Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,2,Mod(362,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.362");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.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: | \(2.89856959337\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{20})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 33) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{20}^{5} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{20}^{6} + \zeta_{20}^{4} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{20}^{7} + \zeta_{20}^{3} \)
|
| \(\beta_{4}\) | \(=\) |
\( -\zeta_{20}^{7} + \zeta_{20}^{3} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{20}^{6} + \zeta_{20}^{4} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + 2\zeta_{20} \)
|
| \(\beta_{7}\) | \(=\) |
\( \zeta_{20}^{6} - \zeta_{20}^{4} + 2\zeta_{20}^{2} - 1 \)
|
| \(\zeta_{20}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} - \beta_1 ) / 2 \)
|
| \(\zeta_{20}^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} + 1 ) / 2 \)
|
| \(\zeta_{20}^{3}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\zeta_{20}^{4}\) | \(=\) |
\( ( \beta_{5} + \beta_{2} ) / 2 \)
|
| \(\zeta_{20}^{5}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{20}^{6}\) | \(=\) |
\( ( -\beta_{5} + \beta_{2} ) / 2 \)
|
| \(\zeta_{20}^{7}\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(244\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 362.1 |
|
−1.90211 | 1.61803 | − | 0.618034i | 1.61803 | 2.61803i | −3.07768 | + | 1.17557i | − | 0.726543i | 0.726543 | 2.23607 | − | 2.00000i | − | 4.97980i | ||||||||||||||||||||||||||||||||||
| 362.2 | −1.90211 | 1.61803 | + | 0.618034i | 1.61803 | − | 2.61803i | −3.07768 | − | 1.17557i | 0.726543i | 0.726543 | 2.23607 | + | 2.00000i | 4.97980i | ||||||||||||||||||||||||||||||||||||
| 362.3 | −1.17557 | −0.618034 | − | 1.61803i | −0.618034 | − | 0.381966i | 0.726543 | + | 1.90211i | 3.07768i | 3.07768 | −2.23607 | + | 2.00000i | 0.449028i | ||||||||||||||||||||||||||||||||||||
| 362.4 | −1.17557 | −0.618034 | + | 1.61803i | −0.618034 | 0.381966i | 0.726543 | − | 1.90211i | − | 3.07768i | 3.07768 | −2.23607 | − | 2.00000i | − | 0.449028i | |||||||||||||||||||||||||||||||||||
| 362.5 | 1.17557 | −0.618034 | − | 1.61803i | −0.618034 | − | 0.381966i | −0.726543 | − | 1.90211i | − | 3.07768i | −3.07768 | −2.23607 | + | 2.00000i | − | 0.449028i | ||||||||||||||||||||||||||||||||||
| 362.6 | 1.17557 | −0.618034 | + | 1.61803i | −0.618034 | 0.381966i | −0.726543 | + | 1.90211i | 3.07768i | −3.07768 | −2.23607 | − | 2.00000i | 0.449028i | |||||||||||||||||||||||||||||||||||||
| 362.7 | 1.90211 | 1.61803 | − | 0.618034i | 1.61803 | 2.61803i | 3.07768 | − | 1.17557i | 0.726543i | −0.726543 | 2.23607 | − | 2.00000i | 4.97980i | |||||||||||||||||||||||||||||||||||||
| 362.8 | 1.90211 | 1.61803 | + | 0.618034i | 1.61803 | − | 2.61803i | 3.07768 | + | 1.17557i | − | 0.726543i | −0.726543 | 2.23607 | + | 2.00000i | − | 4.97980i | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 33.d | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 5T_{2}^{2} + 5 \)
acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 5 T^{2} + 5)^{2} \)
|
| $3$ |
\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 9)^{2} \)
|
| $5$ |
\( (T^{4} + 7 T^{2} + 1)^{2} \)
|
| $7$ |
\( (T^{4} + 10 T^{2} + 5)^{2} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( (T^{4} + 10 T^{2} + 5)^{2} \)
|
| $17$ |
\( (T^{4} - 25 T^{2} + 125)^{2} \)
|
| $19$ |
\( (T^{4} + 25 T^{2} + 125)^{2} \)
|
| $23$ |
\( (T^{4} + 42 T^{2} + 121)^{2} \)
|
| $29$ |
\( (T^{4} - 20 T^{2} + 80)^{2} \)
|
| $31$ |
\( (T^{2} - 5 T - 5)^{4} \)
|
| $37$ |
\( (T - 3)^{8} \)
|
| $41$ |
\( (T^{4} - 10 T^{2} + 5)^{2} \)
|
| $43$ |
\( (T^{4} + 50 T^{2} + 125)^{2} \)
|
| $47$ |
\( (T^{4} + 123 T^{2} + 3721)^{2} \)
|
| $53$ |
\( (T^{4} + 27 T^{2} + 81)^{2} \)
|
| $59$ |
\( (T^{4} + 7 T^{2} + 1)^{2} \)
|
| $61$ |
\( (T^{4} + 25 T^{2} + 125)^{2} \)
|
| $67$ |
\( (T^{2} + T - 61)^{4} \)
|
| $71$ |
\( (T^{4} + 135 T^{2} + 3025)^{2} \)
|
| $73$ |
\( (T^{4} + 320 T^{2} + 20480)^{2} \)
|
| $79$ |
\( (T^{4} + 130 T^{2} + 1805)^{2} \)
|
| $83$ |
\( (T^{4} - 250 T^{2} + 8405)^{2} \)
|
| $89$ |
\( (T^{4} + 90 T^{2} + 25)^{2} \)
|
| $97$ |
\( (T^{2} + 11 T + 29)^{4} \)
|
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