Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,2,Mod(521,1764)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1764.521");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1764.t (of order \(6\), degree \(2\), not 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: | \(14.0856109166\) |
| 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: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 252) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{4} \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{24}^{6} + 2\zeta_{24}^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( -3\zeta_{24}^{5} - 3\zeta_{24}^{3} + 3\zeta_{24} \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\zeta_{24}^{7} - 4\zeta_{24}^{5} + 4\zeta_{24}^{3} + 2\zeta_{24} \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\zeta_{24}^{7} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 4\zeta_{24} \)
|
| \(\beta_{7}\) | \(=\) |
\( 3\zeta_{24}^{7} - 3\zeta_{24} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( -2\beta_{7} + 2\beta_{6} - \beta_{5} ) / 12 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{2} ) / 6 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( ( \beta_{6} + \beta_{5} - 2\beta_{3} ) / 12 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( -2\beta_{7} + \beta_{6} - 2\beta_{5} - 2\beta_{3} ) / 12 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( ( -\beta_{4} + 2\beta_{2} ) / 6 \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{6} - \beta_{5} ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(883\) | \(1081\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 521.1 |
|
0 | 0 | 0 | −1.73205 | − | 3.00000i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 521.2 | 0 | 0 | 0 | −1.73205 | − | 3.00000i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 521.3 | 0 | 0 | 0 | 1.73205 | + | 3.00000i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 521.4 | 0 | 0 | 0 | 1.73205 | + | 3.00000i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1097.1 | 0 | 0 | 0 | −1.73205 | + | 3.00000i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1097.2 | 0 | 0 | 0 | −1.73205 | + | 3.00000i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1097.3 | 0 | 0 | 0 | 1.73205 | − | 3.00000i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1097.4 | 0 | 0 | 0 | 1.73205 | − | 3.00000i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
| 21.g | even | 6 | 1 | inner |
| 21.h | odd | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 12T_{5}^{2} + 144 \)
acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 12 T^{2} + 144)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} - 18 T^{2} + 324)^{2} \)
|
| $13$ |
\( (T^{2} + 24)^{4} \)
|
| $17$ |
\( (T^{4} + 12 T^{2} + 144)^{2} \)
|
| $19$ |
\( (T^{4} - 24 T^{2} + 576)^{2} \)
|
| $23$ |
\( (T^{4} - 18 T^{2} + 324)^{2} \)
|
| $29$ |
\( (T^{2} + 18)^{4} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( (T^{2} - 8 T + 64)^{4} \)
|
| $41$ |
\( (T^{2} - 12)^{4} \)
|
| $43$ |
\( (T + 2)^{8} \)
|
| $47$ |
\( (T^{4} + 48 T^{2} + 2304)^{2} \)
|
| $53$ |
\( (T^{4} - 162 T^{2} + 26244)^{2} \)
|
| $59$ |
\( (T^{4} + 192 T^{2} + 36864)^{2} \)
|
| $61$ |
\( (T^{4} - 96 T^{2} + 9216)^{2} \)
|
| $67$ |
\( (T^{2} + 8 T + 64)^{4} \)
|
| $71$ |
\( (T^{2} + 18)^{4} \)
|
| $73$ |
\( (T^{4} - 24 T^{2} + 576)^{2} \)
|
| $79$ |
\( (T^{2} - 4 T + 16)^{4} \)
|
| $83$ |
\( (T^{2} - 48)^{4} \)
|
| $89$ |
\( (T^{4} + 108 T^{2} + 11664)^{2} \)
|
| $97$ |
\( (T^{2} + 24)^{4} \)
|
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