Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [123,4,Mod(1,123)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(123, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("123.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 123 = 3 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 123.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: | \(7.25723493071\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.183064.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 10x^{2} + 6x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - x^{3} - 10x^{2} + 6x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 8\nu + 4 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} + 8\beta _1 + 1 \)
|
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 |
|
−5.30411 | 3.00000 | 20.1336 | −6.70469 | −15.9123 | −10.2415 | −64.3577 | 9.00000 | 35.5624 | ||||||||||||||||||||||||||||||
| 1.2 | −2.04496 | 3.00000 | −3.81815 | 1.85103 | −6.13487 | −7.32228 | 24.1676 | 9.00000 | −3.78527 | |||||||||||||||||||||||||||||||
| 1.3 | 1.47899 | 3.00000 | −5.81258 | −20.0945 | 4.43698 | 25.0064 | −20.4287 | 9.00000 | −29.7196 | |||||||||||||||||||||||||||||||
| 1.4 | 1.87007 | 3.00000 | −4.50283 | −8.05184 | 5.61021 | −35.4426 | −23.3812 | 9.00000 | −15.0575 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(41\) | \(1\) |
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 | 123.4.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 369.4.a.e | 4 | ||
| 4.b | odd | 2 | 1 | 1968.4.a.k | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 123.4.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 369.4.a.e | 4 | 3.b | odd | 2 | 1 | ||
| 1968.4.a.k | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} - 11T_{2}^{2} - 16T_{2} + 30 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(123))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots + 30 \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} + 33 T^{3} + \cdots - 2008 \)
|
| $7$ |
\( T^{4} + 28 T^{3} + \cdots - 66464 \)
|
| $11$ |
\( T^{4} + 96 T^{3} + \cdots + 52984 \)
|
| $13$ |
\( T^{4} + 39 T^{3} + \cdots - 493664 \)
|
| $17$ |
\( T^{4} + 205 T^{3} + \cdots - 70746142 \)
|
| $19$ |
\( T^{4} - 47 T^{3} + \cdots + 355552 \)
|
| $23$ |
\( T^{4} + 150 T^{3} + \cdots - 59612096 \)
|
| $29$ |
\( T^{4} + 262 T^{3} + \cdots + 518761856 \)
|
| $31$ |
\( T^{4} + 259 T^{3} + \cdots - 140226820 \)
|
| $37$ |
\( T^{4} + 430 T^{3} + \cdots + 810517212 \)
|
| $41$ |
\( (T + 41)^{4} \)
|
| $43$ |
\( T^{4} - 258 T^{3} + \cdots - 339839744 \)
|
| $47$ |
\( T^{4} + \cdots - 67190755624 \)
|
| $53$ |
\( T^{4} + \cdots + 9193158752 \)
|
| $59$ |
\( T^{4} + \cdots + 21063574848 \)
|
| $61$ |
\( T^{4} - 1226 T^{3} + \cdots - 14259108 \)
|
| $67$ |
\( T^{4} + \cdots - 9562633408 \)
|
| $71$ |
\( T^{4} + \cdots - 23205504822 \)
|
| $73$ |
\( T^{4} + \cdots + 7043700302 \)
|
| $79$ |
\( T^{4} + \cdots + 261898271296 \)
|
| $83$ |
\( T^{4} + \cdots - 1520676272 \)
|
| $89$ |
\( T^{4} + \cdots - 41590182024 \)
|
| $97$ |
\( T^{4} + \cdots - 15178118192 \)
|
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