Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,20,Mod(1,5)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 20, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.1");
S:= CuspForms(chi, 20);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.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: | \(11.4408348278\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 22777x - 646704 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3\cdot 5 \) |
| 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 a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - 22777x - 646704 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{2} + 104\nu - 30381 ) / 35 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -6\nu^{2} + 738\nu + 91108 ) / 35 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 1 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -52\beta_{2} + 369\beta _1 + 455663 ) / 30 \)
|
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 |
|
−1240.95 | −64590.2 | 1.01566e6 | 1.95312e6 | 8.01531e7 | −3.47039e7 | −6.09772e8 | 3.00964e9 | −2.42373e9 | |||||||||||||||||||||||||||
| 1.2 | −575.417 | 14082.6 | −193183. | 1.95312e6 | −8.10336e6 | 9.45293e7 | 4.12845e8 | −9.63942e8 | −1.12386e9 | ||||||||||||||||||||||||||||
| 1.3 | 810.365 | −22944.3 | 132403. | 1.95312e6 | −1.85933e7 | −1.14736e8 | −3.17570e8 | −6.35819e8 | 1.58274e9 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
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 | 5.20.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 45.20.a.d | 3 | ||
| 4.b | odd | 2 | 1 | 80.20.a.f | 3 | ||
| 5.b | even | 2 | 1 | 25.20.a.b | 3 | ||
| 5.c | odd | 4 | 2 | 25.20.b.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.20.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 25.20.a.b | 3 | 5.b | even | 2 | 1 | ||
| 25.20.b.b | 6 | 5.c | odd | 4 | 2 | ||
| 45.20.a.d | 3 | 3.b | odd | 2 | 1 | ||
| 80.20.a.f | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 1006T_{2}^{2} - 757856T_{2} - 578651136 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(5))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 1006 T^{2} + \cdots - 578651136 \)
|
| $3$ |
\( T^{3} + \cdots - 20870112837888 \)
|
| $5$ |
\( (T - 1953125)^{3} \)
|
| $7$ |
\( T^{3} + \cdots - 37\!\cdots\!16 \)
|
| $11$ |
\( T^{3} + \cdots - 30\!\cdots\!88 \)
|
| $13$ |
\( T^{3} + \cdots + 87\!\cdots\!12 \)
|
| $17$ |
\( T^{3} + \cdots - 64\!\cdots\!76 \)
|
| $19$ |
\( T^{3} + \cdots + 34\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 58\!\cdots\!88 \)
|
| $29$ |
\( T^{3} + \cdots - 72\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 25\!\cdots\!48 \)
|
| $37$ |
\( T^{3} + \cdots - 10\!\cdots\!96 \)
|
| $41$ |
\( T^{3} + \cdots + 93\!\cdots\!72 \)
|
| $43$ |
\( T^{3} + \cdots - 66\!\cdots\!88 \)
|
| $47$ |
\( T^{3} + \cdots - 16\!\cdots\!56 \)
|
| $53$ |
\( T^{3} + \cdots + 57\!\cdots\!12 \)
|
| $59$ |
\( T^{3} + \cdots - 87\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 21\!\cdots\!12 \)
|
| $67$ |
\( T^{3} + \cdots + 43\!\cdots\!24 \)
|
| $71$ |
\( T^{3} + \cdots + 76\!\cdots\!32 \)
|
| $73$ |
\( T^{3} + \cdots + 55\!\cdots\!12 \)
|
| $79$ |
\( T^{3} + \cdots + 95\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 65\!\cdots\!88 \)
|
| $89$ |
\( T^{3} + \cdots - 14\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 44\!\cdots\!56 \)
|
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