Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2254,4,Mod(1,2254)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2254, 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("2254.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2254 = 2 \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2254.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: | \(132.990305153\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.15384.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 33x - 14 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 322) |
| 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
\( x^{3} - 33x - 14 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 23 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + \beta _1 + 23 \)
|
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 |
|
−2.00000 | −4.94597 | 4.00000 | −17.0281 | 9.89194 | 0 | −8.00000 | −2.53740 | 34.0563 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 1.42659 | 4.00000 | 5.31700 | −2.85319 | 0 | −8.00000 | −24.9648 | −10.6340 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 6.51937 | 4.00000 | 3.71113 | −13.0387 | 0 | −8.00000 | 15.5022 | −7.42226 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 2254.4.a.h | 3 | |
| 7.b | odd | 2 | 1 | 322.4.a.d | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.4.a.d | ✓ | 3 | 7.b | odd | 2 | 1 | |
| 2254.4.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 3T_{3}^{2} - 30T_{3} + 46 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} - 3 T^{2} + \cdots + 46 \)
|
| $5$ |
\( T^{3} + 8 T^{2} + \cdots + 336 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} - 6 T^{2} + \cdots + 152 \)
|
| $13$ |
\( T^{3} + 13 T^{2} + \cdots - 8232 \)
|
| $17$ |
\( T^{3} + 20 T^{2} + \cdots + 768 \)
|
| $19$ |
\( T^{3} - 12 T^{2} + \cdots - 360288 \)
|
| $23$ |
\( (T + 23)^{3} \)
|
| $29$ |
\( T^{3} + 67 T^{2} + \cdots - 6059928 \)
|
| $31$ |
\( T^{3} + 41 T^{2} + \cdots - 2541894 \)
|
| $37$ |
\( T^{3} + 36 T^{2} + \cdots + 2187072 \)
|
| $41$ |
\( T^{3} + 273 T^{2} + \cdots - 749772 \)
|
| $43$ |
\( T^{3} - 564 T^{2} + \cdots - 2697984 \)
|
| $47$ |
\( T^{3} + 1145 T^{2} + \cdots + 34776282 \)
|
| $53$ |
\( T^{3} - 632 T^{2} + \cdots + 5164272 \)
|
| $59$ |
\( T^{3} - 128 T^{2} + \cdots + 120101304 \)
|
| $61$ |
\( T^{3} + 266 T^{2} + \cdots - 9837212 \)
|
| $67$ |
\( T^{3} - 1128 T^{2} + \cdots - 36525328 \)
|
| $71$ |
\( T^{3} - 1075 T^{2} + \cdots - 31062456 \)
|
| $73$ |
\( T^{3} + 59 T^{2} + \cdots + 205303924 \)
|
| $79$ |
\( T^{3} + 744 T^{2} + \cdots - 97811888 \)
|
| $83$ |
\( T^{3} + 2708 T^{2} + \cdots + 489525568 \)
|
| $89$ |
\( T^{3} + 448 T^{2} + \cdots + 76009752 \)
|
| $97$ |
\( T^{3} + \cdots + 1411462508 \)
|
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