Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,3,Mod(40,121)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([7]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.40");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.d (of order \(10\), degree \(4\), 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: | \(3.29701119876\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | 8.0.64000000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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
\( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( 8\beta_{7} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 40.1 |
|
−0.831254 | + | 1.14412i | −0.309017 | + | 0.951057i | 0.618034 | + | 1.90211i | 5.66312 | − | 4.11450i | −0.831254 | − | 1.14412i | 6.72499 | − | 2.18508i | −8.06998 | − | 2.62210i | 6.47214 | + | 4.70228i | 9.89949i | ||||||||||||||||||||||||||
| 40.2 | 0.831254 | − | 1.14412i | −0.309017 | + | 0.951057i | 0.618034 | + | 1.90211i | 5.66312 | − | 4.11450i | 0.831254 | + | 1.14412i | −6.72499 | + | 2.18508i | 8.06998 | + | 2.62210i | 6.47214 | + | 4.70228i | − | 9.89949i | ||||||||||||||||||||||||||
| 94.1 | −1.34500 | + | 0.437016i | 0.809017 | + | 0.587785i | −1.61803 | + | 1.17557i | −2.16312 | + | 6.65740i | −1.34500 | − | 0.437016i | −4.15627 | − | 5.72061i | 4.98752 | − | 6.86474i | −2.47214 | − | 7.60845i | − | 9.89949i | ||||||||||||||||||||||||||
| 94.2 | 1.34500 | − | 0.437016i | 0.809017 | + | 0.587785i | −1.61803 | + | 1.17557i | −2.16312 | + | 6.65740i | 1.34500 | + | 0.437016i | 4.15627 | + | 5.72061i | −4.98752 | + | 6.86474i | −2.47214 | − | 7.60845i | 9.89949i | |||||||||||||||||||||||||||
| 112.1 | −1.34500 | − | 0.437016i | 0.809017 | − | 0.587785i | −1.61803 | − | 1.17557i | −2.16312 | − | 6.65740i | −1.34500 | + | 0.437016i | −4.15627 | + | 5.72061i | 4.98752 | + | 6.86474i | −2.47214 | + | 7.60845i | 9.89949i | |||||||||||||||||||||||||||
| 112.2 | 1.34500 | + | 0.437016i | 0.809017 | − | 0.587785i | −1.61803 | − | 1.17557i | −2.16312 | − | 6.65740i | 1.34500 | − | 0.437016i | 4.15627 | − | 5.72061i | −4.98752 | − | 6.86474i | −2.47214 | + | 7.60845i | − | 9.89949i | ||||||||||||||||||||||||||
| 118.1 | −0.831254 | − | 1.14412i | −0.309017 | − | 0.951057i | 0.618034 | − | 1.90211i | 5.66312 | + | 4.11450i | −0.831254 | + | 1.14412i | 6.72499 | + | 2.18508i | −8.06998 | + | 2.62210i | 6.47214 | − | 4.70228i | − | 9.89949i | ||||||||||||||||||||||||||
| 118.2 | 0.831254 | + | 1.14412i | −0.309017 | − | 0.951057i | 0.618034 | − | 1.90211i | 5.66312 | + | 4.11450i | 0.831254 | − | 1.14412i | −6.72499 | − | 2.18508i | 8.06998 | − | 2.62210i | 6.47214 | − | 4.70228i | 9.89949i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
| 11.c | even | 5 | 3 | inner |
| 11.d | odd | 10 | 3 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 121.3.d.e | 8 | |
| 11.b | odd | 2 | 1 | inner | 121.3.d.e | 8 | |
| 11.c | even | 5 | 1 | 121.3.b.a | ✓ | 2 | |
| 11.c | even | 5 | 3 | inner | 121.3.d.e | 8 | |
| 11.d | odd | 10 | 1 | 121.3.b.a | ✓ | 2 | |
| 11.d | odd | 10 | 3 | inner | 121.3.d.e | 8 | |
| 33.f | even | 10 | 1 | 1089.3.c.a | 2 | ||
| 33.h | odd | 10 | 1 | 1089.3.c.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 121.3.b.a | ✓ | 2 | 11.c | even | 5 | 1 | |
| 121.3.b.a | ✓ | 2 | 11.d | odd | 10 | 1 | |
| 121.3.d.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 121.3.d.e | 8 | 11.b | odd | 2 | 1 | inner | |
| 121.3.d.e | 8 | 11.c | even | 5 | 3 | inner | |
| 121.3.d.e | 8 | 11.d | odd | 10 | 3 | inner | |
| 1089.3.c.a | 2 | 33.f | even | 10 | 1 | ||
| 1089.3.c.a | 2 | 33.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 2T_{2}^{6} + 4T_{2}^{4} - 8T_{2}^{2} + 16 \)
acting on \(S_{3}^{\mathrm{new}}(121, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 2 T^{6} + \cdots + 16 \)
|
| $3$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $5$ |
\( (T^{4} - 7 T^{3} + \cdots + 2401)^{2} \)
|
| $7$ |
\( T^{8} - 50 T^{6} + \cdots + 6250000 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 6879707136 \)
|
| $17$ |
\( T^{8} - 18 T^{6} + \cdots + 104976 \)
|
| $19$ |
\( T^{8} + \cdots + 6879707136 \)
|
| $23$ |
\( (T + 9)^{8} \)
|
| $29$ |
\( T^{8} + \cdots + 68719476736 \)
|
| $31$ |
\( (T^{4} + 49 T^{3} + \cdots + 5764801)^{2} \)
|
| $37$ |
\( (T^{4} + 17 T^{3} + \cdots + 83521)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 6879707136 \)
|
| $43$ |
\( (T^{2} + 2178)^{4} \)
|
| $47$ |
\( (T^{4} + 32 T^{3} + \cdots + 1048576)^{2} \)
|
| $53$ |
\( (T^{4} + 16 T^{3} + \cdots + 65536)^{2} \)
|
| $59$ |
\( (T^{4} - 71 T^{3} + \cdots + 25411681)^{2} \)
|
| $61$ |
\( T^{8} - 128 T^{6} + \cdots + 268435456 \)
|
| $67$ |
\( (T + 31)^{8} \)
|
| $71$ |
\( (T^{4} - 73 T^{3} + \cdots + 28398241)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 6044831973376 \)
|
| $79$ |
\( T^{8} + \cdots + 36\!\cdots\!96 \)
|
| $83$ |
\( T^{8} + \cdots + 2441406250000 \)
|
| $89$ |
\( (T + 9)^{8} \)
|
| $97$ |
\( (T^{4} - 17 T^{3} + \cdots + 83521)^{2} \)
|
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