Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [15,18,Mod(1,15)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 15.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: | \(27.4833131017\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{3} - 37234x - 350700 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3\cdot 5 \) |
| 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
\( x^{3} - 37234x - 350700 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{2} + 58\nu + 24809 ) / 41 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 61\nu^{2} + 16142\nu - 1514169 ) / 41 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 61\beta _1 + 20 ) / 480 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 29\beta_{2} - 8071\beta _1 + 5954740 ) / 240 \)
|
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 |
|
−670.613 | −6561.00 | 318650. | −390625. | 4.39989e6 | −2.70804e7 | −1.25792e8 | 4.30467e7 | 2.61958e8 | |||||||||||||||||||||||||||
| 1.2 | −213.954 | −6561.00 | −85295.6 | −390625. | 1.40375e6 | 7.24373e6 | 4.62928e7 | 4.30467e7 | 8.35759e7 | ||||||||||||||||||||||||||||
| 1.3 | 442.567 | −6561.00 | 64793.8 | −390625. | −2.90368e6 | 2.47993e7 | −2.93326e7 | 4.30467e7 | −1.72878e8 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
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 | 15.18.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 45.18.a.e | 3 | ||
| 5.b | even | 2 | 1 | 75.18.a.e | 3 | ||
| 5.c | odd | 4 | 2 | 75.18.b.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.18.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 45.18.a.e | 3 | 3.b | odd | 2 | 1 | ||
| 75.18.a.e | 3 | 5.b | even | 2 | 1 | ||
| 75.18.b.d | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 442T_{2}^{2} - 248000T_{2} - 63499776 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(15))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 442 T^{2} + \cdots - 63499776 \)
|
| $3$ |
\( (T + 6561)^{3} \)
|
| $5$ |
\( (T + 390625)^{3} \)
|
| $7$ |
\( T^{3} + \cdots + 48\!\cdots\!80 \)
|
| $11$ |
\( T^{3} + \cdots + 32\!\cdots\!96 \)
|
| $13$ |
\( T^{3} + \cdots - 48\!\cdots\!08 \)
|
| $17$ |
\( T^{3} + \cdots + 19\!\cdots\!92 \)
|
| $19$ |
\( T^{3} + \cdots + 30\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 24\!\cdots\!20 \)
|
| $29$ |
\( T^{3} + \cdots + 64\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 53\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 17\!\cdots\!40 \)
|
| $41$ |
\( T^{3} + \cdots - 69\!\cdots\!80 \)
|
| $43$ |
\( T^{3} + \cdots + 41\!\cdots\!56 \)
|
| $47$ |
\( T^{3} + \cdots + 55\!\cdots\!08 \)
|
| $53$ |
\( T^{3} + \cdots + 30\!\cdots\!80 \)
|
| $59$ |
\( T^{3} + \cdots + 38\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 55\!\cdots\!12 \)
|
| $67$ |
\( T^{3} + \cdots + 30\!\cdots\!92 \)
|
| $71$ |
\( T^{3} + \cdots + 11\!\cdots\!92 \)
|
| $73$ |
\( T^{3} + \cdots - 57\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots - 87\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 42\!\cdots\!52 \)
|
| $89$ |
\( T^{3} + \cdots - 28\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 36\!\cdots\!28 \)
|
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