Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [416,4,Mod(1,416)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("416.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 416 = 2^{5} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 416.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: | \(24.5447945624\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1847677.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 18x^{2} + 19x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{4} - 2x^{3} - 18x^{2} + 19x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{3} - 3\nu^{2} - 37\nu + 19 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} + \beta _1 + 42 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} + 10\beta _1 + 31 ) / 2 \)
|
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 | −7.85014 | 0 | −13.2082 | 0 | 7.03277 | 0 | 34.6247 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.83719 | 0 | 6.20824 | 0 | 19.4818 | 0 | −23.6247 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.83719 | 0 | 6.20824 | 0 | −19.4818 | 0 | −23.6247 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 7.85014 | 0 | −13.2082 | 0 | −7.03277 | 0 | 34.6247 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 416.4.a.g | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 416.4.a.g | ✓ | 4 |
| 8.b | even | 2 | 1 | 832.4.a.be | 4 | ||
| 8.d | odd | 2 | 1 | 832.4.a.be | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 416.4.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 416.4.a.g | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 832.4.a.be | 4 | 8.b | even | 2 | 1 | ||
| 832.4.a.be | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 65T_{3}^{2} + 208 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(416))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 65T^{2} + 208 \)
|
| $5$ |
\( (T^{2} + 7 T - 82)^{2} \)
|
| $7$ |
\( T^{4} - 429 T^{2} + 18772 \)
|
| $11$ |
\( T^{4} - 3224 T^{2} + 2381392 \)
|
| $13$ |
\( (T + 13)^{4} \)
|
| $17$ |
\( (T^{2} - 3 T - 2354)^{2} \)
|
| $19$ |
\( T^{4} - 1144 T^{2} + 110032 \)
|
| $23$ |
\( T^{4} - 27456 T^{2} + 76890112 \)
|
| $29$ |
\( (T^{2} + 216 T + 10156)^{2} \)
|
| $31$ |
\( T^{4} - 42692 T^{2} + 1139008 \)
|
| $37$ |
\( (T^{2} + 395 T + 23078)^{2} \)
|
| $41$ |
\( (T^{2} + 194 T + 9032)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 3537534208 \)
|
| $47$ |
\( T^{4} + \cdots + 2863704532 \)
|
| $53$ |
\( (T^{2} + 466 T - 81808)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 40832089168 \)
|
| $61$ |
\( (T^{2} + 622 T - 12232)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 359800746448 \)
|
| $71$ |
\( T^{4} + \cdots + 799203845908 \)
|
| $73$ |
\( (T^{2} - 268 T - 1001452)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 224156937472 \)
|
| $83$ |
\( T^{4} + \cdots + 426005951488 \)
|
| $89$ |
\( (T^{2} - 272 T + 16988)^{2} \)
|
| $97$ |
\( (T^{2} + 564 T - 2098028)^{2} \)
|
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