Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1775,4,Mod(1,1775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1775, 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("1775.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1775 = 5^{2} \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1775.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: | \(104.728390260\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.135041.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 15x^{2} - 9x + 32 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 71) |
| 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
\( x^{4} - 15x^{2} - 9x + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 9\nu + 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 8\nu + 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{3} + 3\beta_{2} + 11\beta _1 + 8 \)
|
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 |
|
−3.36251 | 7.45338 | 3.30648 | 0 | −25.0621 | 9.18078 | 15.7820 | 28.5528 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −0.612007 | 1.79036 | −7.62545 | 0 | −1.09572 | 12.2553 | 9.56289 | −23.7946 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 2.78121 | −6.20340 | −0.264856 | 0 | −17.2530 | 18.0632 | −22.9863 | 11.4822 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 4.19331 | 2.95966 | 9.58382 | 0 | 12.4108 | −4.49924 | 6.64143 | −18.2404 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(71\) | \(-1\) |
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 | 1775.4.a.b | 4 | |
| 5.b | even | 2 | 1 | 71.4.a.b | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 639.4.a.c | 4 | ||
| 20.d | odd | 2 | 1 | 1136.4.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.4.a.b | ✓ | 4 | 5.b | even | 2 | 1 | |
| 639.4.a.c | 4 | 15.d | odd | 2 | 1 | ||
| 1136.4.a.d | 4 | 20.d | odd | 2 | 1 | ||
| 1775.4.a.b | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 3T_{2}^{3} - 14T_{2}^{2} + 32T_{2} + 24 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1775))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots + 24 \)
|
| $3$ |
\( T^{4} - 6 T^{3} + \cdots - 245 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 35 T^{3} + \cdots - 9144 \)
|
| $11$ |
\( T^{4} + 64 T^{3} + \cdots - 163728 \)
|
| $13$ |
\( T^{4} - 101 T^{3} + \cdots - 218344 \)
|
| $17$ |
\( T^{4} - 150 T^{3} + \cdots + 511056 \)
|
| $19$ |
\( T^{4} + 12 T^{3} + \cdots + 823725 \)
|
| $23$ |
\( T^{4} + 321 T^{3} + \cdots - 9940680 \)
|
| $29$ |
\( T^{4} + 83 T^{3} + \cdots + 23423958 \)
|
| $31$ |
\( T^{4} + 321 T^{3} + \cdots - 77700392 \)
|
| $37$ |
\( T^{4} - 97 T^{3} + \cdots + 452768562 \)
|
| $41$ |
\( T^{4} + \cdots + 5259534336 \)
|
| $43$ |
\( T^{4} - 158 T^{3} + \cdots + 82333701 \)
|
| $47$ |
\( T^{4} - 443 T^{3} + \cdots + 759919104 \)
|
| $53$ |
\( T^{4} + \cdots - 17233218192 \)
|
| $59$ |
\( T^{4} + 558 T^{3} + \cdots + 941809680 \)
|
| $61$ |
\( T^{4} + \cdots + 8843856336 \)
|
| $67$ |
\( T^{4} + 428 T^{3} + \cdots - 110522416 \)
|
| $71$ |
\( (T - 71)^{4} \)
|
| $73$ |
\( T^{4} + \cdots - 56384322169 \)
|
| $79$ |
\( T^{4} + \cdots - 119709446390 \)
|
| $83$ |
\( T^{4} + \cdots - 487336824324 \)
|
| $89$ |
\( T^{4} + \cdots + 1511004605607 \)
|
| $97$ |
\( T^{4} + \cdots + 416719798992 \)
|
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