Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [435,4,Mod(1,435)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(435, 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("435.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 435 = 3 \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 435.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: | \(25.6658308525\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 23x^{3} + 38x^{2} + 90x - 116 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 23x^{3} + 38x^{2} + 90x - 116 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 15\nu + 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 23\nu^{2} - 4\nu + 76 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 15\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 23\beta_{2} + 4\beta _1 + 154 \)
|
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 |
|
−4.12552 | −3.00000 | 9.01995 | −5.00000 | 12.3766 | 7.89008 | −4.20783 | 9.00000 | 20.6276 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.12043 | −3.00000 | −3.50376 | −5.00000 | 6.36130 | −1.40136 | 24.3930 | 9.00000 | 10.6022 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.13455 | −3.00000 | −6.71280 | −5.00000 | −3.40364 | −29.0256 | −16.6924 | 9.00000 | −5.67274 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.57832 | −3.00000 | −1.35225 | −5.00000 | −7.73497 | 11.3529 | −24.1131 | 9.00000 | −12.8916 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.53309 | −3.00000 | 12.5489 | −5.00000 | −13.5993 | −17.8159 | 20.6204 | 9.00000 | −22.6654 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 435.4.a.f | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1305.4.a.g | 5 | ||
| 5.b | even | 2 | 1 | 2175.4.a.j | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 435.4.a.f | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1305.4.a.g | 5 | 3.b | odd | 2 | 1 | ||
| 2175.4.a.j | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 2T_{2}^{4} - 23T_{2}^{3} + 38T_{2}^{2} + 90T_{2} - 116 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(435))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 116 \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( (T + 5)^{5} \)
|
| $7$ |
\( T^{5} + 29 T^{4} + \cdots + 64912 \)
|
| $11$ |
\( T^{5} - 15 T^{4} + \cdots + 29068332 \)
|
| $13$ |
\( T^{5} + 31 T^{4} + \cdots + 8403376 \)
|
| $17$ |
\( T^{5} - 119 T^{4} + \cdots + 130071808 \)
|
| $19$ |
\( T^{5} + \cdots + 6222267424 \)
|
| $23$ |
\( T^{5} - 170 T^{4} + \cdots - 31859584 \)
|
| $29$ |
\( (T + 29)^{5} \)
|
| $31$ |
\( T^{5} + \cdots + 22866112000 \)
|
| $37$ |
\( T^{5} - 570 T^{4} + \cdots + 318330576 \)
|
| $41$ |
\( T^{5} + \cdots + 4212050444712 \)
|
| $43$ |
\( T^{5} + \cdots + 66497819328 \)
|
| $47$ |
\( T^{5} + \cdots - 19681644288 \)
|
| $53$ |
\( T^{5} + \cdots - 15607459176 \)
|
| $59$ |
\( T^{5} + \cdots - 2775803904 \)
|
| $61$ |
\( T^{5} + \cdots - 10432735987200 \)
|
| $67$ |
\( T^{5} + \cdots + 3957838742128 \)
|
| $71$ |
\( T^{5} + \cdots - 1075530164736 \)
|
| $73$ |
\( T^{5} + \cdots - 1589482814112 \)
|
| $79$ |
\( T^{5} + \cdots + 5147612247168 \)
|
| $83$ |
\( T^{5} + \cdots + 216952111909256 \)
|
| $89$ |
\( T^{5} + \cdots - 170678758313808 \)
|
| $97$ |
\( T^{5} + \cdots + 79624452250024 \)
|
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