Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,40,Mod(1,1)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 40, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 40);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 40 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1.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: | \(9.63395513897\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 175630027x - 142249227846 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{5}\cdot 5\cdot 13 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 175630027x - 142249227846 \)
:
| \(\beta_{1}\) | \(=\) |
\( 72\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 216\nu^{2} - 262224\nu - 25290723888 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{2} + 3642\beta _1 + 25290723888 ) / 216 \)
|
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 |
|
−799152. | 1.27116e9 | 8.88877e10 | −6.16448e13 | −1.01585e15 | 1.43779e16 | 3.68304e17 | −2.43670e18 | 4.92635e19 | |||||||||||||||||||||||||||
| 1.2 | 241488. | −3.14962e9 | −4.91439e11 | 5.36221e13 | −7.60595e14 | 1.82084e16 | −2.51436e17 | 5.86757e18 | 1.29491e19 | ||||||||||||||||||||||||||||
| 1.3 | 1.10652e6 | 2.98790e9 | 6.74631e11 | 2.54496e13 | 3.30617e15 | −5.05826e16 | 1.38177e17 | 4.87501e18 | 2.81604e19 | ||||||||||||||||||||||||||||
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 | 1.40.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 9.40.a.b | 3 | ||
| 4.b | odd | 2 | 1 | 16.40.a.c | 3 | ||
| 5.b | even | 2 | 1 | 25.40.a.a | 3 | ||
| 5.c | odd | 4 | 2 | 25.40.b.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.40.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 9.40.a.b | 3 | 3.b | odd | 2 | 1 | ||
| 16.40.a.c | 3 | 4.b | odd | 2 | 1 | ||
| 25.40.a.a | 3 | 5.b | even | 2 | 1 | ||
| 25.40.b.a | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{40}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 21\!\cdots\!36 \)
|
| $3$ |
\( T^{3} + \cdots + 11\!\cdots\!88 \)
|
| $5$ |
\( T^{3} + \cdots + 84\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots + 13\!\cdots\!16 \)
|
| $11$ |
\( T^{3} + \cdots - 64\!\cdots\!88 \)
|
| $13$ |
\( T^{3} + \cdots + 27\!\cdots\!88 \)
|
| $17$ |
\( T^{3} + \cdots - 65\!\cdots\!24 \)
|
| $19$ |
\( T^{3} + \cdots - 32\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 11\!\cdots\!88 \)
|
| $29$ |
\( T^{3} + \cdots - 79\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 14\!\cdots\!52 \)
|
| $37$ |
\( T^{3} + \cdots + 22\!\cdots\!96 \)
|
| $41$ |
\( T^{3} + \cdots + 32\!\cdots\!72 \)
|
| $43$ |
\( T^{3} + \cdots + 13\!\cdots\!88 \)
|
| $47$ |
\( T^{3} + \cdots + 71\!\cdots\!56 \)
|
| $53$ |
\( T^{3} + \cdots + 22\!\cdots\!88 \)
|
| $59$ |
\( T^{3} + \cdots - 90\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 34\!\cdots\!12 \)
|
| $67$ |
\( T^{3} + \cdots + 23\!\cdots\!76 \)
|
| $71$ |
\( T^{3} + \cdots - 11\!\cdots\!68 \)
|
| $73$ |
\( T^{3} + \cdots - 48\!\cdots\!12 \)
|
| $79$ |
\( T^{3} + \cdots + 95\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 40\!\cdots\!88 \)
|
| $89$ |
\( T^{3} + \cdots + 24\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 58\!\cdots\!56 \)
|
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