Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [238,4,Mod(1,238)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(238, 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("238.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 238 = 2 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 238.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: | \(14.0424545814\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 91x^{2} - 2x + 1596 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 91x^{2} - 2x + 1596 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 3\nu^{2} - 61\nu - 192 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 6\nu^{2} - 52\nu + 213 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 45 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 6\beta_{2} + 58\beta _1 + 57 \)
|
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 | −7.32515 | 4.00000 | −15.7796 | −14.6503 | −7.00000 | 8.00000 | 26.6578 | −31.5591 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −5.25930 | 4.00000 | 19.1906 | −10.5186 | −7.00000 | 8.00000 | 0.660209 | 38.3812 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 4.62272 | 4.00000 | 3.31015 | 9.24543 | −7.00000 | 8.00000 | −5.63049 | 6.62029 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 8.96173 | 4.00000 | 12.2788 | 17.9235 | −7.00000 | 8.00000 | 53.3125 | 24.5577 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 238.4.a.h | ✓ | 4 |
| 3.b | odd | 2 | 1 | 2142.4.a.w | 4 | ||
| 4.b | odd | 2 | 1 | 1904.4.a.i | 4 | ||
| 7.b | odd | 2 | 1 | 1666.4.a.k | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 238.4.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1666.4.a.k | 4 | 7.b | odd | 2 | 1 | ||
| 1904.4.a.i | 4 | 4.b | odd | 2 | 1 | ||
| 2142.4.a.w | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - T_{3}^{3} - 91T_{3}^{2} - 2T_{3} + 1596 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(238))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} - T^{3} + \cdots + 1596 \)
|
| $5$ |
\( T^{4} - 19 T^{3} + \cdots - 12308 \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( T^{4} - 74 T^{3} + \cdots - 274512 \)
|
| $13$ |
\( T^{4} - 12 T^{3} + \cdots + 4536000 \)
|
| $17$ |
\( (T - 17)^{4} \)
|
| $19$ |
\( T^{4} - 104 T^{3} + \cdots - 7412544 \)
|
| $23$ |
\( T^{4} - 118 T^{3} + \cdots - 41099520 \)
|
| $29$ |
\( T^{4} + 114 T^{3} + \cdots - 119873808 \)
|
| $31$ |
\( T^{4} - 233 T^{3} + \cdots + 372862224 \)
|
| $37$ |
\( T^{4} + 26 T^{3} + \cdots + 7605184 \)
|
| $41$ |
\( T^{4} + 573 T^{3} + \cdots + 33878796 \)
|
| $43$ |
\( T^{4} - 623 T^{3} + \cdots + 234709072 \)
|
| $47$ |
\( T^{4} - 134 T^{3} + \cdots + 707081408 \)
|
| $53$ |
\( T^{4} + \cdots - 13473503004 \)
|
| $59$ |
\( T^{4} + \cdots - 6970666560 \)
|
| $61$ |
\( T^{4} + \cdots - 4148722372 \)
|
| $67$ |
\( T^{4} + \cdots - 2613731376 \)
|
| $71$ |
\( T^{4} + \cdots + 13176860928 \)
|
| $73$ |
\( T^{4} + \cdots + 97461737852 \)
|
| $79$ |
\( T^{4} + \cdots + 258965318720 \)
|
| $83$ |
\( T^{4} + \cdots - 180417397104 \)
|
| $89$ |
\( T^{4} + \cdots + 1046116641264 \)
|
| $97$ |
\( T^{4} + \cdots - 56700484260 \)
|
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