Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,14,Mod(1,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.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: | \(48.2539180284\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{499}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 499 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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 \(\beta = 4\sqrt{499}\). 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 |
|
−49.3532 | 0 | −5756.26 | 15625.0 | 0 | −161968. | 688392. | 0 | −771144. | ||||||||||||||||||||||||
| 1.2 | 129.353 | 0 | 8540.26 | 15625.0 | 0 | −454332. | 45048.4 | 0 | 2.02114e6 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 45.14.a.d | 2 | |
| 3.b | odd | 2 | 1 | 5.14.a.a | ✓ | 2 | |
| 12.b | even | 2 | 1 | 80.14.a.d | 2 | ||
| 15.d | odd | 2 | 1 | 25.14.a.a | 2 | ||
| 15.e | even | 4 | 2 | 25.14.b.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.14.a.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 25.14.a.a | 2 | 15.d | odd | 2 | 1 | ||
| 25.14.b.a | 4 | 15.e | even | 4 | 2 | ||
| 45.14.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 80.14.a.d | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 80T_{2} - 6384 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 80T - 6384 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T - 15625)^{2} \)
|
| $7$ |
\( T^{2} + \cdots + 73587278436 \)
|
| $11$ |
\( T^{2} + \cdots - 48306453989376 \)
|
| $13$ |
\( T^{2} + \cdots - 131305798237756 \)
|
| $17$ |
\( T^{2} + \cdots - 83\!\cdots\!24 \)
|
| $19$ |
\( T^{2} + \cdots + 33\!\cdots\!00 \)
|
| $23$ |
\( T^{2} + \cdots - 62\!\cdots\!16 \)
|
| $29$ |
\( T^{2} + \cdots + 12\!\cdots\!00 \)
|
| $31$ |
\( T^{2} + \cdots - 22\!\cdots\!36 \)
|
| $37$ |
\( T^{2} + \cdots - 19\!\cdots\!44 \)
|
| $41$ |
\( T^{2} + \cdots + 70\!\cdots\!84 \)
|
| $43$ |
\( T^{2} + \cdots + 67\!\cdots\!64 \)
|
| $47$ |
\( T^{2} + \cdots + 13\!\cdots\!96 \)
|
| $53$ |
\( T^{2} + \cdots + 12\!\cdots\!04 \)
|
| $59$ |
\( T^{2} + \cdots - 48\!\cdots\!00 \)
|
| $61$ |
\( T^{2} + \cdots + 55\!\cdots\!24 \)
|
| $67$ |
\( T^{2} + \cdots + 14\!\cdots\!76 \)
|
| $71$ |
\( T^{2} + \cdots - 80\!\cdots\!56 \)
|
| $73$ |
\( T^{2} + \cdots - 36\!\cdots\!16 \)
|
| $79$ |
\( T^{2} + \cdots - 48\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots + 24\!\cdots\!24 \)
|
| $89$ |
\( T^{2} + \cdots - 10\!\cdots\!00 \)
|
| $97$ |
\( T^{2} + \cdots + 81\!\cdots\!96 \)
|
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