Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [44,4,Mod(1,44)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("44.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 44 = 2^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 44.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: | \(2.59608404025\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{97}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{97})\). 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 |
|
0 | −0.424429 | 0 | 10.4244 | 0 | 34.5466 | 0 | −26.8199 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 9.42443 | 0 | 0.575571 | 0 | −24.5466 | 0 | 61.8199 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 44.4.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 396.4.a.i | 2 | ||
| 4.b | odd | 2 | 1 | 176.4.a.g | 2 | ||
| 5.b | even | 2 | 1 | 1100.4.a.e | 2 | ||
| 5.c | odd | 4 | 2 | 1100.4.b.e | 4 | ||
| 7.b | odd | 2 | 1 | 2156.4.a.d | 2 | ||
| 8.b | even | 2 | 1 | 704.4.a.m | 2 | ||
| 8.d | odd | 2 | 1 | 704.4.a.r | 2 | ||
| 11.b | odd | 2 | 1 | 484.4.a.e | 2 | ||
| 12.b | even | 2 | 1 | 1584.4.a.z | 2 | ||
| 44.c | even | 2 | 1 | 1936.4.a.o | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 44.4.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 176.4.a.g | 2 | 4.b | odd | 2 | 1 | ||
| 396.4.a.i | 2 | 3.b | odd | 2 | 1 | ||
| 484.4.a.e | 2 | 11.b | odd | 2 | 1 | ||
| 704.4.a.m | 2 | 8.b | even | 2 | 1 | ||
| 704.4.a.r | 2 | 8.d | odd | 2 | 1 | ||
| 1100.4.a.e | 2 | 5.b | even | 2 | 1 | ||
| 1100.4.b.e | 4 | 5.c | odd | 4 | 2 | ||
| 1584.4.a.z | 2 | 12.b | even | 2 | 1 | ||
| 1936.4.a.o | 2 | 44.c | even | 2 | 1 | ||
| 2156.4.a.d | 2 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 9T_{3} - 4 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(44))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 9T - 4 \)
|
| $5$ |
\( T^{2} - 11T + 6 \)
|
| $7$ |
\( T^{2} - 10T - 848 \)
|
| $11$ |
\( (T - 11)^{2} \)
|
| $13$ |
\( T^{2} + 22T + 24 \)
|
| $17$ |
\( T^{2} + 160T + 6012 \)
|
| $19$ |
\( T^{2} - 12T - 9664 \)
|
| $23$ |
\( T^{2} + 119T - 9288 \)
|
| $29$ |
\( T^{2} + 138T - 38016 \)
|
| $31$ |
\( T^{2} - 401T + 40176 \)
|
| $37$ |
\( T^{2} - 341T + 16242 \)
|
| $41$ |
\( T^{2} + 318T + 17424 \)
|
| $43$ |
\( T^{2} - 450T + 34232 \)
|
| $47$ |
\( T^{2} + 312T - 101376 \)
|
| $53$ |
\( T^{2} + 400T + 30300 \)
|
| $59$ |
\( T^{2} + 21T - 62964 \)
|
| $61$ |
\( T^{2} + 462T + 18344 \)
|
| $67$ |
\( T^{2} - 327T - 192124 \)
|
| $71$ |
\( T^{2} - 603T - 24552 \)
|
| $73$ |
\( T^{2} - 206T - 70968 \)
|
| $79$ |
\( T^{2} - 2026 T + 983392 \)
|
| $83$ |
\( T^{2} + 1046 T + 110472 \)
|
| $89$ |
\( T^{2} - 721T - 647034 \)
|
| $97$ |
\( T^{2} - 1467 T - 603886 \)
|
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