Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.46");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(18.5856016518\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 105) |
| 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(136\) | \(281\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 46.1 |
|
−1.70711 | − | 2.95680i | 0 | −1.82843 | + | 3.16693i | −2.50000 | − | 4.33013i | 0 | 16.7426 | − | 7.91732i | −14.8284 | 0 | −8.53553 | + | 14.7840i | ||||||||||||||||||||
| 46.2 | −0.292893 | − | 0.507306i | 0 | 3.82843 | − | 6.63103i | −2.50000 | − | 4.33013i | 0 | 8.25736 | + | 16.5776i | −9.17157 | 0 | −1.46447 | + | 2.53653i | |||||||||||||||||||||
| 226.1 | −1.70711 | + | 2.95680i | 0 | −1.82843 | − | 3.16693i | −2.50000 | + | 4.33013i | 0 | 16.7426 | + | 7.91732i | −14.8284 | 0 | −8.53553 | − | 14.7840i | |||||||||||||||||||||
| 226.2 | −0.292893 | + | 0.507306i | 0 | 3.82843 | + | 6.63103i | −2.50000 | + | 4.33013i | 0 | 8.25736 | − | 16.5776i | −9.17157 | 0 | −1.46447 | − | 2.53653i | |||||||||||||||||||||
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 | 315.4.j.d | 4 | |
| 3.b | odd | 2 | 1 | 105.4.i.b | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 315.4.j.d | 4 | |
| 7.c | even | 3 | 1 | 2205.4.a.bd | 2 | ||
| 7.d | odd | 6 | 1 | 2205.4.a.bc | 2 | ||
| 21.g | even | 6 | 1 | 735.4.a.n | 2 | ||
| 21.h | odd | 6 | 1 | 105.4.i.b | ✓ | 4 | |
| 21.h | odd | 6 | 1 | 735.4.a.l | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.i.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 105.4.i.b | ✓ | 4 | 21.h | odd | 6 | 1 | |
| 315.4.j.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 315.4.j.d | 4 | 7.c | even | 3 | 1 | inner | |
| 735.4.a.l | 2 | 21.h | odd | 6 | 1 | ||
| 735.4.a.n | 2 | 21.g | even | 6 | 1 | ||
| 2205.4.a.bc | 2 | 7.d | odd | 6 | 1 | ||
| 2205.4.a.bd | 2 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(315, [\chi])\):
|
\( T_{2}^{4} + 4T_{2}^{3} + 14T_{2}^{2} + 8T_{2} + 4 \)
|
|
\( T_{13}^{2} - 14T_{13} - 1303 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots + 4 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} - 50 T^{3} + \cdots + 117649 \)
|
| $11$ |
\( T^{4} + 32 T^{3} + \cdots + 6927424 \)
|
| $13$ |
\( (T^{2} - 14 T - 1303)^{2} \)
|
| $17$ |
\( T^{4} + 20 T^{3} + \cdots + 4892944 \)
|
| $19$ |
\( T^{4} + 18 T^{3} + \cdots + 65788321 \)
|
| $23$ |
\( T^{4} + 68 T^{3} + \cdots + 38416 \)
|
| $29$ |
\( (T^{2} - 332 T + 27484)^{2} \)
|
| $31$ |
\( T^{4} + 66 T^{3} + \cdots + 2259009 \)
|
| $37$ |
\( T^{4} + \cdots + 9699689169 \)
|
| $41$ |
\( (T^{2} + 152 T - 23992)^{2} \)
|
| $43$ |
\( (T^{2} + 842 T + 174353)^{2} \)
|
| $47$ |
\( T^{4} + 212 T^{3} + \cdots + 308494096 \)
|
| $53$ |
\( T^{4} + 368 T^{3} + \cdots + 760877056 \)
|
| $59$ |
\( T^{4} + \cdots + 4133461264 \)
|
| $61$ |
\( T^{4} + \cdots + 16067083536 \)
|
| $67$ |
\( T^{4} + \cdots + 25309309921 \)
|
| $71$ |
\( (T^{2} - 1208 T + 243784)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 467485345441 \)
|
| $79$ |
\( T^{4} + \cdots + 102042552481 \)
|
| $83$ |
\( (T^{2} + 968 T + 213448)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 4742934797584 \)
|
| $97$ |
\( (T^{2} - 1692 T - 367676)^{2} \)
|
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