Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2023,4,Mod(1,2023)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2023, 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("2023.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2023 = 7 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2023.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: | \(119.360863942\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2429.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 14x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 119) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 14x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 10 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 10 \)
|
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 |
|
−4.39364 | −2.93843 | 11.3040 | 3.54480 | 12.9104 | 7.00000 | −14.5168 | −18.3656 | −15.5745 | |||||||||||||||||||||||||||
| 1.2 | 0.293684 | −5.51635 | −7.91375 | 10.8100 | −1.62007 | 7.00000 | −4.67362 | 3.43010 | 3.17474 | ||||||||||||||||||||||||||||
| 1.3 | 3.09995 | 3.45478 | 1.60971 | 4.64517 | 10.7097 | 7.00000 | −19.8096 | −15.0645 | 14.3998 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(17\) | \(1\) |
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 | 2023.4.a.e | 3 | |
| 17.b | even | 2 | 1 | 119.4.a.b | ✓ | 3 | |
| 51.c | odd | 2 | 1 | 1071.4.a.g | 3 | ||
| 68.d | odd | 2 | 1 | 1904.4.a.e | 3 | ||
| 119.d | odd | 2 | 1 | 833.4.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 119.4.a.b | ✓ | 3 | 17.b | even | 2 | 1 | |
| 833.4.a.c | 3 | 119.d | odd | 2 | 1 | ||
| 1071.4.a.g | 3 | 51.c | odd | 2 | 1 | ||
| 1904.4.a.e | 3 | 68.d | odd | 2 | 1 | ||
| 2023.4.a.e | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2023))\):
|
\( T_{2}^{3} + T_{2}^{2} - 14T_{2} + 4 \)
|
|
\( T_{3}^{3} + 5T_{3}^{2} - 13T_{3} - 56 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 14T + 4 \)
|
| $3$ |
\( T^{3} + 5 T^{2} + \cdots - 56 \)
|
| $5$ |
\( T^{3} - 19 T^{2} + \cdots - 178 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} - 6 T^{2} + \cdots + 15904 \)
|
| $13$ |
\( T^{3} + 44 T^{2} + \cdots - 568 \)
|
| $17$ |
\( T^{3} \)
|
| $19$ |
\( T^{3} + 96 T^{2} + \cdots + 2896 \)
|
| $23$ |
\( T^{3} - 98 T^{2} + \cdots + 886272 \)
|
| $29$ |
\( T^{3} - 144 T^{2} + \cdots + 525416 \)
|
| $31$ |
\( T^{3} - 129 T^{2} + \cdots + 319728 \)
|
| $37$ |
\( T^{3} + 118 T^{2} + \cdots + 11688 \)
|
| $41$ |
\( T^{3} + 113 T^{2} + \cdots + 22238 \)
|
| $43$ |
\( T^{3} + 439 T^{2} + \cdots + 380236 \)
|
| $47$ |
\( T^{3} - 268 T^{2} + \cdots + 19307232 \)
|
| $53$ |
\( T^{3} - 79 T^{2} + \cdots + 102235806 \)
|
| $59$ |
\( T^{3} + 988 T^{2} + \cdots - 60012288 \)
|
| $61$ |
\( T^{3} + 387 T^{2} + \cdots - 16852826 \)
|
| $67$ |
\( T^{3} + 357 T^{2} + \cdots - 200256548 \)
|
| $71$ |
\( T^{3} + 64 T^{2} + \cdots - 1288704 \)
|
| $73$ |
\( T^{3} - 1485 T^{2} + \cdots - 28225458 \)
|
| $79$ |
\( T^{3} - 272 T^{2} + \cdots - 34227232 \)
|
| $83$ |
\( T^{3} - 312 T^{2} + \cdots - 33155136 \)
|
| $89$ |
\( T^{3} + 238 T^{2} + \cdots - 449745368 \)
|
| $97$ |
\( T^{3} + 109 T^{2} + \cdots - 435394194 \)
|
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