Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [50,16,Mod(1,50)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.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: | \(71.3467525500\) |
| 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: |
\( x^{4} - 395652x^{2} - 11553784x + 30750871635 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3\cdot 5^{7} \) |
| Twist minimal: | no (minimal twist has level 10) |
| 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
\( x^{4} - 395652x^{2} - 11553784x + 30750871635 \)
:
| \(\beta_{1}\) | \(=\) |
\( 10\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -500\nu^{3} - 53900\nu^{2} + 122544310\nu + 14995490400 ) / 33561 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -500\nu^{3} + 251200\nu^{2} + 109058890\nu - 45361222200 ) / 9153 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} - 11\beta_{2} + 442\beta _1 + 19782600 ) / 100 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -1617\beta_{3} - 27632\beta_{2} + 12016193\beta _1 + 4332669000 ) / 500 \)
|
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 |
|
128.000 | −5042.00 | 16384.0 | 0 | −645376. | 3.81841e6 | 2.09715e6 | 1.10729e7 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 128.000 | −3702.75 | 16384.0 | 0 | −473951. | −2.91723e6 | 2.09715e6 | −638585. | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 128.000 | 2932.14 | 16384.0 | 0 | 375313. | 1.80018e6 | 2.09715e6 | −5.75148e6 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 128.000 | 5604.61 | 16384.0 | 0 | 717390. | −750784. | 2.09715e6 | 1.70628e7 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 50.16.a.k | 4 | |
| 5.b | even | 2 | 1 | 50.16.a.j | 4 | ||
| 5.c | odd | 4 | 2 | 10.16.b.a | ✓ | 8 | |
| 15.e | even | 4 | 2 | 90.16.c.c | 8 | ||
| 20.e | even | 4 | 2 | 80.16.c.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.16.b.a | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 50.16.a.j | 4 | 5.b | even | 2 | 1 | ||
| 50.16.a.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 80.16.c.c | 8 | 20.e | even | 4 | 2 | ||
| 90.16.c.c | 8 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 208T_{3}^{3} - 39548976T_{3}^{2} - 15668002368T_{3} + 306800942592816 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(50))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 128)^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 306800942592816 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 15\!\cdots\!96 \)
|
| $11$ |
\( T^{4} + \cdots + 12\!\cdots\!16 \)
|
| $13$ |
\( T^{4} + \cdots - 49\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots - 26\!\cdots\!84 \)
|
| $19$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots - 10\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 12\!\cdots\!84 \)
|
| $37$ |
\( T^{4} + \cdots + 13\!\cdots\!56 \)
|
| $41$ |
\( T^{4} + \cdots - 28\!\cdots\!84 \)
|
| $43$ |
\( T^{4} + \cdots - 30\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots + 84\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots - 28\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 56\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 11\!\cdots\!84 \)
|
| $67$ |
\( T^{4} + \cdots - 46\!\cdots\!84 \)
|
| $71$ |
\( T^{4} + \cdots - 35\!\cdots\!84 \)
|
| $73$ |
\( T^{4} + \cdots - 13\!\cdots\!44 \)
|
| $79$ |
\( T^{4} + \cdots - 30\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 39\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots - 35\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 52\!\cdots\!76 \)
|
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