Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [368,4,Mod(1,368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("368.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 368 = 2^{4} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 368.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: | \(21.7127028821\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.761.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 6x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 184) |
| 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
\( x^{3} - x^{2} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} - 4\nu - 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 9 ) / 2 \)
|
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 |
|
0 | −7.72680 | 0 | 18.6126 | 0 | −28.1804 | 0 | 32.7035 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0.476220 | 0 | 13.8291 | 0 | −3.57134 | 0 | −26.7732 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 6.25058 | 0 | −16.4417 | 0 | 13.7517 | 0 | 12.0698 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 368.4.a.i | 3 | |
| 4.b | odd | 2 | 1 | 184.4.a.d | ✓ | 3 | |
| 8.b | even | 2 | 1 | 1472.4.a.t | 3 | ||
| 8.d | odd | 2 | 1 | 1472.4.a.s | 3 | ||
| 12.b | even | 2 | 1 | 1656.4.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 184.4.a.d | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 368.4.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
| 1472.4.a.s | 3 | 8.d | odd | 2 | 1 | ||
| 1472.4.a.t | 3 | 8.b | even | 2 | 1 | ||
| 1656.4.a.i | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + T_{3}^{2} - 49T_{3} + 23 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(368))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + T^{2} + \cdots + 23 \)
|
| $5$ |
\( T^{3} - 16 T^{2} + \cdots + 4232 \)
|
| $7$ |
\( T^{3} + 18 T^{2} + \cdots - 1384 \)
|
| $11$ |
\( T^{3} + 40 T^{2} + \cdots - 66056 \)
|
| $13$ |
\( T^{3} + 95 T^{2} + \cdots - 32179 \)
|
| $17$ |
\( T^{3} + 42 T^{2} + \cdots - 56456 \)
|
| $19$ |
\( T^{3} + 26 T^{2} + \cdots - 1143816 \)
|
| $23$ |
\( (T - 23)^{3} \)
|
| $29$ |
\( T^{3} + 181 T^{2} + \cdots - 7111481 \)
|
| $31$ |
\( T^{3} + 705 T^{2} + \cdots + 10237599 \)
|
| $37$ |
\( T^{3} + 80 T^{2} + \cdots - 19357544 \)
|
| $41$ |
\( T^{3} - 281 T^{2} + \cdots - 2573499 \)
|
| $43$ |
\( T^{3} + 248 T^{2} + \cdots - 2773504 \)
|
| $47$ |
\( T^{3} + 517 T^{2} + \cdots - 647501 \)
|
| $53$ |
\( T^{3} + 190 T^{2} + \cdots - 18150072 \)
|
| $59$ |
\( T^{3} + 996 T^{2} + \cdots - 16600384 \)
|
| $61$ |
\( T^{3} + 1214 T^{2} + \cdots + 12905704 \)
|
| $67$ |
\( T^{3} - 776 T^{2} + \cdots + 14223944 \)
|
| $71$ |
\( T^{3} + 229 T^{2} + \cdots + 1059547 \)
|
| $73$ |
\( T^{3} + 333 T^{2} + \cdots + 8715807 \)
|
| $79$ |
\( T^{3} - 26 T^{2} + \cdots + 5488392 \)
|
| $83$ |
\( T^{3} - 474 T^{2} + \cdots - 8830248 \)
|
| $89$ |
\( T^{3} + 1320 T^{2} + \cdots - 655097408 \)
|
| $97$ |
\( T^{3} - 2386 T^{2} + \cdots + 449588984 \)
|
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