Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [238,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("238.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 238 = 2 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(38.1713491155\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 95x^{2} + 164x + 816 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - x^{3} - 95x^{2} + 164x + 816 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{3} + 7\nu^{2} + 189\nu - 536 ) / 52 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{3} + 7\nu^{2} + 293\nu - 536 ) / 52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{3} - 23\nu^{2} + 315\nu + 736 ) / 52 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{3} + 5\beta _1 + 94 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{3} + 63\beta_{2} - 86\beta _1 - 138 ) / 2 \)
|
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 |
|
4.00000 | −21.3411 | 16.0000 | 30.7218 | −85.3643 | −49.0000 | 64.0000 | 212.441 | 122.887 | ||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −5.33058 | 16.0000 | 84.7835 | −21.3223 | −49.0000 | 64.0000 | −214.585 | 339.134 | |||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 3.51224 | 16.0000 | −26.6876 | 14.0490 | −49.0000 | 64.0000 | −230.664 | −106.750 | |||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 15.1594 | 16.0000 | −52.8177 | 60.6376 | −49.0000 | 64.0000 | −13.1923 | −211.271 | |||||||||||||||||||||||||||||||
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 | 238.6.a.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 238.6.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 8T_{3}^{3} - 331T_{3}^{2} - 704T_{3} + 6057 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(238))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{4} \)
|
| $3$ |
\( T^{4} + 8 T^{3} + \cdots + 6057 \)
|
| $5$ |
\( T^{4} - 36 T^{3} + \cdots + 3671531 \)
|
| $7$ |
\( (T + 49)^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 19240366704 \)
|
| $13$ |
\( T^{4} + \cdots + 71860160304 \)
|
| $17$ |
\( (T - 289)^{4} \)
|
| $19$ |
\( T^{4} + \cdots + 3315149762496 \)
|
| $23$ |
\( T^{4} + \cdots - 58271878368560 \)
|
| $29$ |
\( T^{4} + \cdots + 411575028758064 \)
|
| $31$ |
\( T^{4} + \cdots - 106801810312751 \)
|
| $37$ |
\( T^{4} + \cdots - 169695073288000 \)
|
| $41$ |
\( T^{4} + \cdots + 98\!\cdots\!31 \)
|
| $43$ |
\( T^{4} + \cdots + 73\!\cdots\!35 \)
|
| $47$ |
\( T^{4} + \cdots - 14\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots - 35\!\cdots\!91 \)
|
| $59$ |
\( T^{4} + \cdots + 64\!\cdots\!12 \)
|
| $61$ |
\( T^{4} + \cdots + 24\!\cdots\!63 \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!67 \)
|
| $71$ |
\( T^{4} + \cdots - 10\!\cdots\!68 \)
|
| $73$ |
\( T^{4} + \cdots + 76\!\cdots\!47 \)
|
| $79$ |
\( T^{4} + \cdots - 43\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 19\!\cdots\!60 \)
|
| $89$ |
\( T^{4} + \cdots + 52\!\cdots\!40 \)
|
| $97$ |
\( T^{4} + \cdots - 11\!\cdots\!61 \)
|
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