Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(361,945)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.361");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 945.k (of order \(3\), degree \(2\), not 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: | \(7.54586299101\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{13})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 4x^{2} + 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 315) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 4x^{2} + 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 7 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{3} - 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).
| \(n\) | \(136\) | \(596\) | \(757\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
−1.15139 | + | 1.99426i | 0 | −1.65139 | − | 2.86029i | 1.00000 | 0 | −2.00000 | + | 1.73205i | 3.00000 | 0 | −1.15139 | + | 1.99426i | ||||||||||||||||||||||
| 361.2 | 0.651388 | − | 1.12824i | 0 | 0.151388 | + | 0.262211i | 1.00000 | 0 | −2.00000 | + | 1.73205i | 3.00000 | 0 | 0.651388 | − | 1.12824i | |||||||||||||||||||||||
| 856.1 | −1.15139 | − | 1.99426i | 0 | −1.65139 | + | 2.86029i | 1.00000 | 0 | −2.00000 | − | 1.73205i | 3.00000 | 0 | −1.15139 | − | 1.99426i | |||||||||||||||||||||||
| 856.2 | 0.651388 | + | 1.12824i | 0 | 0.151388 | − | 0.262211i | 1.00000 | 0 | −2.00000 | − | 1.73205i | 3.00000 | 0 | 0.651388 | + | 1.12824i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.g | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 945.2.k.a | 4 | |
| 3.b | odd | 2 | 1 | 315.2.k.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | 945.2.l.a | 4 | ||
| 9.c | even | 3 | 1 | 945.2.l.a | 4 | ||
| 9.d | odd | 6 | 1 | 315.2.l.a | yes | 4 | |
| 21.h | odd | 6 | 1 | 315.2.l.a | yes | 4 | |
| 63.g | even | 3 | 1 | inner | 945.2.k.a | 4 | |
| 63.n | odd | 6 | 1 | 315.2.k.a | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.2.k.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 315.2.k.a | ✓ | 4 | 63.n | odd | 6 | 1 | |
| 315.2.l.a | yes | 4 | 9.d | odd | 6 | 1 | |
| 315.2.l.a | yes | 4 | 21.h | odd | 6 | 1 | |
| 945.2.k.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 945.2.k.a | 4 | 63.g | even | 3 | 1 | inner | |
| 945.2.l.a | 4 | 7.c | even | 3 | 1 | ||
| 945.2.l.a | 4 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} + 4 T^{2} + \cdots + 9 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T - 1)^{4} \)
|
| $7$ |
\( (T^{2} + 4 T + 7)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 4 T^{3} + \cdots + 81 \)
|
| $17$ |
\( T^{4} - 4 T^{3} + \cdots + 81 \)
|
| $19$ |
\( T^{4} + 13T^{2} + 169 \)
|
| $23$ |
\( (T^{2} + 4 T - 48)^{2} \)
|
| $29$ |
\( T^{4} - 2 T^{3} + \cdots + 2601 \)
|
| $31$ |
\( T^{4} + 13T^{2} + 169 \)
|
| $37$ |
\( T^{4} + 13T^{2} + 169 \)
|
| $41$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $43$ |
\( T^{4} - 6 T^{3} + \cdots + 1849 \)
|
| $47$ |
\( T^{4} + 2 T^{3} + \cdots + 2601 \)
|
| $53$ |
\( T^{4} - 8 T^{3} + \cdots + 9 \)
|
| $59$ |
\( T^{4} + 16 T^{3} + \cdots + 2601 \)
|
| $61$ |
\( T^{4} - 6 T^{3} + \cdots + 1849 \)
|
| $67$ |
\( T^{4} - 6 T^{3} + \cdots + 1849 \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} + 13T^{2} + 169 \)
|
| $79$ |
\( T^{4} + 4 T^{3} + \cdots + 12769 \)
|
| $83$ |
\( T^{4} + 14 T^{3} + \cdots + 9 \)
|
| $89$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $97$ |
\( T^{4} + 20 T^{3} + \cdots + 7569 \)
|
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