Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3330,2,Mod(2071,3330)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3330, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3330.2071");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3330.h (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: | \(26.5901838731\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.399424.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 370) |
| 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
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{3} - 5\beta_{2} - \beta _1 + 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).
| \(n\) | \(371\) | \(631\) | \(667\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2071.1 |
|
− | 1.00000i | 0 | −1.00000 | 1.00000i | 0 | −0.342923 | 1.00000i | 0 | 1.00000 | |||||||||||||||||||||||||||||||||||
| 2071.2 | − | 1.00000i | 0 | −1.00000 | 1.00000i | 0 | 1.52932 | 1.00000i | 0 | 1.00000 | ||||||||||||||||||||||||||||||||||||
| 2071.3 | − | 1.00000i | 0 | −1.00000 | 1.00000i | 0 | 3.81361 | 1.00000i | 0 | 1.00000 | ||||||||||||||||||||||||||||||||||||
| 2071.4 | 1.00000i | 0 | −1.00000 | − | 1.00000i | 0 | −0.342923 | − | 1.00000i | 0 | 1.00000 | |||||||||||||||||||||||||||||||||||
| 2071.5 | 1.00000i | 0 | −1.00000 | − | 1.00000i | 0 | 1.52932 | − | 1.00000i | 0 | 1.00000 | |||||||||||||||||||||||||||||||||||
| 2071.6 | 1.00000i | 0 | −1.00000 | − | 1.00000i | 0 | 3.81361 | − | 1.00000i | 0 | 1.00000 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3330.2.h.n | 6 | |
| 3.b | odd | 2 | 1 | 370.2.d.c | ✓ | 6 | |
| 12.b | even | 2 | 1 | 2960.2.p.g | 6 | ||
| 15.d | odd | 2 | 1 | 1850.2.d.f | 6 | ||
| 15.e | even | 4 | 1 | 1850.2.c.i | 6 | ||
| 15.e | even | 4 | 1 | 1850.2.c.j | 6 | ||
| 37.b | even | 2 | 1 | inner | 3330.2.h.n | 6 | |
| 111.d | odd | 2 | 1 | 370.2.d.c | ✓ | 6 | |
| 444.g | even | 2 | 1 | 2960.2.p.g | 6 | ||
| 555.b | odd | 2 | 1 | 1850.2.d.f | 6 | ||
| 555.n | even | 4 | 1 | 1850.2.c.i | 6 | ||
| 555.n | even | 4 | 1 | 1850.2.c.j | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 370.2.d.c | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 370.2.d.c | ✓ | 6 | 111.d | odd | 2 | 1 | |
| 1850.2.c.i | 6 | 15.e | even | 4 | 1 | ||
| 1850.2.c.i | 6 | 555.n | even | 4 | 1 | ||
| 1850.2.c.j | 6 | 15.e | even | 4 | 1 | ||
| 1850.2.c.j | 6 | 555.n | even | 4 | 1 | ||
| 1850.2.d.f | 6 | 15.d | odd | 2 | 1 | ||
| 1850.2.d.f | 6 | 555.b | odd | 2 | 1 | ||
| 2960.2.p.g | 6 | 12.b | even | 2 | 1 | ||
| 2960.2.p.g | 6 | 444.g | even | 2 | 1 | ||
| 3330.2.h.n | 6 | 1.a | even | 1 | 1 | trivial | |
| 3330.2.h.n | 6 | 37.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3330, [\chi])\):
|
\( T_{7}^{3} - 5T_{7}^{2} + 4T_{7} + 2 \)
|
|
\( T_{11}^{3} - T_{11}^{2} - 16T_{11} - 16 \)
|
|
\( T_{13}^{6} + 36T_{13}^{4} + 320T_{13}^{2} + 256 \)
|
|
\( T_{41}^{3} + T_{41}^{2} - 56T_{41} - 172 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{2} + 1)^{3} \)
|
| $7$ |
\( (T^{3} - 5 T^{2} + 4 T + 2)^{2} \)
|
| $11$ |
\( (T^{3} - T^{2} - 16 T - 16)^{2} \)
|
| $13$ |
\( T^{6} + 36 T^{4} + \cdots + 256 \)
|
| $17$ |
\( T^{6} + 81 T^{4} + \cdots + 16384 \)
|
| $19$ |
\( T^{6} + 36 T^{4} + \cdots + 16 \)
|
| $23$ |
\( T^{6} + 36 T^{4} + \cdots + 256 \)
|
| $29$ |
\( T^{6} + 49 T^{4} + \cdots + 16 \)
|
| $31$ |
\( T^{6} + 57 T^{4} + \cdots + 6724 \)
|
| $37$ |
\( T^{6} + 10 T^{5} + \cdots + 50653 \)
|
| $41$ |
\( (T^{3} + T^{2} - 56 T - 172)^{2} \)
|
| $43$ |
\( T^{6} + 33 T^{4} + \cdots + 256 \)
|
| $47$ |
\( (T^{3} - 2 T^{2} - 14 T + 32)^{2} \)
|
| $53$ |
\( (T^{3} + T^{2} - 104 T - 352)^{2} \)
|
| $59$ |
\( T^{6} + 96 T^{4} + \cdots + 64 \)
|
| $61$ |
\( T^{6} + 121 T^{4} + \cdots + 15376 \)
|
| $67$ |
\( (T^{3} - 4 T^{2} + \cdots - 124)^{2} \)
|
| $71$ |
\( (T^{3} - 10 T^{2} + \cdots + 16)^{2} \)
|
| $73$ |
\( (T^{3} + 6 T^{2} + \cdots - 344)^{2} \)
|
| $79$ |
\( T^{6} + 80 T^{4} + \cdots + 4096 \)
|
| $83$ |
\( (T^{3} - 10 T^{2} - 62 T - 8)^{2} \)
|
| $89$ |
\( T^{6} + 256 T^{4} + \cdots + 262144 \)
|
| $97$ |
\( T^{6} + 289 T^{4} + \cdots + 29584 \)
|
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