Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1120,4,Mod(1,1120)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1120, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1120.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1120 = 2^{5} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1120.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: | \(66.0821392064\) |
| 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: |
\( x^{5} - 99x^{3} - 98x^{2} + 924x + 168 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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
\( x^{5} - 99x^{3} - 98x^{2} + 924x + 168 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 4\nu^{3} + 83\nu^{2} - 198\nu - 348 ) / 18 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 2\nu^{3} - 95\nu^{2} - 252\nu + 456 ) / 18 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} + 2\beta_{3} + 3\beta_{2} + 77\beta _1 + 62 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{4} + 91\beta_{3} - 6\beta_{2} + 193\beta _1 + 3220 \)
|
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 | −9.70695 | 0 | −5.00000 | 0 | 7.00000 | 0 | 67.2248 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −4.87339 | 0 | −5.00000 | 0 | 7.00000 | 0 | −3.25002 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.17903 | 0 | −5.00000 | 0 | 7.00000 | 0 | −25.6099 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.79130 | 0 | −5.00000 | 0 | 7.00000 | 0 | −23.7912 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 8.96807 | 0 | −5.00000 | 0 | 7.00000 | 0 | 53.4263 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 1120.4.a.r | ✓ | 5 |
| 4.b | odd | 2 | 1 | 1120.4.a.s | yes | 5 | |
| 8.b | even | 2 | 1 | 2240.4.a.co | 5 | ||
| 8.d | odd | 2 | 1 | 2240.4.a.cn | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1120.4.a.r | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1120.4.a.s | yes | 5 | 4.b | odd | 2 | 1 | |
| 2240.4.a.cn | 5 | 8.d | odd | 2 | 1 | ||
| 2240.4.a.co | 5 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1120))\):
|
\( T_{3}^{5} + 5T_{3}^{4} - 89T_{3}^{3} - 385T_{3}^{2} + 436T_{3} + 896 \)
|
|
\( T_{11}^{5} + 35T_{11}^{4} - 3817T_{11}^{3} - 112999T_{11}^{2} + 3316452T_{11} + 65668224 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 5 T^{4} + \cdots + 896 \)
|
| $5$ |
\( (T + 5)^{5} \)
|
| $7$ |
\( (T - 7)^{5} \)
|
| $11$ |
\( T^{5} + 35 T^{4} + \cdots + 65668224 \)
|
| $13$ |
\( T^{5} + 39 T^{4} + \cdots - 32631244 \)
|
| $17$ |
\( T^{5} + 29 T^{4} + \cdots - 60507308 \)
|
| $19$ |
\( T^{5} + \cdots + 1070681600 \)
|
| $23$ |
\( T^{5} - 128 T^{4} + \cdots + 542924800 \)
|
| $29$ |
\( T^{5} + \cdots - 44854213484 \)
|
| $31$ |
\( T^{5} + \cdots - 118773702656 \)
|
| $37$ |
\( T^{5} + 236 T^{4} + \cdots - 290827392 \)
|
| $41$ |
\( T^{5} + \cdots + 20074389792 \)
|
| $43$ |
\( T^{5} + \cdots + 521644284928 \)
|
| $47$ |
\( T^{5} + \cdots + 260469006336 \)
|
| $53$ |
\( T^{5} + \cdots - 17732875299072 \)
|
| $59$ |
\( T^{5} + \cdots - 8244134363136 \)
|
| $61$ |
\( T^{5} + \cdots - 5296911597024 \)
|
| $67$ |
\( T^{5} + \cdots - 5266967707648 \)
|
| $71$ |
\( T^{5} + \cdots - 8718444505088 \)
|
| $73$ |
\( T^{5} + \cdots - 7501067288992 \)
|
| $79$ |
\( T^{5} + \cdots + 224450049813600 \)
|
| $83$ |
\( T^{5} + \cdots - 31391683936256 \)
|
| $89$ |
\( T^{5} + \cdots - 161865734411232 \)
|
| $97$ |
\( T^{5} + \cdots + 47617191825876 \)
|
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