Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(541,945)]
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([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.541");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 945.j (of order \(3\), 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: | \(7.54586299101\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.1783323.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 19\nu^{2} + 12\nu - 60 ) / 83 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{5} - 20\nu^{4} + 17\nu^{3} - 76\nu^{2} + 48\nu - 240 ) / 83 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -20\nu^{5} + 17\nu^{4} - 85\nu^{3} - 35\nu^{2} - 323\nu + 204 ) / 249 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -16\nu^{5} - 3\nu^{4} - 68\nu^{3} - 28\nu^{2} - 275\nu - 36 ) / 83 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} - \beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 4\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{5} + 12\beta_{4} - \beta _1 - 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( -6\beta_{5} + 3\beta_{4} + 6\beta_{3} - 17\beta_{2} - 17\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).
| \(n\) | \(136\) | \(596\) | \(757\) |
| \(\chi(n)\) | \(-1 + \beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 541.1 |
|
−1.09935 | + | 1.90412i | 0 | −1.41712 | − | 2.45453i | 0.500000 | − | 0.866025i | 0 | 2.43359 | + | 1.03810i | 1.83424 | 0 | 1.09935 | + | 1.90412i | ||||||||||||||||||||||||||
| 541.2 | −0.356769 | + | 0.617942i | 0 | 0.745432 | + | 1.29113i | 0.500000 | − | 0.866025i | 0 | −2.63409 | − | 0.248083i | −2.49086 | 0 | 0.356769 | + | 0.617942i | |||||||||||||||||||||||||||
| 541.3 | 0.956115 | − | 1.65604i | 0 | −0.828310 | − | 1.43468i | 0.500000 | − | 0.866025i | 0 | −0.799494 | − | 2.52206i | 0.656620 | 0 | −0.956115 | − | 1.65604i | |||||||||||||||||||||||||||
| 676.1 | −1.09935 | − | 1.90412i | 0 | −1.41712 | + | 2.45453i | 0.500000 | + | 0.866025i | 0 | 2.43359 | − | 1.03810i | 1.83424 | 0 | 1.09935 | − | 1.90412i | |||||||||||||||||||||||||||
| 676.2 | −0.356769 | − | 0.617942i | 0 | 0.745432 | − | 1.29113i | 0.500000 | + | 0.866025i | 0 | −2.63409 | + | 0.248083i | −2.49086 | 0 | 0.356769 | − | 0.617942i | |||||||||||||||||||||||||||
| 676.3 | 0.956115 | + | 1.65604i | 0 | −0.828310 | + | 1.43468i | 0.500000 | + | 0.866025i | 0 | −0.799494 | + | 2.52206i | 0.656620 | 0 | −0.956115 | + | 1.65604i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | 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.j.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 945.2.j.d | yes | 6 | |
| 7.c | even | 3 | 1 | inner | 945.2.j.c | ✓ | 6 |
| 7.c | even | 3 | 1 | 6615.2.a.ba | 3 | ||
| 7.d | odd | 6 | 1 | 6615.2.a.bb | 3 | ||
| 21.g | even | 6 | 1 | 6615.2.a.y | 3 | ||
| 21.h | odd | 6 | 1 | 945.2.j.d | yes | 6 | |
| 21.h | odd | 6 | 1 | 6615.2.a.z | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 945.2.j.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 945.2.j.c | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 945.2.j.d | yes | 6 | 3.b | odd | 2 | 1 | |
| 945.2.j.d | yes | 6 | 21.h | odd | 6 | 1 | |
| 6615.2.a.y | 3 | 21.g | even | 6 | 1 | ||
| 6615.2.a.z | 3 | 21.h | odd | 6 | 1 | ||
| 6615.2.a.ba | 3 | 7.c | even | 3 | 1 | ||
| 6615.2.a.bb | 3 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} + 5T_{2}^{4} + 2T_{2}^{3} + 19T_{2}^{2} + 12T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + 5 T^{4} + \cdots + 9 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{2} - T + 1)^{3} \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 343 \)
|
| $11$ |
\( T^{6} - T^{5} + \cdots + 81 \)
|
| $13$ |
\( (T^{3} + 2 T^{2} - 9 T - 19)^{2} \)
|
| $17$ |
\( T^{6} + 11 T^{5} + \cdots + 729 \)
|
| $19$ |
\( T^{6} - 13 T^{5} + \cdots + 2401 \)
|
| $23$ |
\( T^{6} - 10 T^{5} + \cdots + 9 \)
|
| $29$ |
\( (T^{3} + 7 T^{2} - 8 T - 81)^{2} \)
|
| $31$ |
\( T^{6} + 8 T^{5} + \cdots + 81 \)
|
| $37$ |
\( T^{6} + 2 T^{5} + \cdots + 321489 \)
|
| $41$ |
\( (T^{3} - 10 T^{2} + \cdots - 21)^{2} \)
|
| $43$ |
\( (T^{3} - 6 T^{2} + \cdots + 344)^{2} \)
|
| $47$ |
\( T^{6} + 17 T^{5} + \cdots + 301401 \)
|
| $53$ |
\( T^{6} - 7 T^{5} + \cdots + 45369 \)
|
| $59$ |
\( T^{6} + 17 T^{5} + \cdots + 301401 \)
|
| $61$ |
\( T^{6} - T^{5} + \cdots + 2025 \)
|
| $67$ |
\( T^{6} - 15 T^{5} + \cdots + 32761 \)
|
| $71$ |
\( (T^{3} - 2 T^{2} + \cdots - 120)^{2} \)
|
| $73$ |
\( T^{6} + 107 T^{4} + \cdots + 32761 \)
|
| $79$ |
\( T^{6} - 10 T^{5} + \cdots + 1 \)
|
| $83$ |
\( (T^{3} - 16 T^{2} + \cdots + 24)^{2} \)
|
| $89$ |
\( T^{6} + 24 T^{5} + \cdots + 5625 \)
|
| $97$ |
\( (T^{3} - 5 T^{2} - 44 T + 97)^{2} \)
|
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