Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [333,2,Mod(184,333)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(333, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("333.184");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 333 = 3^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 333.q (of order \(6\), 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: | \(2.65901838731\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.592240896.6 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{6} + 7x^{4} - 12x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{6} + 7x^{4} - 12x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + \nu^{4} - \nu^{2} + 8 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - \nu^{5} + 13\nu^{3} - 8\nu ) / 24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 5\nu^{5} - 5\nu^{3} + 22\nu ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + \nu^{5} + 3\nu^{3} ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{5} - 2\nu^{3} + \nu ) / 6 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{4} - 3\nu^{2} + 8 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{3} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{6} + \beta_{5} + \beta_{4} + 5\beta_{3} ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 3\beta_{2} + \beta _1 - 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7\beta_{6} + 8\beta_{5} - \beta_{4} - 2\beta_{3} ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{7} + 9\beta_{2} - 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -4\beta_{6} + 13\beta_{5} - 2\beta_{4} - 13\beta_{3} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/333\mathbb{Z}\right)^\times\).
| \(n\) | \(38\) | \(298\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 184.1 |
|
−1.79641 | + | 1.03716i | − | 1.73205i | 1.15139 | − | 1.99426i | 0.543909 | + | 0.314026i | 1.79641 | + | 3.11147i | −1.80278 | − | 3.12250i | 0.628052i | −3.00000 | −1.30278 | |||||||||||||||||||||||||||||||
| 184.2 | −0.723131 | + | 0.417500i | − | 1.73205i | −0.651388 | + | 1.12824i | −2.38834 | − | 1.37891i | 0.723131 | + | 1.25250i | 1.80278 | + | 3.12250i | − | 2.75782i | −3.00000 | 2.30278 | |||||||||||||||||||||||||||||||
| 184.3 | 0.723131 | − | 0.417500i | − | 1.73205i | −0.651388 | + | 1.12824i | 2.38834 | + | 1.37891i | −0.723131 | − | 1.25250i | 1.80278 | + | 3.12250i | 2.75782i | −3.00000 | 2.30278 | ||||||||||||||||||||||||||||||||
| 184.4 | 1.79641 | − | 1.03716i | − | 1.73205i | 1.15139 | − | 1.99426i | −0.543909 | − | 0.314026i | −1.79641 | − | 3.11147i | −1.80278 | − | 3.12250i | − | 0.628052i | −3.00000 | −1.30278 | |||||||||||||||||||||||||||||||
| 295.1 | −1.79641 | − | 1.03716i | 1.73205i | 1.15139 | + | 1.99426i | 0.543909 | − | 0.314026i | 1.79641 | − | 3.11147i | −1.80278 | + | 3.12250i | − | 0.628052i | −3.00000 | −1.30278 | ||||||||||||||||||||||||||||||||
| 295.2 | −0.723131 | − | 0.417500i | 1.73205i | −0.651388 | − | 1.12824i | −2.38834 | + | 1.37891i | 0.723131 | − | 1.25250i | 1.80278 | − | 3.12250i | 2.75782i | −3.00000 | 2.30278 | |||||||||||||||||||||||||||||||||
| 295.3 | 0.723131 | + | 0.417500i | 1.73205i | −0.651388 | − | 1.12824i | 2.38834 | − | 1.37891i | −0.723131 | + | 1.25250i | 1.80278 | − | 3.12250i | − | 2.75782i | −3.00000 | 2.30278 | ||||||||||||||||||||||||||||||||
| 295.4 | 1.79641 | + | 1.03716i | 1.73205i | 1.15139 | + | 1.99426i | −0.543909 | + | 0.314026i | −1.79641 | + | 3.11147i | −1.80278 | + | 3.12250i | 0.628052i | −3.00000 | −1.30278 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
| 37.b | even | 2 | 1 | inner |
| 333.q | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 333.2.q.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 999.2.q.a | 8 | ||
| 9.c | even | 3 | 1 | inner | 333.2.q.a | ✓ | 8 |
| 9.d | odd | 6 | 1 | 999.2.q.a | 8 | ||
| 37.b | even | 2 | 1 | inner | 333.2.q.a | ✓ | 8 |
| 111.d | odd | 2 | 1 | 999.2.q.a | 8 | ||
| 333.n | odd | 6 | 1 | 999.2.q.a | 8 | ||
| 333.q | even | 6 | 1 | inner | 333.2.q.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 333.2.q.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 333.2.q.a | ✓ | 8 | 9.c | even | 3 | 1 | inner |
| 333.2.q.a | ✓ | 8 | 37.b | even | 2 | 1 | inner |
| 333.2.q.a | ✓ | 8 | 333.q | even | 6 | 1 | inner |
| 999.2.q.a | 8 | 3.b | odd | 2 | 1 | ||
| 999.2.q.a | 8 | 9.d | odd | 6 | 1 | ||
| 999.2.q.a | 8 | 111.d | odd | 2 | 1 | ||
| 999.2.q.a | 8 | 333.n | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 5T_{2}^{6} + 22T_{2}^{4} - 15T_{2}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(333, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 5 T^{6} + \cdots + 9 \)
|
| $3$ |
\( (T^{2} + 3)^{4} \)
|
| $5$ |
\( T^{8} - 8 T^{6} + \cdots + 9 \)
|
| $7$ |
\( (T^{4} + 13 T^{2} + 169)^{2} \)
|
| $11$ |
\( (T^{2} - 3 T + 9)^{4} \)
|
| $13$ |
\( T^{8} - 24 T^{6} + \cdots + 729 \)
|
| $17$ |
\( (T^{4} + 20 T^{2} + 48)^{2} \)
|
| $19$ |
\( (T^{4} + 60 T^{2} + 432)^{2} \)
|
| $23$ |
\( T^{8} - 80 T^{6} + \cdots + 2518569 \)
|
| $29$ |
\( T^{8} - 32 T^{6} + \cdots + 59049 \)
|
| $31$ |
\( T^{8} - 24 T^{6} + \cdots + 729 \)
|
| $37$ |
\( (T^{4} + 2 T^{3} + \cdots + 1369)^{2} \)
|
| $41$ |
\( (T^{2} - 3 T + 9)^{4} \)
|
| $43$ |
\( T^{8} - 240 T^{6} + \cdots + 204004089 \)
|
| $47$ |
\( (T^{2} - 3 T + 9)^{4} \)
|
| $53$ |
\( (T + 12)^{8} \)
|
| $59$ |
\( T^{8} - 80 T^{6} + \cdots + 729 \)
|
| $61$ |
\( T^{8} - 72 T^{6} + \cdots + 59049 \)
|
| $67$ |
\( (T^{4} + 13 T^{2} + 169)^{2} \)
|
| $71$ |
\( (T^{2} + 6 T - 108)^{4} \)
|
| $73$ |
\( (T^{2} - 12 T - 16)^{4} \)
|
| $79$ |
\( T^{8} - 240 T^{6} + \cdots + 59049 \)
|
| $83$ |
\( (T^{4} - 12 T^{3} + \cdots + 6561)^{2} \)
|
| $89$ |
\( (T^{4} + 236 T^{2} + 13872)^{2} \)
|
| $97$ |
\( T^{8} - 240 T^{6} + \cdots + 204004089 \)
|
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