Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [33,8,Mod(1,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.a (trivial) |
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: | yes |
| Analytic conductor: | \(10.3087058410\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{177}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 44 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−16.1521 | 27.0000 | 132.889 | −269.779 | −436.106 | 941.418 | −78.9720 | 729.000 | 4357.48 | ||||||||||||||||||||||||
| 1.2 | −2.84793 | 27.0000 | −119.889 | 235.779 | −76.8942 | −1107.42 | 705.972 | 729.000 | −671.481 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 33.8.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 99.8.a.d | 2 | ||
| 4.b | odd | 2 | 1 | 528.8.a.f | 2 | ||
| 11.b | odd | 2 | 1 | 363.8.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.8.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 99.8.a.d | 2 | 3.b | odd | 2 | 1 | ||
| 363.8.a.d | 2 | 11.b | odd | 2 | 1 | ||
| 528.8.a.f | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 19T_{2} + 46 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 19T + 46 \)
|
| $3$ |
\( (T - 27)^{2} \)
|
| $5$ |
\( T^{2} + 34T - 63608 \)
|
| $7$ |
\( T^{2} + 166 T - 1042544 \)
|
| $11$ |
\( (T + 1331)^{2} \)
|
| $13$ |
\( T^{2} + 12670 T + 6020608 \)
|
| $17$ |
\( T^{2} + 62344 T + 927796876 \)
|
| $19$ |
\( T^{2} + 37980 T + 328498848 \)
|
| $23$ |
\( T^{2} + 90686 T + 282938824 \)
|
| $29$ |
\( T^{2} + \cdots - 38836484052 \)
|
| $31$ |
\( T^{2} + \cdots + 1286906368 \)
|
| $37$ |
\( T^{2} + \cdots - 6524218716 \)
|
| $41$ |
\( T^{2} + \cdots - 322827909452 \)
|
| $43$ |
\( T^{2} + \cdots - 207765596544 \)
|
| $47$ |
\( T^{2} + \cdots - 176077767704 \)
|
| $53$ |
\( T^{2} + \cdots + 397572445128 \)
|
| $59$ |
\( T^{2} + \cdots - 24663435104 \)
|
| $61$ |
\( T^{2} + \cdots - 604169699352 \)
|
| $67$ |
\( T^{2} + \cdots - 8829893772944 \)
|
| $71$ |
\( T^{2} + \cdots - 24503996328 \)
|
| $73$ |
\( T^{2} + \cdots - 5666837293628 \)
|
| $79$ |
\( T^{2} + \cdots - 21513626700752 \)
|
| $83$ |
\( T^{2} + \cdots + 8122054073616 \)
|
| $89$ |
\( T^{2} + \cdots - 15754651515708 \)
|
| $97$ |
\( T^{2} + \cdots + 69751828200124 \)
|
| show more | |
| show less |