 | − | 2.80747i | −2.74625 | + | 1.20752i | −0.317456 | + | 3.14630i | ||||||||||||||||||||||||||
| 239.2 | −1.19503 | + | 0.756243i | −0.356193 | + | 1.69503i | 0.856193 | − | 1.80747i | −1.00000 | + | 2.00000i | −0.856193 | − | 2.29498i | 1.00000 | 0.343707 | + | 2.80747i | −2.74625 | − | 1.20752i | −0.317456 | − | 3.14630i | |||||||||||||||||||||||||||
| 239.3 | −0.835949 | − | 1.14070i | 1.10238 | + | 1.33595i | −0.602380 | + | 1.90713i | −1.00000 | + | 2.00000i | 0.602380 | − | 2.37427i | 1.00000 | 2.67901 | − | 0.907128i | −0.569517 | + | 2.94545i | 3.11734 | − | 0.531200i | |||||||||||||||||||||||||||
| 239.4 | −0.835949 | + | 1.14070i | 1.10238 | − | 1.33595i | −0.602380 | − | 1.90713i | −1.00000 | − | 2.00000i | 0.602380 | + | 2.37427i | 1.00000 | 2.67901 | + | 0.907128i | −0.569517 | − | 2.94545i | 3.11734 | + | 0.531200i | |||||||||||||||||||||||||||
| 239.5 | 0.621372 | − | 1.27039i | 1.72779 | + | 0.121372i | −1.22779 | − | 1.57877i | −1.00000 | − | 2.00000i | 1.22779 | − | 2.11956i | 1.00000 | −2.76858 | + | 0.578773i | 2.97054 | + | 0.419412i | −3.16216 | + | 0.0276478i | |||||||||||||||||||||||||||
| 239.6 | 0.621372 | + | 1.27039i | 1.72779 | − | 0.121372i | −1.22779 | + | 1.57877i | −1.00000 | + | 2.00000i | 1.22779 | + | 2.11956i | 1.00000 | −2.76858 | − | 0.578773i | 2.97054 | − | 0.419412i | −3.16216 | − | 0.0276478i | |||||||||||||||||||||||||||
| 239.7 | 1.40961 | − | 0.114062i | −1.47398 | + | 0.909606i | 1.97398 | − | 0.321565i | −1.00000 | − | 2.00000i | −1.97398 | + | 1.45031i | 1.00000 | 2.74586 | − | 0.678435i | 1.34523 | − | 2.68148i | −1.63773 | − | 2.70515i | |||||||||||||||||||||||||||
| 239.8 | 1.40961 | + | 0.114062i | −1.47398 | − | 0.909606i | 1.97398 | + | 0.321565i | −1.00000 | + | 2.00000i | −1.97398 | − | 1.45031i | 1.00000 | 2.74586 | + | 0.678435i | 1.34523 | + | 2.68148i | −1.63773 | + | 2.70515i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 60.h | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 420.2.l.e | yes | 8 |
| 3.b | odd | 2 | 1 | 420.2.l.f | yes | 8 | |
| 4.b | odd | 2 | 1 | 420.2.l.c | ✓ | 8 | |
| 5.b | even | 2 | 1 | 420.2.l.d | yes | 8 | |
| 12.b | even | 2 | 1 | 420.2.l.d | yes | 8 | |
| 15.d | odd | 2 | 1 | 420.2.l.c | ✓ | 8 | |
| 20.d | odd | 2 | 1 | 420.2.l.f | yes | 8 | |
| 60.h | even | 2 | 1 | inner | 420.2.l.e | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 420.2.l.c | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 420.2.l.c | ✓ | 8 | 15.d | odd | 2 | 1 | |
| 420.2.l.d | yes | 8 | 5.b | even | 2 | 1 | |
| 420.2.l.d | yes | 8 | 12.b | even | 2 | 1 | |
| 420.2.l.e | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 420.2.l.e | yes | 8 | 60.h | even | 2 | 1 | inner |
| 420.2.l.f | yes | 8 | 3.b | odd | 2 | 1 | |
| 420.2.l.f | yes | 8 | 20.d | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\):
|
\( T_{11}^{4} + 2T_{11}^{3} - 11T_{11}^{2} - 16T_{11} + 16 \)
|
|
\( T_{17}^{4} - 2T_{17}^{3} - 35T_{17}^{2} + 52T_{17} + 100 \)
|
|
\( T_{43}^{4} + 4T_{43}^{3} - 56T_{43}^{2} - 192T_{43} - 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{6} + \cdots + 16 \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 81 \)
|
| $5$ |
\( (T^{2} + 2 T + 5)^{4} \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( (T^{4} + 2 T^{3} - 11 T^{2} + \cdots + 16)^{2} \)
|
| $13$ |
\( T^{8} + 70 T^{6} + \cdots + 6400 \)
|
| $17$ |
\( (T^{4} - 2 T^{3} + \cdots + 100)^{2} \)
|
| $19$ |
\( T^{8} + 88 T^{6} + \cdots + 1024 \)
|
| $23$ |
\( T^{8} + 96 T^{6} + \cdots + 6400 \)
|
| $29$ |
\( T^{8} + 70 T^{6} + \cdots + 6400 \)
|
| $31$ |
\( T^{8} + 192 T^{6} + \cdots + 12544 \)
|
| $37$ |
\( T^{8} + 136 T^{6} + \cdots + 4096 \)
|
| $41$ |
\( T^{8} + 104 T^{6} + \cdots + 65536 \)
|
| $43$ |
\( (T^{4} + 4 T^{3} + \cdots - 128)^{2} \)
|
| $47$ |
\( T^{8} + 22 T^{6} + \cdots + 16 \)
|
| $53$ |
\( (T^{4} - 200 T^{2} + \cdots + 5392)^{2} \)
|
| $59$ |
\( (T^{4} + 4 T^{3} + \cdots - 2240)^{2} \)
|
| $61$ |
\( (T^{4} + 8 T^{3} + \cdots - 1312)^{2} \)
|
| $67$ |
\( (T^{4} + 12 T^{3} + \cdots - 6848)^{2} \)
|
| $71$ |
\( (T - 8)^{8} \)
|
| $73$ |
\( (T^{2} + 64)^{4} \)
|
| $79$ |
\( T^{8} + 502 T^{6} + \cdots + 753424 \)
|
| $83$ |
\( T^{8} + 432 T^{6} + \cdots + 29073664 \)
|
| $89$ |
\( T^{8} + 352 T^{6} + \cdots + 6553600 \)
|
| $97$ |
\( T^{8} + 694 T^{6} + \cdots + 16777216 \)
|
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