Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(46,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("315.46");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.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: | \(2.51528766367\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.4406832.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 6\nu^{4} - 36\nu^{3} + 24\nu^{2} + 5\nu - 30 ) / 149 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -6\nu^{5} + 36\nu^{4} - 67\nu^{3} + 144\nu^{2} + 30\nu + 416 ) / 149 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 30\nu^{5} - 31\nu^{4} + 186\nu^{3} + 174\nu^{2} + 744\nu + 155 ) / 149 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 89\nu^{5} - 87\nu^{4} + 522\nu^{3} + 695\nu^{2} + 2088\nu + 435 ) / 149 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 6\beta_{2} - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{5} + 19\beta_{4} + 6\beta_{3} - 12\beta _1 - 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{5} + 42\beta_{4} + 43\beta_{2} - 43\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(136\) | \(281\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 46.1 |
|
−1.32772 | − | 2.29968i | 0 | −2.52569 | + | 4.37462i | −0.500000 | − | 0.866025i | 0 | −2.35341 | + | 1.20891i | 8.10275 | 0 | −1.32772 | + | 2.29968i | ||||||||||||||||||||||||||
| 46.2 | −0.605378 | − | 1.04855i | 0 | 0.267035 | − | 0.462518i | −0.500000 | − | 0.866025i | 0 | 1.16166 | − | 2.37709i | −3.06814 | 0 | −0.605378 | + | 1.04855i | |||||||||||||||||||||||||||
| 46.3 | 0.933099 | + | 1.61618i | 0 | −0.741348 | + | 1.28405i | −0.500000 | − | 0.866025i | 0 | 1.69175 | + | 2.03420i | 0.965392 | 0 | 0.933099 | − | 1.61618i | |||||||||||||||||||||||||||
| 226.1 | −1.32772 | + | 2.29968i | 0 | −2.52569 | − | 4.37462i | −0.500000 | + | 0.866025i | 0 | −2.35341 | − | 1.20891i | 8.10275 | 0 | −1.32772 | − | 2.29968i | |||||||||||||||||||||||||||
| 226.2 | −0.605378 | + | 1.04855i | 0 | 0.267035 | + | 0.462518i | −0.500000 | + | 0.866025i | 0 | 1.16166 | + | 2.37709i | −3.06814 | 0 | −0.605378 | − | 1.04855i | |||||||||||||||||||||||||||
| 226.3 | 0.933099 | − | 1.61618i | 0 | −0.741348 | − | 1.28405i | −0.500000 | + | 0.866025i | 0 | 1.69175 | − | 2.03420i | 0.965392 | 0 | 0.933099 | + | 1.61618i | |||||||||||||||||||||||||||
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 | 315.2.j.f | ✓ | 6 |
| 3.b | odd | 2 | 1 | 315.2.j.g | yes | 6 | |
| 7.c | even | 3 | 1 | inner | 315.2.j.f | ✓ | 6 |
| 7.c | even | 3 | 1 | 2205.2.a.be | 3 | ||
| 7.d | odd | 6 | 1 | 2205.2.a.bd | 3 | ||
| 21.g | even | 6 | 1 | 2205.2.a.bc | 3 | ||
| 21.h | odd | 6 | 1 | 315.2.j.g | yes | 6 | |
| 21.h | odd | 6 | 1 | 2205.2.a.bb | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.2.j.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 315.2.j.f | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 315.2.j.g | yes | 6 | 3.b | odd | 2 | 1 | |
| 315.2.j.g | yes | 6 | 21.h | odd | 6 | 1 | |
| 2205.2.a.bb | 3 | 21.h | odd | 6 | 1 | ||
| 2205.2.a.bc | 3 | 21.g | even | 6 | 1 | ||
| 2205.2.a.bd | 3 | 7.d | odd | 6 | 1 | ||
| 2205.2.a.be | 3 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 2T_{2}^{5} + 8T_{2}^{4} + 4T_{2}^{3} + 28T_{2}^{2} + 24T_{2} + 36 \)
acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots + 36 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{2} + T + 1)^{3} \)
|
| $7$ |
\( T^{6} - T^{5} + \cdots + 343 \)
|
| $11$ |
\( T^{6} + 10 T^{5} + \cdots + 36 \)
|
| $13$ |
\( (T^{3} - 7 T^{2} + 9 T + 11)^{2} \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots + 36 \)
|
| $19$ |
\( T^{6} - T^{5} + \cdots + 1849 \)
|
| $23$ |
\( T^{6} + 10 T^{5} + \cdots + 66564 \)
|
| $29$ |
\( (T^{3} - 4 T^{2} - 14 T + 54)^{2} \)
|
| $31$ |
\( T^{6} - T^{5} + \cdots + 9801 \)
|
| $37$ |
\( T^{6} + 11 T^{5} + \cdots + 81 \)
|
| $41$ |
\( (T^{3} - 2 T^{2} - 58 T - 18)^{2} \)
|
| $43$ |
\( (T^{3} + 3 T^{2} - 41 T - 49)^{2} \)
|
| $47$ |
\( T^{6} - 2 T^{5} + \cdots + 28224 \)
|
| $53$ |
\( T^{6} + 4 T^{5} + \cdots + 2304 \)
|
| $59$ |
\( T^{6} + 4 T^{5} + \cdots + 49284 \)
|
| $61$ |
\( T^{6} - 22 T^{5} + \cdots + 63504 \)
|
| $67$ |
\( T^{6} - 15 T^{5} + \cdots + 22801 \)
|
| $71$ |
\( (T^{3} - 16 T^{2} + \cdots + 882)^{2} \)
|
| $73$ |
\( T^{6} + 9 T^{5} + \cdots + 1369 \)
|
| $79$ |
\( T^{6} - 7 T^{5} + \cdots + 5329 \)
|
| $83$ |
\( (T^{3} - 14 T^{2} + \cdots - 18)^{2} \)
|
| $89$ |
\( T^{6} - 30 T^{5} + \cdots + 285156 \)
|
| $97$ |
\( (T^{3} + 4 T^{2} + \cdots - 212)^{2} \)
|
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