Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,22,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, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| 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: | \(13.9738672144\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 1929606x^{2} - 743130000x + 239341586400 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3\cdot 5^{2}\cdot 7 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 1929606x^{2} - 743130000x + 239341586400 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{3} - 3089\nu^{2} - 6622602\nu + 187959204 ) / 324 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 877\nu^{2} - 1174314\nu + 287631324 ) / 324 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5\beta_{3} + \beta_{2} + 1159\beta _1 + 3858634 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3089\beta_{3} + 877\beta_{2} + 3365071\beta _1 + 2233496722 ) / 4 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2315.74 | 45067.6 | 3.26550e6 | 9.76562e6 | −1.04365e8 | −6.93632e8 | −2.70560e9 | −8.42927e9 | −2.26147e10 | ||||||||||||||||||||||||||||||
| 1.2 | 307.836 | 62164.3 | −2.00239e6 | 9.76562e6 | 1.91364e7 | 8.89757e8 | −1.26199e9 | −6.59596e9 | 3.00621e9 | |||||||||||||||||||||||||||||||
| 1.3 | 2416.74 | 157402. | 3.74348e6 | 9.76562e6 | 3.80399e8 | −6.35419e8 | 3.97874e9 | 1.43149e10 | 2.36010e10 | |||||||||||||||||||||||||||||||
| 1.4 | 2501.16 | −181393. | 4.15867e6 | 9.76562e6 | −4.53695e8 | 9.51907e8 | 5.15621e9 | 2.24432e10 | 2.44254e10 | |||||||||||||||||||||||||||||||
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 | 5.22.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 45.22.a.f | 4 | ||
| 4.b | odd | 2 | 1 | 80.22.a.g | 4 | ||
| 5.b | even | 2 | 1 | 25.22.a.c | 4 | ||
| 5.c | odd | 4 | 2 | 25.22.b.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.22.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 25.22.a.c | 4 | 5.b | even | 2 | 1 | ||
| 25.22.b.c | 8 | 5.c | odd | 4 | 2 | ||
| 45.22.a.f | 4 | 3.b | odd | 2 | 1 | ||
| 80.22.a.g | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2910T_{2}^{3} - 4542888T_{2}^{2} + 15642931840T_{2} - 4309053579264 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(5))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots - 4309053579264 \)
|
| $3$ |
\( T^{4} + \cdots - 79\!\cdots\!04 \)
|
| $5$ |
\( (T - 9765625)^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 37\!\cdots\!96 \)
|
| $11$ |
\( T^{4} + \cdots + 17\!\cdots\!36 \)
|
| $13$ |
\( T^{4} + \cdots - 19\!\cdots\!04 \)
|
| $17$ |
\( T^{4} + \cdots + 20\!\cdots\!16 \)
|
| $19$ |
\( T^{4} + \cdots - 27\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots - 10\!\cdots\!04 \)
|
| $29$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 37\!\cdots\!24 \)
|
| $37$ |
\( T^{4} + \cdots + 70\!\cdots\!56 \)
|
| $41$ |
\( T^{4} + \cdots + 44\!\cdots\!96 \)
|
| $43$ |
\( T^{4} + \cdots - 72\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots - 58\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 14\!\cdots\!96 \)
|
| $59$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 37\!\cdots\!36 \)
|
| $67$ |
\( T^{4} + \cdots + 21\!\cdots\!16 \)
|
| $71$ |
\( T^{4} + \cdots - 28\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots - 21\!\cdots\!04 \)
|
| $79$ |
\( T^{4} + \cdots + 73\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 23\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots - 46\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 31\!\cdots\!76 \)
|
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