Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [333,3,Mod(233,333)]
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([3, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("333.233");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 333 = 3^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 333.m (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: | \(9.07359280320\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{4} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{6} + \zeta_{24}^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{7} + \zeta_{24} \)
|
| \(\beta_{4}\) | \(=\) |
\( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{24}^{7} + \zeta_{24}^{5} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 2\beta_{3} ) / 3 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{2} ) / 3 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( ( -2\beta_{7} + \beta_{6} + 3\beta_{5} - \beta_{3} ) / 3 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( -\beta_{7} + 2\beta_{6} + \beta_{3} ) / 3 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( ( -\beta_{4} + 2\beta_{2} ) / 3 \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} + \beta_{3} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/333\mathbb{Z}\right)^\times\).
| \(n\) | \(38\) | \(298\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 233.1 |
|
−0.965926 | − | 1.67303i | 0 | 0.133975 | − | 0.232051i | 2.63896 | − | 4.57081i | 0 | 3.69615 | − | 6.40192i | −8.24504 | 0 | −10.1962 | ||||||||||||||||||||||||||||||||||
| 233.2 | −0.258819 | − | 0.448288i | 0 | 1.86603 | − | 3.23205i | −0.189469 | + | 0.328169i | 0 | −6.69615 | + | 11.5981i | −4.00240 | 0 | 0.196152 | |||||||||||||||||||||||||||||||||||
| 233.3 | 0.258819 | + | 0.448288i | 0 | 1.86603 | − | 3.23205i | 0.189469 | − | 0.328169i | 0 | −6.69615 | + | 11.5981i | 4.00240 | 0 | 0.196152 | |||||||||||||||||||||||||||||||||||
| 233.4 | 0.965926 | + | 1.67303i | 0 | 0.133975 | − | 0.232051i | −2.63896 | + | 4.57081i | 0 | 3.69615 | − | 6.40192i | 8.24504 | 0 | −10.1962 | |||||||||||||||||||||||||||||||||||
| 323.1 | −0.965926 | + | 1.67303i | 0 | 0.133975 | + | 0.232051i | 2.63896 | + | 4.57081i | 0 | 3.69615 | + | 6.40192i | −8.24504 | 0 | −10.1962 | |||||||||||||||||||||||||||||||||||
| 323.2 | −0.258819 | + | 0.448288i | 0 | 1.86603 | + | 3.23205i | −0.189469 | − | 0.328169i | 0 | −6.69615 | − | 11.5981i | −4.00240 | 0 | 0.196152 | |||||||||||||||||||||||||||||||||||
| 323.3 | 0.258819 | − | 0.448288i | 0 | 1.86603 | + | 3.23205i | 0.189469 | + | 0.328169i | 0 | −6.69615 | − | 11.5981i | 4.00240 | 0 | 0.196152 | |||||||||||||||||||||||||||||||||||
| 323.4 | 0.965926 | − | 1.67303i | 0 | 0.133975 | + | 0.232051i | −2.63896 | − | 4.57081i | 0 | 3.69615 | + | 6.40192i | 8.24504 | 0 | −10.1962 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 37.e | even | 6 | 1 | inner |
| 111.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 333.3.m.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 333.3.m.a | ✓ | 8 |
| 37.e | even | 6 | 1 | inner | 333.3.m.a | ✓ | 8 |
| 111.h | odd | 6 | 1 | inner | 333.3.m.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 333.3.m.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 333.3.m.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 333.3.m.a | ✓ | 8 | 37.e | even | 6 | 1 | inner |
| 333.3.m.a | ✓ | 8 | 111.h | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 4T_{2}^{6} + 15T_{2}^{4} + 4T_{2}^{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(333, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 4 T^{6} + \cdots + 1 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 28 T^{6} + \cdots + 16 \)
|
| $7$ |
\( (T^{4} + 6 T^{3} + \cdots + 9801)^{2} \)
|
| $11$ |
\( (T^{4} + 372 T^{2} + 19044)^{2} \)
|
| $13$ |
\( (T^{4} - 6 T^{3} + \cdots + 19881)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 15352201216 \)
|
| $19$ |
\( (T^{2} - 6 T + 12)^{4} \)
|
| $23$ |
\( (T^{4} - 1324 T^{2} + 131044)^{2} \)
|
| $29$ |
\( (T^{2} - 8)^{4} \)
|
| $31$ |
\( (T^{4} + 798 T^{2} + 62001)^{2} \)
|
| $37$ |
\( (T + 37)^{8} \)
|
| $41$ |
\( T^{8} + \cdots + 2821109907456 \)
|
| $43$ |
\( (T^{4} + 3678 T^{2} + 2852721)^{2} \)
|
| $47$ |
\( (T^{4} + 3504 T^{2} + 304704)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 182940976656 \)
|
| $59$ |
\( T^{8} + \cdots + 821700341379856 \)
|
| $61$ |
\( (T^{4} + 48 T^{3} + \cdots + 147456)^{2} \)
|
| $67$ |
\( (T^{4} + 30 T^{3} + \cdots + 5625)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 151613669376 \)
|
| $73$ |
\( (T^{2} - 162 T + 5361)^{4} \)
|
| $79$ |
\( (T^{4} + 78 T^{3} + \cdots + 31102929)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 12\!\cdots\!96 \)
|
| $89$ |
\( T^{8} + \cdots + 91506250000 \)
|
| $97$ |
\( (T^{4} + 45702 T^{2} + 509088969)^{2} \)
|
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