Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,4,Mod(3,121)]
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([8]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.c (of order \(5\), 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: | \(7.13923111069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.324000000.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 27 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{7} ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 9\beta_{5} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 27\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 27\beta_{7} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−3.61153 | + | 2.62393i | −2.30653 | − | 7.09878i | 3.68602 | − | 11.3444i | 6.84760 | + | 4.97507i | 26.9569 | + | 19.5853i | −6.25720 | + | 19.2577i | 5.41889 | + | 16.6776i | −23.2292 | + | 16.8770i | −37.7846 | ||||||||||||||||||||||||||
| 3.2 | 1.99350 | − | 1.44836i | −0.165602 | − | 0.509670i | −0.595848 | + | 1.83383i | 1.24257 | + | 0.902778i | −1.06831 | − | 0.776175i | 8.72933 | − | 26.8661i | 7.55982 | + | 23.2667i | 21.6111 | − | 15.7014i | 3.78461 | |||||||||||||||||||||||||||
| 9.1 | −0.761449 | + | 2.34350i | 0.433551 | − | 0.314993i | 1.55995 | + | 1.13337i | −0.474619 | − | 1.46073i | 0.408059 | + | 1.25588i | −22.8537 | − | 16.6042i | −19.7919 | + | 14.3796i | −8.25471 | + | 25.4054i | 3.78461 | |||||||||||||||||||||||||||
| 9.2 | 1.37948 | − | 4.24561i | 6.03859 | − | 4.38729i | −9.65012 | − | 7.01122i | −2.61555 | − | 8.04984i | −10.2966 | − | 31.6897i | 16.3816 | + | 11.9019i | −14.1868 | + | 10.3073i | 8.87275 | − | 27.3075i | −37.7846 | |||||||||||||||||||||||||||
| 27.1 | −0.761449 | − | 2.34350i | 0.433551 | + | 0.314993i | 1.55995 | − | 1.13337i | −0.474619 | + | 1.46073i | 0.408059 | − | 1.25588i | −22.8537 | + | 16.6042i | −19.7919 | − | 14.3796i | −8.25471 | − | 25.4054i | 3.78461 | |||||||||||||||||||||||||||
| 27.2 | 1.37948 | + | 4.24561i | 6.03859 | + | 4.38729i | −9.65012 | + | 7.01122i | −2.61555 | + | 8.04984i | −10.2966 | + | 31.6897i | 16.3816 | − | 11.9019i | −14.1868 | − | 10.3073i | 8.87275 | + | 27.3075i | −37.7846 | |||||||||||||||||||||||||||
| 81.1 | −3.61153 | − | 2.62393i | −2.30653 | + | 7.09878i | 3.68602 | + | 11.3444i | 6.84760 | − | 4.97507i | 26.9569 | − | 19.5853i | −6.25720 | − | 19.2577i | 5.41889 | − | 16.6776i | −23.2292 | − | 16.8770i | −37.7846 | |||||||||||||||||||||||||||
| 81.2 | 1.99350 | + | 1.44836i | −0.165602 | + | 0.509670i | −0.595848 | − | 1.83383i | 1.24257 | − | 0.902778i | −1.06831 | + | 0.776175i | 8.72933 | + | 26.8661i | 7.55982 | − | 23.2667i | 21.6111 | + | 15.7014i | 3.78461 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 3 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 121.4.c.d | 8 | |
| 11.b | odd | 2 | 1 | 121.4.c.g | 8 | ||
| 11.c | even | 5 | 1 | 121.4.a.e | yes | 2 | |
| 11.c | even | 5 | 3 | inner | 121.4.c.d | 8 | |
| 11.d | odd | 10 | 1 | 121.4.a.b | ✓ | 2 | |
| 11.d | odd | 10 | 3 | 121.4.c.g | 8 | ||
| 33.f | even | 10 | 1 | 1089.4.a.x | 2 | ||
| 33.h | odd | 10 | 1 | 1089.4.a.k | 2 | ||
| 44.g | even | 10 | 1 | 1936.4.a.z | 2 | ||
| 44.h | odd | 10 | 1 | 1936.4.a.y | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 121.4.a.b | ✓ | 2 | 11.d | odd | 10 | 1 | |
| 121.4.a.e | yes | 2 | 11.c | even | 5 | 1 | |
| 121.4.c.d | 8 | 1.a | even | 1 | 1 | trivial | |
| 121.4.c.d | 8 | 11.c | even | 5 | 3 | inner | |
| 121.4.c.g | 8 | 11.b | odd | 2 | 1 | ||
| 121.4.c.g | 8 | 11.d | odd | 10 | 3 | ||
| 1089.4.a.k | 2 | 33.h | odd | 10 | 1 | ||
| 1089.4.a.x | 2 | 33.f | even | 10 | 1 | ||
| 1936.4.a.y | 2 | 44.h | odd | 10 | 1 | ||
| 1936.4.a.z | 2 | 44.g | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} + 15T_{2}^{6} + 52T_{2}^{5} + 269T_{2}^{4} - 572T_{2}^{3} + 1815T_{2}^{2} - 2662T_{2} + 14641 \)
acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 14641 \)
|
| $3$ |
\( T^{8} - 8 T^{7} + \cdots + 256 \)
|
| $5$ |
\( T^{8} - 10 T^{7} + \cdots + 28561 \)
|
| $7$ |
\( T^{8} + \cdots + 107049369856 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 315040099898161 \)
|
| $17$ |
\( T^{8} + \cdots + 217206315402241 \)
|
| $19$ |
\( T^{8} + \cdots + 24591257856 \)
|
| $23$ |
\( (T^{2} + 128 T - 8972)^{4} \)
|
| $29$ |
\( T^{8} + \cdots + 68\!\cdots\!81 \)
|
| $31$ |
\( T^{8} + \cdots + 10\!\cdots\!96 \)
|
| $37$ |
\( T^{8} + \cdots + 25\!\cdots\!41 \)
|
| $41$ |
\( T^{8} + \cdots + 13\!\cdots\!41 \)
|
| $43$ |
\( (T^{2} - 264 T + 11616)^{4} \)
|
| $47$ |
\( T^{8} + \cdots + 29\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 41\!\cdots\!41 \)
|
| $59$ |
\( T^{8} + \cdots + 26\!\cdots\!16 \)
|
| $61$ |
\( T^{8} + \cdots + 39\!\cdots\!76 \)
|
| $67$ |
\( (T^{2} + 656 T + 106132)^{4} \)
|
| $71$ |
\( T^{8} + \cdots + 90\!\cdots\!36 \)
|
| $73$ |
\( T^{8} + \cdots + 59\!\cdots\!96 \)
|
| $79$ |
\( T^{8} + \cdots + 97\!\cdots\!96 \)
|
| $83$ |
\( T^{8} + \cdots + 15\!\cdots\!36 \)
|
| $89$ |
\( (T^{2} + 1074 T - 83343)^{4} \)
|
| $97$ |
\( T^{8} + \cdots + 15\!\cdots\!41 \)
|
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