Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1530,2,Mod(919,1530)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1530, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1530.919");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1530.d (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: | \(12.2171115093\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.350464.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 30\nu^{2} - 32\nu + 13 ) / 23 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -9\nu^{5} + 3\nu^{4} + 10\nu^{3} - 32\nu^{2} - 74\nu - 3 ) / 23 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -10\nu^{5} + 11\nu^{4} - 17\nu^{3} - 10\nu^{2} - 72\nu - 11 ) / 23 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 12\nu^{5} - 27\nu^{4} + 25\nu^{3} + 12\nu^{2} + 68\nu - 65 ) / 23 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -19\nu^{5} + 37\nu^{4} - 30\nu^{3} - 42\nu^{2} - 54\nu + 55 ) / 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{2} - 4\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - 3\beta_{3} + 3\beta_{2} - 4\beta _1 - 4 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{5} - 5\beta_{4} - 5\beta_{3} + \beta_{2} - 14 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -11\beta_{5} - 11\beta_{4} - 5\beta_{3} - 5\beta_{2} + 18\beta _1 - 18 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times\).
| \(n\) | \(307\) | \(1261\) | \(1361\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 919.1 |
|
− | 1.00000i | 0 | −1.00000 | −2.17009 | − | 0.539189i | 0 | − | 1.07838i | 1.00000i | 0 | −0.539189 | + | 2.17009i | ||||||||||||||||||||||||||||||
| 919.2 | − | 1.00000i | 0 | −1.00000 | −0.311108 | + | 2.21432i | 0 | 4.42864i | 1.00000i | 0 | 2.21432 | + | 0.311108i | ||||||||||||||||||||||||||||||||
| 919.3 | − | 1.00000i | 0 | −1.00000 | 1.48119 | − | 1.67513i | 0 | − | 3.35026i | 1.00000i | 0 | −1.67513 | − | 1.48119i | |||||||||||||||||||||||||||||||
| 919.4 | 1.00000i | 0 | −1.00000 | −2.17009 | + | 0.539189i | 0 | 1.07838i | − | 1.00000i | 0 | −0.539189 | − | 2.17009i | ||||||||||||||||||||||||||||||||
| 919.5 | 1.00000i | 0 | −1.00000 | −0.311108 | − | 2.21432i | 0 | − | 4.42864i | − | 1.00000i | 0 | 2.21432 | − | 0.311108i | |||||||||||||||||||||||||||||||
| 919.6 | 1.00000i | 0 | −1.00000 | 1.48119 | + | 1.67513i | 0 | 3.35026i | − | 1.00000i | 0 | −1.67513 | + | 1.48119i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1530.2.d.h | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1530.2.d.j | yes | 6 | |
| 5.b | even | 2 | 1 | inner | 1530.2.d.h | ✓ | 6 |
| 5.c | odd | 4 | 1 | 7650.2.a.dk | 3 | ||
| 5.c | odd | 4 | 1 | 7650.2.a.dp | 3 | ||
| 15.d | odd | 2 | 1 | 1530.2.d.j | yes | 6 | |
| 15.e | even | 4 | 1 | 7650.2.a.di | 3 | ||
| 15.e | even | 4 | 1 | 7650.2.a.dn | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1530.2.d.h | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1530.2.d.h | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 1530.2.d.j | yes | 6 | 3.b | odd | 2 | 1 | |
| 1530.2.d.j | yes | 6 | 15.d | odd | 2 | 1 | |
| 7650.2.a.di | 3 | 15.e | even | 4 | 1 | ||
| 7650.2.a.dk | 3 | 5.c | odd | 4 | 1 | ||
| 7650.2.a.dn | 3 | 15.e | even | 4 | 1 | ||
| 7650.2.a.dp | 3 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1530, [\chi])\):
|
\( T_{7}^{6} + 32T_{7}^{4} + 256T_{7}^{2} + 256 \)
|
|
\( T_{11} - 4 \)
|
|
\( T_{29}^{3} - 2T_{29}^{2} - 12T_{29} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 2 T^{5} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 32 T^{4} + \cdots + 256 \)
|
| $11$ |
\( (T - 4)^{6} \)
|
| $13$ |
\( T^{6} + 44 T^{4} + \cdots + 64 \)
|
| $17$ |
\( (T^{2} + 1)^{3} \)
|
| $19$ |
\( (T^{3} + 4 T^{2} - 48 T - 64)^{2} \)
|
| $23$ |
\( T^{6} + 44 T^{4} + \cdots + 64 \)
|
| $29$ |
\( (T^{3} - 2 T^{2} - 12 T + 8)^{2} \)
|
| $31$ |
\( (T^{3} - 6 T^{2} - 4 T + 40)^{2} \)
|
| $37$ |
\( T^{6} + 92 T^{4} + \cdots + 1600 \)
|
| $41$ |
\( (T^{3} - 2 T^{2} + \cdots + 104)^{2} \)
|
| $43$ |
\( T^{6} + 108 T^{4} + \cdots + 64 \)
|
| $47$ |
\( T^{6} + 128 T^{4} + \cdots + 16384 \)
|
| $53$ |
\( T^{6} + 460 T^{4} + \cdots + 3474496 \)
|
| $59$ |
\( (T^{3} + 6 T^{2} + \cdots - 632)^{2} \)
|
| $61$ |
\( (T^{3} - 12 T^{2} + \cdots + 944)^{2} \)
|
| $67$ |
\( T^{6} + 204 T^{4} + \cdots + 53824 \)
|
| $71$ |
\( (T^{3} + 8 T^{2} - 64 T + 80)^{2} \)
|
| $73$ |
\( T^{6} + 428 T^{4} + \cdots + 1459264 \)
|
| $79$ |
\( (T^{3} + 10 T^{2} + \cdots - 1432)^{2} \)
|
| $83$ |
\( T^{6} + 112 T^{4} + \cdots + 4096 \)
|
| $89$ |
\( (T^{3} - 12 T^{2} + \cdots + 320)^{2} \)
|
| $97$ |
\( T^{6} + 524 T^{4} + \cdots + 287296 \)
|
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